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. . . 1523 A Atomic Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1531 B Selected Radioactive Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1537 C Useful Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1541 D Glossary of Key Symbols and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1659 This content is available for free at http://cnx.org/content/col11844/1.13 Preface 1 PREFACE The OpenStax College Physics: AP® Edition program has been developed with several goals in mind: accessibility, customization, and student engagement—all while encouraging science students toward high levels of academic scholarship. Instructors and students alike will find that this program offers a strong foundation in physics in an accessible format. Welcome! About OpenStax College OpenStax College is a nonprofit organization committed to improving student access to quality learning materials. Our free textbooks are developed and peer-reviewed by educators to ensure they are readable, accurate, and meet the scope and sequence requirements of today’s high school courses. Unlike traditional textbooks, OpenStax College resources live online and are owned by the community of educators using them. Through our partnerships with companies and foundations committed to reducing costs for students, OpenStax College is working to improve access to education for all. OpenStax College is an initiative of Rice University and is made possible through the generous support of several philanthropic foundations. OpenStax College resources provide quality academic instruction. Three key features set our materials apart from others: they can be customized by instructors for each class, they are a “living” resource that grows online through contributions from science educators, and they are available for free or at minimal cost. Customization OpenStax College learning resources are designed to be customized for each course. Our textbooks provide a solid foundation on which instructors can build, and our resources are conceived and written with flexibility in mind. Instructors can simply select the sections most relevant to their curricula and create a textbook that speaks directly to the needs of their classes and students. Teachers are encouraged to expand on existing examples by adding unique context via geographically localized applications and topical connections. This customization feature will help bring physics to life for students and will ensure that your textbook truly reflects the goals of your course. Curation To broaden access and encourage community curation, OpenStax College Physics: AP® Edition is “open source” licensed under a Creative Commons Attribution (CC-BY) license. The scientific community is invited to submit examples, emerging research, and other feedback to enhance and strengthen the material and keep it current and relevant for today’s students. Submit your suggestions to info@openstaxcollege.org, and find information on edition status, alternate versions, errata, and news on the StaxDash at http://openstaxcollege.org (http://openstaxcollege.org) . Cost Our textbooks are available for free online and in low-cost print and e-book editions. About OpenStax College Physics: AP® Edition In 2012, OpenStax College published College Physics as part of a series that offers free and open college textbooks for higher education. College Physics was quickly adopted for science courses all around the country, and as word about this valuable resource spread, advanced placement teachers around the country started utilizing the book in AP® courses too. Physics: AP® Edition is the result of an effort to better serve these teachers and students. Based on College Physics—a program based on the teaching and research experience of numerous physicists—Physics: AP® Edition focuses on and emphasizes the new AP® curriculum's concepts and practices. Alignment to the AP® curriculum The new AP® Physics curriculum framework outlines the two full-year physics courses AP® Physics 1: Algebra-Based and AP® Physics 2: Algebra-Based. These two courses replaced the one-year AP® Physics B course, which over the years had become a fast-paced survey of physics facts and formulas that did not provide in-depth conceptual understanding of major physics ideas and the connections between them. The new AP® Physics 1 and 2 courses focus on the big ideas typically included in the first and second semesters of an algebrabased, introductory college-level physics course, providing students with the essential knowledge and skills required to support future advanced course work in physics. The AP® Physics 1 curriculum includes mechanics, mechanical waves, sound, and electrostatics. The AP® Physics 2 curriculum focuses on thermodynamics, fluid statics, dynamics, electromagnetism, geometric and physical optics, quantum physics, atomic physics, and nuclear physics. Seven unifying themes of physics called the Big Ideas each include three to seven Enduring Understandings (EU), which are themselves composed of Essential Knowledge (EK) that provides details and context for students as they explore physics. AP® Science Practices emphasize inquiry-based learning and development of critical thinking and reasoning skills. Inquiry usually uses a series of steps to gain new knowledge, beginning with an observation and following with a hypothesis to explain the observation; then experiments are conducted to test the hypothesis, gather results, and draw conclusions from data. The 2 Preface AP® framework has identified seven major science practices, which can be described by short phrases: using representations and models to communicate information and solve problems; using mathematics appropriately; engaging in questioning; planning and implementing data collection strategies; analyzing and evaluating data; justifying scientific explanations; and connecting concepts. The framework’s Learning Objectives merge content (EU and EK) with one or more of the seven science practices that students should develop as they prepare for the AP® Physics exam. Each chapter of OpenStax College Physics: AP® Edition begins with a Connection for AP® Courses introduction that explains how the content in the chapter sections align to the Big Ideas, Enduring Understandings, and Essential Knowledge in the AP® framework. Physics: AP® Edition contains a wealth of information and the Connection for AP® Courses sections will help you distill the required AP® content from material that, although interesting, exceeds the scope of an introductory-level course. Each section opens with the program’s learning objectives as well as the AP® learning objectives and science practices addressed. We have also developed Real World Connections features and Applying the Science Practices features that highlight concepts, examples, and practices in the framework. Pedagogical Foundation and Features OpenStax College Physics: AP® Edition is organized such that topics are introduced conceptually with a steady progression to precise definitions and analytical applications. The analytical aspect (problem solving) is tied back to the conceptual before moving on to another topic. Each introductory chapter, for example, opens with an engaging photograph relevant to the subject of the chapter and interesting applications that are easy for most students to visualize. Our features include: • Connections for AP® Courses introduce each chapter and explain how its content addresses the AP® curriculum. • Worked examples promote both analytical and conceptual skills. They are introduced using an application of interest followed by a strategy that emphasizes the concepts involved, a mathematical solution, and a discussion. • Problem-solving strategies are presented independently and subsequently appear at crucial points in the text where students can benefit most from them. • Misconception Alerts address common misconceptions that students may bring to class. • Take Home Investigations provide the opportunity for students to apply or explore what they have learned with a hands- on activity. • Real World Connections highlight important concepts and examples in the AP® framework. • Applying the Science Practices includes activities and challenging questions that engage students while they apply the AP® science practices. • Things Great and Small explain macroscopic phenomena (such as air pressure) with submicroscopic phenomena (such as atoms bouncing off walls). • Simulations direct students to further explore the physics concepts they have learned about in the module through the interactive PHeT physics simulations developed by the University of Colorado. Assessment Physics: AP® Edition offers a wealth of assessment options that include: • End-of-Module Problems include conceptual questions that challenge students’ ability to explain what they have learned conceptually, independent of the mathematical details, and problems and exercises that challenge students to apply both concepts and skills to solve mathematical physics problems. Integrated Concept Problems challenge students to apply concepts and skills to solve a problem. • • Unreasonable Results encourage students to analyze the answer with respect to how likely or realistic it really is. • Construct Your Own Problem requires students to construct the details of a problem, justify their starting assumptions, show specific steps in the problem’s solution, and finally discuss the meaning of the result. • Test Prep for AP® Courses consists of end-of-module problems that include assessment items with the format and rigor found in the AP® exam to help prepare
students. About Our Team Physics: AP® Edition would not be possible if not for the tremendous contributions of the authors and community reviewing team. Contributors to OpenStax College Physics: AP® Edition This content is available for free at http://cnx.org/content/col11844/1.13 Preface Senior Contributors 3 Irna Lyublinskaya CUNY College of Staten Island, Staten Island, NY Gregg Wolfe Avonworth High School, Pittsburgh, PA Douglas Ingram TCU Department of Physics and Astronomy, Fort Worth, TX Liza Pujji Manukau Institute of Technology (MIT), New Zealand Sudhi Oberoi Visiting Research Student, QuIC Lab, Raman Research Institute, India Nathan Czuba Sabio Academy, Chicago, IL Julie Kretchman Science Writer, BS, University of Toronto, Canada John Stoke Science Writer, MS, University of Chicago, IL David Anderson Science Writer, PhD, College of William and Mary, Williamsburg, VA Erika Gasper Science Writer, MA, University of California, Santa Cruz, CA Advanced Placement Teacher Reviewers Faculty Reviewers Michelle Burgess Avon Lake High School, Avon Lake, OH Alexander Lavy Xavier High School, New York, NY Brian Hastings Spring Grove Area School District, York, PA John Boehringer Prosper High School, Prosper, TX Victor Brazil Petaluma High School, Petaluma, CA Jerome Mass Glastonbury Public Schools, Glastonbury, CT Bryan Callow Lindenwold High School, Lindenwold, NJ Anand Batra Howard University, Washington, DC John Aiken Georgia Institute of Technology, Atlanta, GA Robert Arts University of Pikeville, Pikeville, KY Ulrich Zurcher Cleveland State University, Cleveland, OH Michael Ottinger Missouri Western State University, Kansas City, MO James Smith Caldwell University, Caldwell, NJ Additional Resources Preparing for the AP® Physics 1 Exam Rice Online’s dynamic new course, available on edX, is fully integrated with Physics for AP® Courses for free. Developed by nationally recognized Rice Professor Dr. Jason Hafner and AP® Physics teachers Gigi Nevils-Noe and Matt Wilson the course combines innovative learning technologies with engaging, professionally-produced Concept Trailers™, inquiry based labs, practice problems, lectures, demonstrations, assessments, and other compelling resources to promote engagement and longterm retention of AP® Physics 1 concepts and application. Learn more at online.rice.edu. Other learning resources (powerpoint slides, testbanks, online homework etc) are updated frequently and can be viewed by going to https://openstaxcollege.org. To the AP® Physics Student The fundamental goal of physics is to discover and understand the “laws” that govern observed phenomena in the world around us. Why study physics? If you plan to become a physicist, the answer is obvious—introductory physics provides the foundation for your career; or if you want to become an engineer, physics provides the basis for the engineering principles used to solve applied and practical problems. For example, after the discovery of the photoelectric effect by physicists, engineers developed photocells that are used in solar panels to convert sunlight to electricity. What if you are an aspiring medical doctor? Although the applications of the laws of physics may not be obvious, their understanding is tremendously valuable. Physics is involved in medical diagnostics, such as x-rays, magnetic resonance imaging (MRI), and ultrasonic blood flow measurements. Medical therapy sometimes directly involves physics; cancer radiotherapy uses ionizing radiation. What if you are planning a nonscience career? Learning physics provides you with a well-rounded education and the ability to make important decisions, such as evaluating the pros and cons of energy production sources or voting on decisions about nuclear waste disposal. 4 Preface This AP® Physics 1 course begins with kinematics, the study of motion without considering its causes. Motion is everywhere: from the vibration of atoms to the planetary revolutions around the Sun. Understanding motion is key to understanding other concepts in physics. You will then study dynamics, which considers the forces that affect the motion of moving objects and systems. Newton’s laws of motion are the foundation of dynamics. These laws provide an example of the breadth and simplicity of the principles under which nature functions. One of the most remarkable simplifications in physics is that only four distinct forces account for all known phenomena. Your journey will continue as you learn about energy. Energy plays an essential role both in everyday events and in scientific phenomena. You can likely name many forms of energy, from that provided by our foods, to the energy we use to run our cars, to the sunlight that warms us on the beach. The next stop is learning about oscillatory motion and waves. All oscillations involve force and energy: you push a child in a swing to get the motion started and you put energy into a guitar string when you pluck it. Some oscillations create waves. For example, a guitar creates sound waves. You will conclude this first physics course with the study of static electricity and electric currents. Many of the characteristics of static electricity can be explored by rubbing things together. Rubbing creates the spark you get from walking across a wool carpet, for example. Similarly, lightning results from air movements under certain weather conditions. In AP® Physics 2 course you will continue your journey by studying fluid dynamics, which explains why rising smoke curls and twists and how the body regulates blood flow. The next stop is thermodynamics, the study of heat transfer—energy in transit—that can be used to do work. Basic physical laws govern how heat transfers and its efficiency. Then you will learn more about electric phenomena as you delve into electromagnetism. An electric current produces a magnetic field; similarly, a magnetic field produces a current. This phenomenon, known as magnetic induction, is essential to our technological society. The generators in cars and nuclear plants use magnetism to generate a current. Other devices that use magnetism to induce currents include pickup coils in electric guitars, transformers of every size, certain microphones, airport security gates, and damping mechanisms on sensitive chemical balances. From electromagnetism you will continue your journey to optics, the study of light. You already know that visible light is the type of electromagnetic waves to which our eyes respond. Through vision, light can evoke deep emotions, such as when we view a magnificent sunset or glimpse a rainbow breaking through the clouds. Optics is concerned with the generation and propagation of light. The quantum mechanics, atomic physics, and nuclear physics are at the end of your journey. These areas of physics have been developed at the end of the 19th and early 20th centuries and deal with submicroscopic objects. Because these objects are smaller than we can observe directly with our senses and generally must be observed with the aid of instruments, parts of these physics areas may seem foreign and bizarre to you at first. However, we have experimentally confirmed most of the ideas in these areas of physics. AP® Physics is a challenging course. After all, you are taking physics at the introductory college level. You will discover that some concepts are more difficult to understand than others; most students, for example, struggle to understand rotational motion and angular momentum or particle-wave duality. The AP® curriculum promotes depth of understanding over breadth of content, and to make your exploration of topics more manageable, concepts are organized around seven major themes called the Big Ideas that apply to all levels of physical systems and interactions between them (see web diagram below). Each Big Idea identifies Enduring Understandings (EU), Essential Knowledge (EK), and illustrative examples that support key concepts and content. Simple descriptions define the focus of each Big Idea. • Big Idea 1: Objects and systems have properties. • Big Idea 2: Fields explain interactions. • Big Idea 3: The interactions are described by forces. • Big Idea 4: Interactions result in changes. • Big Idea 5: Changes are constrained by conservation laws. • Big Idea 6: Waves can transfer energy and momentum. • Big Idea 7: The mathematics of probability can to describe the behavior of complex and quantum mechanical systems. Doing college work is not easy, but completion of AP® classes is a reliable predictor of college success and prepares you for subsequent courses. The more you engage in the subject, the easier your journey through the curriculum will be. Bring your enthusiasm to class every day along with your notebook, pencil, and calculator. Prepare for class the day before, and review concepts daily. Form a peer study group and ask your teacher for extra help if necessary. The AP® lab program focuses on more open-ended, student-directed, and inquiry-based lab investigations designed to make you think, ask questions, and analyze data like scientists. You will develop critical thinking and reasoning skills and apply different means of communicating information. By the time you sit for the AP® exam in May, you will be fluent in the language of physics; because you have been doing real science, you will be ready to show what you have learned. Along the way, you will find the study of the world around us to be one of the most relevant and enjoyable experiences of your high school career. Irina Lyublinskaya, PhD Professor of Science Education To the AP® Physics Teacher The AP® curriculum was designed to allow instructors flexibility in their approach to teaching the physics courses. OpenStax College Physics: AP® Edition helps you orient students as they delve deeper into the world of physics. Each chapter includes a Connection for AP® Courses introduction that describes the AP® Physics Big Ideas, Enduring Understandings, and Essential Knowl
edge addressed in that chapter. Each section starts with specific AP® learning objectives and includes essential concepts, illustrative examples, and science practices, along with suggestions for applying the learning objectives through take home experiments, virtual lab investigations, and activities and questions for preparation and review. At the end of each section, students will find the Test Prep for AP® courses with multiple-choice and open-response questions addressing AP® learning objectives to help them prepare for the AP® exam. This content is available for free at http://cnx.org/content/col11844/1.13 Preface 5 OpenStax College Physics: AP® Edition has been written to engage students in their exploration of physics and help them relate what they learn in the classroom to their lives outside of it. Physics underlies much of what is happening today in other sciences and in technology. Thus, the book content includes interesting facts and ideas that go beyond the scope of the AP® course. The AP® Connection in each chapter directs students to the material they should focus on for the AP® exam, and what content—although interesting—is not part of the AP® curriculum. Physics is a beautiful and fascinating science. It is in your hands to engage and inspire your students to dive into an amazing world of physics, so they can enjoy it beyond just preparation for the AP® exam. Irina Lyublinskaya, PhD Professor of Science Education The concept map showing major links between Big Ideas and Enduring Understandings is provided below for visual reference. 6 Preface This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 \| Introduction: The Nature of Science and Physics 7 INTRODUCTION: THE NATURE OF 1 SCIENCE AND PHYSICS Figure 1.1 Galaxies are as immense as atoms are small. Yet the same laws of physics describe both, and all the rest of nature—an indication of the underlying unity in the universe. The laws of physics are surprisingly few in number, implying an underlying simplicity to nature's apparent complexity. (credit: NASA, JPL-Caltech, P. Barmby, Harvard-Smithsonian Center for Astrophysics) Chapter Outline 1.1. Physics: An Introduction 1.2. Physical Quantities and Units 1.3. Accuracy, Precision, and Significant Figures 1.4. Approximation Connection for AP® Courses What is your first reaction when you hear the word “physics”? Did you imagine working through difficult equations or memorizing formulas that seem to have no real use in life outside the physics classroom? Many people come to the subject of physics with a bit of fear. But as you begin your exploration of this broad-ranging subject, you may soon come to realize that physics plays a much larger role in your life than you first thought, no matter your life goals or career choice. For example, take a look at the image above. This image is of the Andromeda Galaxy, which contains billions of individual stars, huge clouds of gas, and dust. Two smaller galaxies are also visible as bright blue spots in the background. At a staggering 2.5 million light years from Earth, this galaxy is the nearest one to our own galaxy (which is called the Milky Way). The stars and planets that make up Andromeda might seem to be the furthest thing from most people's regular, everyday lives. But Andromeda is a great starting point to think about the forces that hold together the universe. The forces that cause Andromeda to act as it does are the same forces we contend with here on Earth, whether we are planning to send a rocket into space or simply raise the walls for a new home. The same gravity that causes the stars of Andromeda to rotate and revolve also causes water to flow over hydroelectric dams here on Earth. Tonight, take a moment to look up at the stars. The forces out there are the same as the ones here on Earth. Through a study of physics, you may gain a greater understanding of the interconnectedness of everything we can see and know in this universe. Think now about all of the technological devices that you use on a regular basis. Computers, smart phones, GPS systems, MP3 players, and satellite radio might come to mind. Next, think about the most exciting modern technologies that you have heard about in the news, such as trains that levitate above tracks, “invisibility cloaks” that bend light around them, and microscopic robots that fight cancer cells in our bodies. All of these groundbreaking advancements, commonplace or unbelievable, rely on the principles of physics. Aside from playing a significant role in technology, professionals such as engineers, pilots, physicians, physical therapists, electricians, and computer programmers apply physics concepts in their daily work. For example, a pilot must understand how wind forces affect a flight path and a physical therapist must understand how the muscles in the body experience forces as they move and bend. As you will learn in this text, physics principles are propelling new, exciting technologies, and these principles are applied in a wide range of careers. In this text, you will begin to explore the history of the formal study of physics, beginning with natural philosophy and the ancient Greeks, and leading up through a review of Sir Isaac Newton and the laws of physics that bear his name. You will also be introduced to the standards scientists use when they study physical quantities and the interrelated system of measurements most of the scientific community uses to communicate in a single mathematical language. Finally, you will study the limits of our ability to be accurate and precise, and the reasons scientists go to painstaking lengths to be as clear as possible regarding their own limitations. 8 Chapter 1 \| Introduction: The Nature of Science and Physics Chapter 1 introduces many fundamental skills and understandings needed for success with the AP® Learning Objectives. While this chapter does not directly address any Big Ideas, its content will allow for a more meaningful understanding when these Big Ideas are addressed in future chapters. For instance, the discussion of models, theories, and laws will assist you in understanding the concept of fields as addressed in Big Idea 2, and the section titled ‘The Evolution of Natural Philosophy into Modern Physics' will help prepare you for the statistical topics addressed in Big Idea 7. This chapter will also prepare you to understand the Science Practices. In explicitly addressing the role of models in representing and communicating scientific phenomena, Section 1.1 supports Science Practice 1. Additionally, anecdotes about historical investigations and the inset on the scientific method will help you to engage in the scientific questioning referenced in Science Practice 3. The appropriate use of mathematics, as called for in Science Practice 2, is a major focus throughout sections 1.2, 1.3, and 1.4. 1.1 Physics: An Introduction Figure 1.2 The flight formations of migratory birds such as Canada geese are governed by the laws of physics. (credit: David Merrett) Learning Objectives By the end of this section, you will be able to: • Explain the difference between a principle and a law. • Explain the difference between a model and a theory. The physical universe is enormously complex in its detail. Every day, each of us observes a great variety of objects and phenomena. Over the centuries, the curiosity of the human race has led us collectively to explore and catalog a tremendous wealth of information. From the flight of birds to the colors of flowers, from lightning to gravity, from quarks to clusters of galaxies, from the flow of time to the mystery of the creation of the universe, we have asked questions and assembled huge arrays of facts. In the face of all these details, we have discovered that a surprisingly small and unified set of physical laws can explain what we observe. As humans, we make generalizations and seek order. We have found that nature is remarkably cooperative—it exhibits the underlying order and simplicity we so value. It is the underlying order of nature that makes science in general, and physics in particular, so enjoyable to study. For example, what do a bag of chips and a car battery have in common? Both contain energy that can be converted to other forms. The law of conservation of energy (which says that energy can change form but is never lost) ties together such topics as food calories, batteries, heat, light, and watch springs. Understanding this law makes it easier to learn about the various forms energy takes and how they relate to one another. Apparently unrelated topics are connected through broadly applicable physical laws, permitting an understanding beyond just the memorization of lists of facts. The unifying aspect of physical laws and the basic simplicity of nature form the underlying themes of this text. In learning to apply these laws, you will, of course, study the most important topics in physics. More importantly, you will gain analytical abilities that will enable you to apply these laws far beyond the scope of what can be included in a single book. These analytical skills will help you to excel academically, and they will also help you to think critically in any professional career you choose to pursue. This module discusses the realm of physics (to define what physics is), some applications of physics (to illustrate its relevance to other disciplines), and more precisely what constitutes a physical law (to illuminate the importance of experimentation to theory). Science and the Realm of Physics Science consists of the theories and laws that are the general truths of nature as well as the body of knowledge they encompass. Scientists are continually trying to expand this body of knowledge and to perfect the expression of the laws that describe it. Physics is concerned with describing the interactions of energy, matter, space, and time, and it is especially interested in what fundamental me
chanisms underlie every phenomenon. The concern for describing the basic phenomena in nature essentially defines the realm of physics. Physics aims to describe the function of everything around us, from the movement of tiny charged particles to the motion of people, cars, and spaceships. In fact, almost everything around you can be described quite accurately by the laws of physics. Consider a smart phone (Figure 1.3). Physics describes how electricity interacts with the various circuits inside the device. This This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 \| Introduction: The Nature of Science and Physics 9 knowledge helps engineers select the appropriate materials and circuit layout when building the smart phone. Next, consider a GPS system. Physics describes the relationship between the speed of an object, the distance over which it travels, and the time it takes to travel that distance. When you use a GPS device in a vehicle, it utilizes these physics equations to determine the travel time from one location to another. Figure 1.3 The Apple “iPhone” is a common smart phone with a GPS function. Physics describes the way that electricity flows through the circuits of this device. Engineers use their knowledge of physics to construct an iPhone with features that consumers will enjoy. One specific feature of an iPhone is the GPS function. GPS uses physics equations to determine the driving time between two locations on a map. (credit: @gletham GIS, Social, Mobile Tech Images) Applications of Physics You need not be a scientist to use physics. On the contrary, knowledge of physics is useful in everyday situations as well as in nonscientific professions. It can help you understand how microwave ovens work, why metals should not be put into them, and why they might affect pacemakers. (See Figure 1.4 and Figure 1.5.) Physics allows you to understand the hazards of radiation and rationally evaluate these hazards more easily. Physics also explains the reason why a black car radiator helps remove heat in a car engine, and it explains why a white roof helps keep the inside of a house cool. Similarly, the operation of a car's ignition system as well as the transmission of electrical signals through our body's nervous system are much easier to understand when you think about them in terms of basic physics. Physics is the foundation of many important disciplines and contributes directly to others. Chemistry, for example—since it deals with the interactions of atoms and molecules—is rooted in atomic and molecular physics. Most branches of engineering are applied physics. In architecture, physics is at the heart of structural stability, and is involved in the acoustics, heating, lighting, and cooling of buildings. Parts of geology rely heavily on physics, such as radioactive dating of rocks, earthquake analysis, and heat transfer in the Earth. Some disciplines, such as biophysics and geophysics, are hybrids of physics and other disciplines. Physics has many applications in the biological sciences. On the microscopic level, it helps describe the properties of cell walls and cell membranes (Figure 1.6 and Figure 1.7). On the macroscopic level, it can explain the heat, work, and power associated with the human body. Physics is involved in medical diagnostics, such as x-rays, magnetic resonance imaging (MRI), and ultrasonic blood flow measurements. Medical therapy sometimes directly involves physics; for example, cancer radiotherapy uses ionizing radiation. Physics can also explain sensory phenomena, such as how musical instruments make sound, how the eye detects color, and how lasers can transmit information. It is not necessary to formally study all applications of physics. What is most useful is knowledge of the basic laws of physics and a skill in the analytical methods for applying them. The study of physics also can improve your problem-solving skills. Furthermore, physics has retained the most basic aspects of science, so it is used by all of the sciences, and the study of physics makes other sciences easier to understand. Figure 1.4 The laws of physics help us understand how common appliances work. For example, the laws of physics can help explain how microwave ovens heat up food, and they also help us understand why it is dangerous to place metal objects in a microwave oven. (credit: MoneyBlogNewz) 10 Chapter 1 \| Introduction: The Nature of Science and Physics Figure 1.5 These two applications of physics have more in common than meets the eye. Microwave ovens use electromagnetic waves to heat food. Magnetic resonance imaging (MRI) also uses electromagnetic waves to yield an image of the brain, from which the exact location of tumors can be determined. (credit: Rashmi Chawla, Daniel Smith, and Paul E. Marik) Figure 1.6 Physics, chemistry, and biology help describe the properties of cell walls in plant cells, such as the onion cells seen here. (credit: Umberto Salvagnin) Figure 1.7 An artist's rendition of the the structure of a cell membrane. Membranes form the boundaries of animal cells and are complex in structure and function. Many of the most fundamental properties of life, such as the firing of nerve cells, are related to membranes. The disciplines of biology, chemistry, and physics all help us understand the membranes of animal cells. (credit: Mariana Ruiz) Models, Theories, and Laws; The Role of Experimentation The laws of nature are concise descriptions of the universe around us; they are human statements of the underlying laws or rules that all natural processes follow. Such laws are intrinsic to the universe; humans did not create them and so cannot change them. We can only discover and understand them. Their discovery is a very human endeavor, with all the elements of mystery, imagination, struggle, triumph, and disappointment inherent in any creative effort. (See Figure 1.8 and Figure 1.9.) The cornerstone of discovering natural laws is observation; science must describe the universe as it is, not as we may imagine it to be. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 \| Introduction: The Nature of Science and Physics 11 Figure 1.8 Isaac Newton (1642–1727) was very reluctant to publish his revolutionary work and had to be convinced to do so. In his later years, he stepped down from his academic post and became exchequer of the Royal Mint. He took this post seriously, inventing reeding (or creating ridges) on the edge of coins to prevent unscrupulous people from trimming the silver off of them before using them as currency. (credit: Arthur Shuster and Arthur E. Shipley: Britain's Heritage of Science. London, 1917.) Figure 1.9 Marie Curie (1867–1934) sacrificed monetary assets to help finance her early research and damaged her physical well-being with radiation exposure. She is the only person to win Nobel prizes in both physics and chemistry. One of her daughters also won a Nobel Prize. (credit: Wikimedia Commons) We all are curious to some extent. We look around, make generalizations, and try to understand what we see—for example, we look up and wonder whether one type of cloud signals an oncoming storm. As we become serious about exploring nature, we become more organized and formal in collecting and analyzing data. We attempt greater precision, perform controlled experiments (if we can), and write down ideas about how the data may be organized and unified. We then formulate models, theories, and laws based on the data we have collected and analyzed to generalize and communicate the results of these experiments. A model is a representation of something that is often too difficult (or impossible) to display directly. While a model is justified with experimental proof, it is only accurate under limited situations. An example is the planetary model of the atom in which electrons are pictured as orbiting the nucleus, analogous to the way planets orbit the Sun. (See Figure 1.10.) We cannot observe electron orbits directly, but the mental image helps explain the observations we can make, such as the emission of light from hot gases (atomic spectra). Physicists use models for a variety of purposes. For example, models can help physicists analyze a scenario and perform a calculation, or they can be used to represent a situation in the form of a computer simulation. A theory is an explanation for patterns in nature that is supported by scientific evidence and verified multiple times by various groups of researchers. Some theories include models to help visualize phenomena, whereas others do not. Newton's theory of gravity, for example, does not require a model or mental image, because we can observe the objects directly with our own senses. The kinetic theory of gases, on the other hand, is a model in which a gas is viewed as being composed of atoms and molecules. Atoms and molecules are too small to be observed directly with our senses—thus, we picture them mentally to understand what our instruments tell us about the behavior of gases. A law uses concise language to describe a generalized pattern in nature that is supported by scientific evidence and repeated experiments. Often, a law can be expressed in the form of a single mathematical equation. Laws and theories are similar in that they are both scientific statements that result from a tested hypothesis and are supported by scientific evidence. However, the designation law is reserved for a concise and very general statement that describes phenomena in nature, such as the law that 12 Chapter 1 \| Introduction: The Nature of Science and Physics energy is conserved during any process, or Newton's second law of motion, which relates force, mass, and acceleration by the simple equation F = a . A theory, in contrast, is a less concise statement of observed phenomena. For example, the Theory of Evolution and the Theory of Relativity cannot be expressed concisely enough to be considered a law
. The biggest difference between a law and a theory is that a theory is much more complex and dynamic. A law describes a single action, whereas a theory explains an entire group of related phenomena. And, whereas a law is a postulate that forms the foundation of the scientific method, a theory is the end result of that process. Less broadly applicable statements are usually called principles (such as Pascal's principle, which is applicable only in fluids), but the distinction between laws and principles often is not carefully made. Figure 1.10 What is a model? This planetary model of the atom shows electrons orbiting the nucleus. It is a drawing that we use to form a mental image of the atom that we cannot see directly with our eyes because it is too small. Models, Theories, and Laws Models, theories, and laws are used to help scientists analyze the data they have already collected. However, often after a model, theory, or law has been developed, it points scientists toward new discoveries they would not otherwise have made. The models, theories, and laws we devise sometimes imply the existence of objects or phenomena as yet unobserved. These predictions are remarkable triumphs and tributes to the power of science. It is the underlying order in the universe that enables scientists to make such spectacular predictions. However, if experiment does not verify our predictions, then the theory or law is wrong, no matter how elegant or convenient it is. Laws can never be known with absolute certainty because it is impossible to perform every imaginable experiment in order to confirm a law in every possible scenario. Physicists operate under the assumption that all scientific laws and theories are valid until a counterexample is observed. If a good-quality, verifiable experiment contradicts a well-established law, then the law must be modified or overthrown completely. The study of science in general and physics in particular is an adventure much like the exploration of uncharted ocean. Discoveries are made; models, theories, and laws are formulated; and the beauty of the physical universe is made more sublime for the insights gained. The Scientific Method As scientists inquire and gather information about the world, they follow a process called the scientific method. This process typically begins with an observation and question that the scientist will research. Next, the scientist typically performs some research about the topic and then devises a hypothesis. Then, the scientist will test the hypothesis by performing an experiment. Finally, the scientist analyzes the results of the experiment and draws a conclusion. Note that the scientific method can be applied to many situations that are not limited to science, and this method can be modified to suit the situation. Consider an example. Let us say that you try to turn on your car, but it will not start. You undoubtedly wonder: Why will the car not start? You can follow a scientific method to answer this question. First off, you may perform some research to determine a variety of reasons why the car will not start. Next, you will state a hypothesis. For example, you may believe that the car is not starting because it has no engine oil. To test this, you open the hood of the car and examine the oil level. You observe that the oil is at an acceptable level, and you thus conclude that the oil level is not contributing to your car issue. To troubleshoot the issue further, you may devise a new hypothesis to test and then repeat the process again. The Evolution of Natural Philosophy into Modern Physics Physics was not always a separate and distinct discipline. It remains connected to other sciences to this day. The word physics comes from Greek, meaning nature. The study of nature came to be called “natural philosophy.” From ancient times through the Renaissance, natural philosophy encompassed many fields, including astronomy, biology, chemistry, physics, mathematics, and medicine. Over the last few centuries, the growth of knowledge has resulted in ever-increasing specialization and branching of natural philosophy into separate fields, with physics retaining the most basic facets. (See Figure 1.11, Figure 1.12, and Figure 1.13.) Physics as it developed from the Renaissance to the end of the 19th century is called classical physics. It was transformed into modern physics by revolutionary discoveries made starting at the beginning of the 20th century. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 \| Introduction: The Nature of Science and Physics 13 Figure 1.11 Over the centuries, natural philosophy has evolved into more specialized disciplines, as illustrated by the contributions of some of the greatest minds in history. The Greek philosopher Aristotle (384–322 B.C.) wrote on a broad range of topics including physics, animals, the soul, politics, and poetry. (credit: Jastrow (2006)/Ludovisi Collection) Figure 1.12 Galileo Galilei (1564–1642) laid the foundation of modern experimentation and made contributions in mathematics, physics, and astronomy. (credit: Domenico Tintoretto) Figure 1.13 Niels Bohr (1885–1962) made fundamental contributions to the development of quantum mechanics, one part of modern physics. (credit: United States Library of Congress Prints and Photographs Division) Classical physics is not an exact description of the universe, but it is an excellent approximation under the following conditions: Matter must be moving at speeds less than about 1% of the speed of light, the objects dealt with must be large enough to be seen with a microscope, and only weak gravitational fields, such as the field generated by the Earth, can be involved. Because humans live under such circumstances, classical physics seems intuitively reasonable, while many aspects of modern physics seem bizarre. This is why models are so useful in modern physics—they let us conceptualize phenomena we do not ordinarily experience. We can relate to models in human terms and visualize what happens when objects move at high speeds or imagine what objects too small to observe with our senses might be like. For example, we can understand an atom's properties because we can picture it in our minds, although we have never seen an atom with our eyes. New tools, of course, allow us to better picture phenomena we cannot see. In fact, new instrumentation has allowed us in recent years to actually “picture” the atom. 14 Chapter 1 \| Introduction: The Nature of Science and Physics Limits on the Laws of Classical Physics For the laws of classical physics to apply, the following criteria must be met: Matter must be moving at speeds less than about 1% of the speed of light, the objects dealt with must be large enough to be seen with a microscope, and only weak gravitational fields (such as the field generated by the Earth) can be involved. Figure 1.14 Using a scanning tunneling microscope (STM), scientists can see the individual atoms that compose this sheet of gold. (credit: Erwinrossen) Some of the most spectacular advances in science have been made in modern physics. Many of the laws of classical physics have been modified or rejected, and revolutionary changes in technology, society, and our view of the universe have resulted. Like science fiction, modern physics is filled with fascinating objects beyond our normal experiences, but it has the advantage over science fiction of being very real. Why, then, is the majority of this text devoted to topics of classical physics? There are two main reasons: Classical physics gives an extremely accurate description of the universe under a wide range of everyday circumstances, and knowledge of classical physics is necessary to understand modern physics. Modern physics itself consists of the two revolutionary theories, relativity and quantum mechanics. These theories deal with the very fast and the very small, respectively. Relativity must be used whenever an object is traveling at greater than about 1% of the speed of light or experiences a strong gravitational field such as that near the Sun. Quantum mechanics must be used for objects smaller than can be seen with a microscope. The combination of these two theories is relativistic quantum mechanics, and it describes the behavior of small objects traveling at high speeds or experiencing a strong gravitational field. Relativistic quantum mechanics is the best universally applicable theory we have. Because of its mathematical complexity, it is used only when necessary, and the other theories are used whenever they will produce sufficiently accurate results. We will find, however, that we can do a great deal of modern physics with the algebra and trigonometry used in this text. Check Your Understanding A friend tells you he has learned about a new law of nature. What can you know about the information even before your friend describes the law? How would the information be different if your friend told you he had learned about a scientific theory rather than a law? Solution Without knowing the details of the law, you can still infer that the information your friend has learned conforms to the requirements of all laws of nature: it will be a concise description of the universe around us; a statement of the underlying rules that all natural processes follow. If the information had been a theory, you would be able to infer that the information will be a large-scale, broadly applicable generalization. PhET Explorations: Equation Grapher Learn about graphing polynomials. The shape of the curve changes as the constants are adjusted. View the curves for the individual terms (e.g. = ) to see how they add to generate the polynomial curve. Figure 1.15 Equation Grapher (http://cnx.org/content/m54764/1.2/equation-grapher_en.jar) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 \| Introduction: The Nature of Science and Physics 15 1.2 Physical Quantities an
d Units Figure 1.16 The distance from Earth to the Moon may seem immense, but it is just a tiny fraction of the distances from Earth to other celestial bodies. (credit: NASA) By the end of this section, you will be able to: Learning Objectives • Perform unit conversions both in the SI and English units. • Explain the most common prefixes in the SI units and be able to write them in scientific notation. The range of objects and phenomena studied in physics is immense. From the incredibly short lifetime of a nucleus to the age of Earth, from the tiny sizes of sub-nuclear particles to the vast distance to the edges of the known universe, from the force exerted by a jumping flea to the force between Earth and the Sun, there are enough factors of 10 to challenge the imagination of even the most experienced scientist. Giving numerical values for physical quantities and equations for physical principles allows us to understand nature much more deeply than does qualitative description alone. To comprehend these vast ranges, we must also have accepted units in which to express them. And we shall find that (even in the potentially mundane discussion of meters, kilograms, and seconds) a profound simplicity of nature appears—all physical quantities can be expressed as combinations of only four fundamental physical quantities: length, mass, time, and electric current. We define a physical quantity either by specifying how it is measured or by stating how it is calculated from other measurements. For example, we define distance and time by specifying methods for measuring them, whereas we define average speed by stating that it is calculated as distance traveled divided by time of travel. Measurements of physical quantities are expressed in terms of units, which are standardized values. For example, the length of a race, which is a physical quantity, can be expressed in units of meters (for sprinters) or kilometers (for distance runners). Without standardized units, it would be extremely difficult for scientists to express and compare measured values in a meaningful way. (See Figure 1.17.) Figure 1.17 Distances given in unknown units are maddeningly useless. There are two major systems of units used in the world: SI units (also known as the metric system) and English units (also known as the customary or imperial system). English units were historically used in nations once ruled by the British Empire and are still widely used in the United States. Virtually every other country in the world now uses SI units as the standard; the metric system is also the standard system agreed upon by scientists and mathematicians. The acronym “SI” is derived from the French Système International. 16 Chapter 1 \| Introduction: The Nature of Science and Physics SI Units: Fundamental and Derived Units Table 1.1 gives the fundamental SI units that are used throughout this textbook. This text uses non-SI units in a few applications where they are in very common use, such as the measurement of blood pressure in millimeters of mercury (mm Hg). Whenever non-SI units are discussed, they will be tied to SI units through conversions. Table 1.1 Fundamental SI Units Length Mass Time Electric Charge meter (m) kilogram (kg) second (s) coulomb (c) It is an intriguing fact that some physical quantities are more fundamental than others and that the most fundamental physical quantities can be defined only in terms of the procedure used to measure them. The units in which they are measured are thus called fundamental units. In this textbook, the fundamental physical quantities are taken to be length, mass, time, and electric charge. (Note that electric current will not be introduced until much later in this text.) All other physical quantities, such as force and electric current, can be expressed as algebraic combinations of length, mass, time, and current (for example, speed is length divided by time); these units are called derived units. Units of Time, Length, and Mass: The Second, Meter, and Kilogram The Second The SI unit for time, the second(abbreviated s), has a long history. For many years it was defined as 1/86,400 of a mean solar day. More recently, a new standard was adopted to gain greater accuracy and to define the second in terms of a non-varying, or constant, physical phenomenon (because the solar day is getting longer due to very gradual slowing of the Earth's rotation). Cesium atoms can be made to vibrate in a very steady way, and these vibrations can be readily observed and counted. In 1967 the second was redefined as the time required for 9,192,631,770 of these vibrations. (See Figure 1.18.) Accuracy in the fundamental units is essential, because all measurements are ultimately expressed in terms of fundamental units and can be no more accurate than are the fundamental units themselves. Figure 1.18 An atomic clock such as this one uses the vibrations of cesium atoms to keep time to a precision of better than a microsecond per year. The fundamental unit of time, the second, is based on such clocks. This image is looking down from the top of an atomic fountain nearly 30 feet tall! (credit: Steve Jurvetson/Flickr) The Meter The SI unit for length is the meter (abbreviated m); its definition has also changed over time to become more accurate and precise. The meter was first defined in 1791 as 1/10,000,000 of the distance from the equator to the North Pole. This measurement was improved in 1889 by redefining the meter to be the distance between two engraved lines on a platinum-iridium bar now kept near Paris. By 1960, it had become possible to define the meter even more accurately in terms of the wavelength of light, so it was again redefined as 1,650,763.73 wavelengths of orange light emitted by krypton atoms. In 1983, the meter was given its present definition (partly for greater accuracy) as the distance light travels in a vacuum in 1/299,792,458 of a second. (See Figure 1.19.) This change defines the speed of light to be exactly 299,792,458 meters per second. The length of the meter will change if the speed of light is someday measured with greater accuracy. The Kilogram The SI unit for mass is the kilogram (abbreviated kg); it is defined to be the mass of a platinum-iridium cylinder kept with the old meter standard at the International Bureau of Weights and Measures near Paris. Exact replicas of the standard kilogram are also kept at the United States' National Institute of Standards and Technology, or NIST, located in Gaithersburg, Maryland outside of Washington D.C., and at other locations around the world. The determination of all other masses can be ultimately traced to a comparison with the standard mass. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 \| Introduction: The Nature of Science and Physics 17 Figure 1.19 The meter is defined to be the distance light travels in 1/299,792,458 of a second in a vacuum. Distance traveled is speed multiplied by time. Electric current and its accompanying unit, the ampere, will be introduced in Introduction to Electric Current, Resistance, and Ohm's Law when electricity and magnetism are covered. The initial modules in this textbook are concerned with mechanics, fluids, heat, and waves. In these subjects all pertinent physical quantities can be expressed in terms of the fundamental units of length, mass, and time. Metric Prefixes SI units are part of the metric system. The metric system is convenient for scientific and engineering calculations because the units are categorized by factors of 10. Table 1.2 gives metric prefixes and symbols used to denote various factors of 10. Metric systems have the advantage that conversions of units involve only powers of 10. There are 100 centimeters in a meter, 1000 meters in a kilometer, and so on. In nonmetric systems, such as the system of U.S. customary units, the relationships are not as simple—there are 12 inches in a foot, 5280 feet in a mile, and so on. Another advantage of the metric system is that the same unit can be used over extremely large ranges of values simply by using an appropriate metric prefix. For example, distances in meters are suitable in construction, while distances in kilometers are appropriate for air travel, and the tiny measure of nanometers are convenient in optical design. With the metric system there is no need to invent new units for particular applications. The term order of magnitude refers to the scale of a value expressed in the metric system. Each power of 10 in the metric system represents a different order of magnitude. For example, 101 102 103 magnitude. All quantities that can be expressed as a product of a specific power of 10 are said to be of the same order of magnitude. For example, the number 800 can be written as 8×102 , and the number 450 can be written as 4.5×102. Thus, the numbers 800 and 450 are of the same order of magnitude: 102. Order of magnitude can be thought of as a ballpark estimate for the scale of a value. The diameter of an atom is on the order of 10−9 m, while the diameter of the Sun is on the order of 109 m. , and so forth are all different orders of The Quest for Microscopic Standards for Basic Units The fundamental units described in this chapter are those that produce the greatest accuracy and precision in measurement. There is a sense among physicists that, because there is an underlying microscopic substructure to matter, it would be most satisfying to base our standards of measurement on microscopic objects and fundamental physical phenomena such as the speed of light. A microscopic standard has been accomplished for the standard of time, which is based on the oscillations of the cesium atom. The standard for length was once based on the wavelength of light (a small-scale length) emitted by a certain type of atom, but it has been supplanted by the more precise measurement of the speed of light. If it becomes possible to measure the mass of atoms or a pa
rticular arrangement of atoms such as a silicon sphere to greater precision than the kilogram standard, it may become possible to base mass measurements on the small scale. There are also possibilities that electrical phenomena on the small scale may someday allow us to base a unit of charge on the charge of electrons and protons, but at present current and charge are related to large-scale currents and forces between wires. 18 Chapter 1 \| Introduction: The Nature of Science and Physics Table 1.2 Metric Prefixes for Powers of 10 and their Symbols Value[1] Symbol Prefix Example (some are approximate) exa peta tera giga mega kilo hecto deka — deci centi milli micro nano pico femto atto E P T G M k h da — d c m µ n p f a 1018 1015 1012 109 106 103 102 101 100 (=1) 10−1 10−2 10−3 10−6 10−9 10−12 10−15 10−18 exameter Em 1018 m distance light travels in a century petasecond Ps 1015 s 30 million years terawatt TW 1012 W powerful laser output gigahertz GHz 109 Hz a microwave frequency megacurie MCi 106 Ci high radioactivity kilometer km 103 m about 6/10 mile hectoliter hL 102 L 26 gallons dekagram dag 101 g teaspoon of butter deciliter dL 10−1 L less than half a soda centimeter cm 10−2 m fingertip thickness millimeter mm 10−3 m flea at its shoulders micrometer µm 10−6 m detail in microscope nanogram ng 10−9 g small speck of dust picofarad pF 10−12 F small capacitor in radio femtometer fm 10−15 m size of a proton attosecond as 10−18 s time light crosses an atom Known Ranges of Length, Mass, and Time The vastness of the universe and the breadth over which physics applies are illustrated by the wide range of examples of known lengths, masses, and times in Table 1.3. Examination of this table will give you some feeling for the range of possible topics and numerical values. (See Figure 1.20 and Figure 1.21.) Figure 1.20 Tiny phytoplankton swims among crystals of ice in the Antarctic Sea. They range from a few micrometers to as much as 2 millimeters in length. (credit: Prof. Gordon T. Taylor, Stony Brook University; NOAA Corps Collections) 1. See Appendix A for a discussion of powers of 10. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 \| Introduction: The Nature of Science and Physics 19 Figure 1.21 Galaxies collide 2.4 billion light years away from Earth. The tremendous range of observable phenomena in nature challenges the imagination. (credit: NASA/CXC/UVic./A. Mahdavi et al. Optical/lensing: CFHT/UVic./H. Hoekstra et al.) Unit Conversion and Dimensional Analysis It is often necessary to convert from one type of unit to another. For example, if you are reading a European cookbook, some quantities may be expressed in units of liters and you need to convert them to cups. Or, perhaps you are reading walking directions from one location to another and you are interested in how many miles you will be walking. In this case, you will need to convert units of feet to miles. Let us consider a simple example of how to convert units. Let us say that we want to convert 80 meters (m) to kilometers (km). The first thing to do is to list the units that you have and the units that you want to convert to. In this case, we have units in meters and we want to convert to kilometers. Next, we need to determine a conversion factor relating meters to kilometers. A conversion factor is a ratio expressing how many of one unit are equal to another unit. For example, there are 12 inches in 1 foot, 100 centimeters in 1 meter, 60 seconds in 1 minute, and so on. In this case, we know that there are 1,000 meters in 1 kilometer. Now we can set up our unit conversion. We will write the units that we have and then multiply them by the conversion factor so that the units cancel out, as shown: 80m× 1 km 1000m = 0.080 km. (1.1) Note that the unwanted m unit cancels, leaving only the desired km unit. You can use this method to convert between any types of unit. Click Appendix C for a more complete list of conversion factors. 20 Chapter 1 \| Introduction: The Nature of Science and Physics Table 1.3 Approximate Values of Length, Mass, and Time Lengths in meters Masses in kilograms (more precise values in parentheses) Times in seconds (more precise values in parentheses) 10−18 Present experimental limit to smallest observable detail 10−30 Mass of an electron 9.11×10−31 kg 10−23 Time for light to cross a proton 10−15 Diameter of a proton 10−27 Mass of a hydrogen atom 1.67×10−27 kg 10−22 Mean life of an extremely unstable nucleus 10−14 Diameter of a uranium nucleus 10−15 Mass of a bacterium 10−10 Diameter of a hydrogen atom 10−5 Mass of a mosquito 10−15 Time for one oscillation of visible light 10−13 Time for one vibration of an atom in a solid 10−8 Thickness of membranes in cells of living organisms 10−2 Mass of a hummingbird 10−8 Time for one oscillation of an FM radio wave Mass of a liter of water (about a quart) 10−3 Duration of a nerve impulse 10−6 Wavelength of visible light 10−3 Size of a grain of sand Height of a 4-year-old child Length of a football field 1 102 103 108 Mass of a person Mass of a car Mass of a large ship Greatest ocean depth 1012 Mass of a large iceberg Diameter of the Earth 1015 Mass of the nucleus of a comet 1011 Recorded history 1 105 107 109 Time for one heartbeat One day 8.64×104 s One year (y) 3.16×107 s About half the life expectancy of a human 1017 1018 Age of the Earth Age of the universe 1 102 104 107 1011 Distance from the Earth to the Sun 1023 Mass of the Moon 7.35×1022 kg 1016 Distance traveled by light in 1 year (a light year) 1025 Mass of the Earth 5.97×1024 kg 1021 1022 1026 Diameter of the Milky Way galaxy 1030 Mass of the Sun 1.99×1030 kg Distance from the Earth to the nearest large galaxy (Andromeda) Distance from the Earth to the edges of the known universe 1042 Mass of the Milky Way galaxy (current upper limit) 1053 Mass of the known universe (current upper limit) Example 1.1 Unit Conversions: A Short Drive Home Suppose that you drive the 10.0 km from your university to home in 20.0 min. Calculate your average speed (a) in kilometers per hour (km/h) and (b) in meters per second (m/s). (Note: Average speed is distance traveled divided by time of travel.) Strategy First we calculate the average speed using the given units. Then we can get the average speed into the desired units by picking the correct conversion factor and multiplying by it. The correct conversion factor is the one that cancels the unwanted unit and leaves the desired unit in its place. Solution for (a) (1) Calculate average speed. Average speed is distance traveled divided by time of travel. (Take this definition as a given for now—average speed and other motion concepts will be covered in a later module.) In equation form, average speed =distance time . (1.2) (2) Substitute the given values for distance and time. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 \| Introduction: The Nature of Science and Physics average speed = 10.0 km 20.0 min = 0.500 km min . (3) Convert km/min to km/h: multiply by the conversion factor that will cancel minutes and leave hours. That conversion factor is 60 min/hr . Thus, average speed =0.500 km min × 60 min 1 h = 30.0 km h . Discussion for (a) To check your answer, consider the following: 21 (1.3) (1.4) (1) Be sure that you have properly cancelled the units in the unit conversion. If you have written the unit conversion factor upside down, the units will not cancel properly in the equation. If you accidentally get the ratio upside down, then the units will not cancel; rather, they will give you the wrong units as follows: = 1 60 × 1 hr 60 min km min , km ⋅ hr min2 (1.5) which are obviously not the desired units of km/h. (2) Check that the units of the final answer are the desired units. The problem asked us to solve for average speed in units of km/h and we have indeed obtained these units. (3) Check the significant figures. Because each of the values given in the problem has three significant figures, the answer should also have three significant figures. The answer 30.0 km/hr does indeed have three significant figures, so this is appropriate. Note that the significant figures in the conversion factor are not relevant because an hour is defined to be 60 minutes, so the precision of the conversion factor is perfect. (4) Next, check whether the answer is reasonable. Let us consider some information from the problem—if you travel 10 km in a third of an hour (20 min), you would travel three times that far in an hour. The answer does seem reasonable. Solution for (b) There are several ways to convert the average speed into meters per second. (1) Start with the answer to (a) and convert km/h to m/s. Two conversion factors are needed—one to convert hours to seconds, and another to convert kilometers to meters. (2) Multiplying by these yields Average speed = 30.0km h 3,600 s Average speed = 8.33m s . × 1 h × 1,000 m 1 km , (1.6) (1.7) Discussion for (b) If we had started with 0.500 km/min, we would have needed different conversion factors, but the answer would have been the same: 8.33 m/s. You may have noted that the answers in the worked example just covered were given to three digits. Why? When do you need to be concerned about the number of digits in something you calculate? Why not write down all the digits your calculator produces? The module Accuracy, Precision, and Significant Figures will help you answer these questions. Nonstandard Units While there are numerous types of units that we are all familiar with, there are others that are much more obscure. For example, a firkin is a unit of volume that was once used to measure beer. One firkin equals about 34 liters. To learn more about nonstandard units, use a dictionary or encyclopedia to research different “weights and measures.” Take note of any unusual units, such as a barleycorn, that are not listed in the text. Think about how the unit is defined and state its relationship to SI units. Chec
k Your Understanding Some hummingbirds beat their wings more than 50 times per second. A scientist is measuring the time it takes for a hummingbird to beat its wings once. Which fundamental unit should the scientist use to describe the measurement? Which factor of 10 is the scientist likely to use to describe the motion precisely? Identify the metric prefix that corresponds to this factor of 10. Solution 22 Chapter 1 \| Introduction: The Nature of Science and Physics The scientist will measure the time between each movement using the fundamental unit of seconds. Because the wings beat so fast, the scientist will probably need to measure in milliseconds, or 10−3 20 milliseconds per beat.) seconds. (50 beats per second corresponds to Check Your Understanding One cubic centimeter is equal to one milliliter. What does this tell you about the different units in the SI metric system? Solution The fundamental unit of length (meter) is probably used to create the derived unit of volume (liter). The measure of a milliliter is dependent on the measure of a centimeter. 1.3 Accuracy, Precision, and Significant Figures Figure 1.22 A double-pan mechanical balance is used to compare different masses. Usually an object with unknown mass is placed in one pan and objects of known mass are placed in the other pan. When the bar that connects the two pans is horizontal, then the masses in both pans are equal. The “known masses” are typically metal cylinders of standard mass such as 1 gram, 10 grams, and 100 grams. (credit: Serge Melki) Figure 1.23 Many mechanical balances, such as double-pan balances, have been replaced by digital scales, which can typically measure the mass of an object more precisely. Whereas a mechanical balance may only read the mass of an object to the nearest tenth of a gram, many digital scales can measure the mass of an object up to the nearest thousandth of a gram. (credit: Karel Jakubec) By the end of this section, you will be able to: Learning Objectives • Determine the appropriate number of significant figures in both addition and subtraction, as well as multiplication and division calculations. • Calculate the percent uncertainty of a measurement. Accuracy and Precision of a Measurement Science is based on observation and experiment—that is, on measurements. Accuracy is how close a measurement is to the correct value for that measurement. For example, let us say that you are measuring the length of standard computer paper. The This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 \| Introduction: The Nature of Science and Physics 23 packaging in which you purchased the paper states that it is 11.0 inches long. You measure the length of the paper three times and obtain the following measurements: 11.1 in., 11.2 in., and 10.9 in. These measurements are quite accurate because they are very close to the correct value of 11.0 inches. In contrast, if you had obtained a measurement of 12 inches, your measurement would not be very accurate. The precision of a measurement system is refers to how close the agreement is between repeated measurements (which are repeated under the same conditions). Consider the example of the paper measurements. The precision of the measurements refers to the spread of the measured values. One way to analyze the precision of the measurements would be to determine the range, or difference, between the lowest and the highest measured values. In that case, the lowest value was 10.9 in. and the highest value was 11.2 in. Thus, the measured values deviated from each other by at most 0.3 in. These measurements were relatively precise because they did not vary too much in value. However, if the measured values had been 10.9, 11.1, and 11.9, then the measurements would not be very precise because there would be significant variation from one measurement to another. The measurements in the paper example are both accurate and precise, but in some cases, measurements are accurate but not precise, or they are precise but not accurate. Let us consider an example of a GPS system that is attempting to locate the position of a restaurant in a city. Think of the restaurant location as existing at the center of a bull's-eye target, and think of each GPS attempt to locate the restaurant as a black dot. In Figure 1.24, you can see that the GPS measurements are spread out far apart from each other, but they are all relatively close to the actual location of the restaurant at the center of the target. This indicates a low precision, high accuracy measuring system. However, in Figure 1.25, the GPS measurements are concentrated quite closely to one another, but they are far away from the target location. This indicates a high precision, low accuracy measuring system. Figure 1.24 A GPS system attempts to locate a restaurant at the center of the bull's-eye. The black dots represent each attempt to pinpoint the location of the restaurant. The dots are spread out quite far apart from one another, indicating low precision, but they are each rather close to the actual location of the restaurant, indicating high accuracy. (credit: Dark Evil) Figure 1.25 In this figure, the dots are concentrated rather closely to one another, indicating high precision, but they are rather far away from the actual location of the restaurant, indicating low accuracy. (credit: Dark Evil) Accuracy, Precision, and Uncertainty The degree of accuracy and precision of a measuring system are related to the uncertainty in the measurements. Uncertainty is a quantitative measure of how much your measured values deviate from a standard or expected value. If your measurements are not very accurate or precise, then the uncertainty of your values will be very high. In more general terms, uncertainty can be thought of as a disclaimer for your measured values. For example, if someone asked you to provide the mileage on your car, you might say that it is 45,000 miles, plus or minus 500 miles. The plus or minus amount is the uncertainty in your value. That is, you are indicating that the actual mileage of your car might be as low as 44,500 miles or as high as 45,500 miles, or anywhere in between. All measurements contain some amount of uncertainty. In our example of measuring the length of the paper, we might say that the length of the paper is 11 in., plus or minus 0.2 in. The uncertainty in a measurement, , is often denoted as (“delta ”), so the measurement result would be recorded as ± . In our paper example, the length of the paper could be expressed as 11 in. ± 0.2. The factors contributing to uncertainty in a measurement include: 24 Chapter 1 \| Introduction: The Nature of Science and Physics 1. Limitations of the measuring device, 2. The skill of the person making the measurement, 3. Irregularities in the object being measured, 4. Any other factors that affect the outcome (highly dependent on the situation). In our example, such factors contributing to the uncertainty could be the following: the smallest division on the ruler is 0.1 in., the person using the ruler has bad eyesight, or one side of the paper is slightly longer than the other. At any rate, the uncertainty in a measurement must be based on a careful consideration of all the factors that might contribute and their possible effects. Making Connections: Real-World Connections—Fevers or Chills? Uncertainty is a critical piece of information, both in physics and in many other real-world applications. Imagine you are caring for a sick child. You suspect the child has a fever, so you check his or her temperature with a thermometer. What if the uncertainty of the thermometer were 3.0ºC ? If the child's temperature reading was 37.0ºC (which is normal body temperature), the “true” temperature could be anywhere from a hypothermic 34.0ºC to a dangerously high 40.0ºC . A thermometer with an uncertainty of 3.0ºC would be useless. Percent Uncertainty One method of expressing uncertainty is as a percent of the measured value. If a measurement is expressed with uncertainty, , the percent uncertainty (%unc) is defined to be % unc = ×100%. (1.8) Example 1.2 Calculating Percent Uncertainty: A Bag of Apples A grocery store sells 5 lb bags of apples. You purchase four bags over the course of a month and weigh the apples each time. You obtain the following measurements: • Week 1 weight: 4.8 lb • Week 2 weight: 5.3 lb • Week 3 weight: 4.9 lb • Week 4 weight: 5.4 lb You determine that the weight of the 5 lb bag has an uncertainty of ±0.4 lb . What is the percent uncertainty of the bag's weight? Strategy First, observe that the expected value of the bag's weight, , is 5 lb. The uncertainty in this value, , is 0.4 lb. We can use the following equation to determine the percent uncertainty of the weight: Solution Plug the known values into the equation: Discussion % unc = ×100%. % unc =0.4 lb 5 lb ×100% = 8%. (1.9) (1.10) We can conclude that the weight of the apple bag is 5 lb ± 8% . Consider how this percent uncertainty would change if the bag of apples were half as heavy, but the uncertainty in the weight remained the same. Hint for future calculations: when calculating percent uncertainty, always remember that you must multiply the fraction by 100%. If you do not do this, you will have a decimal quantity, not a percent value. Uncertainties in Calculations There is an uncertainty in anything calculated from measured quantities. For example, the area of a floor calculated from measurements of its length and width has an uncertainty because the length and width have uncertainties. How big is the uncertainty in something you calculate by multiplication or division? If the measurements going into the calculation have small uncertainties (a few percent or less), then the method of adding percents can be used for multiplication or division. This method says that the percent uncertainty in a quantity calculated by multiplication or division is the sum of the percent This content is
available for free at http://cnx.org/content/col11844/1.13 Chapter 1 \| Introduction: The Nature of Science and Physics 25 uncertainties in the items used to make the calculation. For example, if a floor has a length of 4.00 m and a width of 3.00 m , with uncertainties of 2% and 1% , respectively, then the area of the floor is 12.0 m2 and has an uncertainty of 3% . (Expressed as an area this is 0.36 m2 , which we round to 0.4 m2 since the area of the floor is given to a tenth of a square meter.) Check Your Understanding A high school track coach has just purchased a new stopwatch. The stopwatch manual states that the stopwatch has an uncertainty of ±0.05 s . Runners on the track coach's team regularly clock 100 m sprints of 11.49 s to 15.01 s . At the school's last track meet, the first-place sprinter came in at 12.04 s and the second-place sprinter came in at 12.07 s . Will the coach's new stopwatch be helpful in timing the sprint team? Why or why not? Solution No, the uncertainty in the stopwatch is too great to effectively differentiate between the sprint times. Precision of Measuring Tools and Significant Figures An important factor in the accuracy and precision of measurements involves the precision of the measuring tool. In general, a precise measuring tool is one that can measure values in very small increments. For example, a standard ruler can measure length to the nearest millimeter, while a caliper can measure length to the nearest 0.01 millimeter. The caliper is a more precise measuring tool because it can measure extremely small differences in length. The more precise the measuring tool, the more precise and accurate the measurements can be. When we express measured values, we can only list as many digits as we initially measured with our measuring tool. For example, if you use a standard ruler to measure the length of a stick, you may measure it to be 36.7 cm . You could not express this value as 36.71 cm because your measuring tool was not precise enough to measure a hundredth of a centimeter. It should be noted that the last digit in a measured value has been estimated in some way by the person performing the measurement. For example, the person measuring the length of a stick with a ruler notices that the stick length seems to be somewhere in between 36.6 cm and 36.7 cm , and he or she must estimate the value of the last digit. Using the method of significant figures, the rule is that the last digit written down in a measurement is the first digit with some uncertainty. In order to determine the number of significant digits in a value, start with the first measured value at the left and count the number of digits through the last digit written on the right. For example, the measured value 36.7 cm has three digits, or significant figures. Significant figures indicate the precision of a measuring tool that was used to measure a value. Zeros Special consideration is given to zeros when counting significant figures. The zeros in 0.053 are not significant, because they are only placekeepers that locate the decimal point. There are two significant figures in 0.053. The zeros in 10.053 are not placekeepers but are significant—this number has five significant figures. The zeros in 1300 may or may not be significant depending on the style of writing numbers. They could mean the number is known to the last digit, or they could be placekeepers. So 1300 could have two, three, or four significant figures. (To avoid this ambiguity, write 1300 in scientific notation.) Zeros are significant except when they serve only as placekeepers. Check Your Understanding Determine the number of significant figures in the following measurements: a. 0.0009 b. 15,450.0 c. 6×103 d. 87.990 e. 30.42 Solution (a) 1; the zeros in this number are placekeepers that indicate the decimal point (b) 6; here, the zeros indicate that a measurement was made to the 0.1 decimal point, so the zeros are significant (c) 1; the value 103 signifies the decimal place, not the number of measured values (d) 5; the final zero indicates that a measurement was made to the 0.001 decimal point, so it is significant (e) 4; any zeros located in between significant figures in a number are also significant 26 Chapter 1 \| Introduction: The Nature of Science and Physics Significant Figures in Calculations When combining measurements with different degrees of accuracy and precision, the number of significant digits in the final answer can be no greater than the number of significant digits in the least precise measured value. There are two different rules, one for multiplication and division and the other for addition and subtraction, as discussed below. 1. For multiplication and division: The result should have the same number of significant figures as the quantity having the least significant figures entering into the calculation. For example, the area of a circle can be calculated from its radius using = 2 . Let us see how many significant figures the area has if the radius has only two—say, = 1.2 m . Then, = 2 = (3.1415927...)×(1.2 m)2 = 4.5238934 m2 (1.11) is what you would get using a calculator that has an eight-digit output. But because the radius has only two significant figures, it limits the calculated quantity to two significant figures or =4.5 m2, (1.12) even though is good to at least eight digits. 2. For addition and subtraction: The answer can contain no more decimal places than the least precise measurement. Suppose that you buy 7.56 kg of potatoes in a grocery store as measured with a scale with precision 0.01 kg. Then you drop off 6.052 kg of potatoes at your laboratory as measured by a scale with precision 0.001 kg. Finally, you go home and add 13.7 kg of potatoes as measured by a bathroom scale with precision 0.1 kg. How many kilograms of potatoes do you now have, and how many significant figures are appropriate in the answer? The mass is found by simple addition and subtraction: 7.56 kg - 6.052 kg kg +13.7 15.208 kg = 15.2 kg. (1.13) Next, we identify the least precise measurement: 13.7 kg. This measurement is expressed to the 0.1 decimal place, so our final answer must also be expressed to the 0.1 decimal place. Thus, the answer is rounded to the tenths place, giving us 15.2 kg. Significant Figures in this Text In this text, most numbers are assumed to have three significant figures. Furthermore, consistent numbers of significant figures are used in all worked examples. You will note that an answer given to three digits is based on input good to at least three digits, for example. If the input has fewer significant figures, the answer will also have fewer significant figures. Care is also taken that the number of significant figures is reasonable for the situation posed. In some topics, particularly in optics, more accurate numbers are needed and more than three significant figures will be used. Finally, if a number is exact, such as the two in the formula for the circumference of a circle, = 2π , it does not affect the number of significant figures in a calculation. Check Your Understanding Perform the following calculations and express your answer using the correct number of significant digits. (a) A woman has two bags weighing 13.5 pounds and one bag with a weight of 10.2 pounds. What is the total weight of the bags? (b) The force on an object is equal to its mass multiplied by its acceleration . If a wagon with mass 55 kg accelerates at a rate of 0.0255 m/s2 , what is the force on the wagon? (The unit of force is called the newton, and it is expressed with the symbol N.) Solution (a) 37.2 pounds; Because the number of bags is an exact value, it is not considered in the significant figures. (b) 1.4 N; Because the value 55 kg has only two significant figures, the final value must also contain two significant figures. PhET Explorations: Estimation Explore size estimation in one, two, and three dimensions! Multiple levels of difficulty allow for progressive skill improvement. Figure 1.26 Estimation (http://cnx.org/content/m54766/1.7/estimation_en.jar) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 \| Introduction: The Nature of Science and Physics 27 1.4 Approximation Learning Objectives By the end of this section, you will be able to: • Make reasonable approximations based on given data. On many occasions, physicists, other scientists, and engineers need to make approximations or “guesstimates” for a particular quantity. What is the distance to a certain destination? What is the approximate density of a given item? About how large a current will there be in a circuit? Many approximate numbers are based on formulae in which the input quantities are known only to a limited accuracy. As you develop problem-solving skills (that can be applied to a variety of fields through a study of physics), you will also develop skills at approximating. You will develop these skills through thinking more quantitatively, and by being willing to take risks. As with any endeavor, experience helps, as well as familiarity with units. These approximations allow us to rule out certain scenarios or unrealistic numbers. Approximations also allow us to challenge others and guide us in our approaches to our scientific world. Let us do two examples to illustrate this concept. Example 1.3 Approximate the Height of a Building Can you approximate the height of one of the buildings on your campus, or in your neighborhood? Let us make an approximation based upon the height of a person. In this example, we will calculate the height of a 39-story building. Strategy Think about the average height of an adult male. We can approximate the height of the building by scaling up from the height of a person. Solution Based on information in the example, we know there are 39 stories in the building. If we use the fact that the height of one story is approximately equal to about the length of two adult humans (each human is about 2 m tall), then
we can estimate the total height of the building to be 2 m 1 person × 2 person 1 story ×39 stories = 156 m. (1.14) Discussion You can use known quantities to determine an approximate measurement of unknown quantities. If your hand measures 10 cm across, how many hand lengths equal the width of your desk? What other measurements can you approximate besides length? Example 1.4 Approximating Vast Numbers: a Trillion Dollars Figure 1.27 A bank stack contains one-hundred $100 bills, and is worth $10,000. How many bank stacks make up a trillion dollars? (credit: Andrew Magill) The U.S. federal deficit in the 2008 fiscal year was a little greater than $10 trillion. Most of us do not have any concept of how much even one trillion actually is. Suppose that you were given a trillion dollars in $100 bills. If you made 100-bill stacks and used them to evenly cover a football field (between the end zones), make an approximation of how high the money pile 28 Chapter 1 \| Introduction: The Nature of Science and Physics would become. (We will use feet/inches rather than meters here because football fields are measured in yards.) One of your friends says 3 in., while another says 10 ft. What do you think? Strategy When you imagine the situation, you probably envision thousands of small stacks of 100 wrapped $100 bills, such as you might see in movies or at a bank. Since this is an easy-to-approximate quantity, let us start there. We can find the volume of a stack of 100 bills, find out how many stacks make up one trillion dollars, and then set this volume equal to the area of the football field multiplied by the unknown height. Solution (1) Calculate the volume of a stack of 100 bills. The dimensions of a single bill are approximately 3 in. by 6 in. A stack of 100 of these is about 0.5 in. thick. So the total volume of a stack of 100 bills is: volume of stack = length×width×height, volume of stack = 6 in.×3 in.×0.5 in. volume of stack = 9 in.3 . (1.15) (2) Calculate the number of stacks. Note that a trillion dollars is equal to $1×1012, and a stack of one-hundred $100 bills is equal to $10000 or $1×104 . The number of stacks you will have is: $1×1012(a trillion dollars)/ $1×104 per stack = 1×108 stacks. (3) Calculate the area of a football field in square inches. The area of a football field is 100 yd×50 yd, which gives 5,000 yd2. Because we are working in inches, we need to convert square yards to square inches: Area = 5,000 yd2× 3 ft 1 yd × 3 ft 1 yd × 12 in. 1 ft × 12 in. 1 ft = 6,480000 in.2 Area ≈ 6×106 in.2 . (1.16) (1.17) This conversion gives us 6×106 in.2 for the area of the field. (Note that we are using only one significant figure in these calculations.) (4) Calculate the total volume of the bills. The volume of all the $100 -bill stacks is 9 in.3 / stack×108 stacks = 9×108 in.3. (5) Calculate the height. To determine the height of the bills, use the equation: = area of field×height of money: volume of bills Height of money = volume of bills area of field Height of money = 9×108in.3 6×106in.2 = 1.33×102 in., Height of money ≈ 1×102 in. = 100 in. The height of the money will be about 100 in. high. Converting this value to feet gives 100 in.× 1 ft 12 in. = 8.33 ft ≈ 8 ft. Discussion (1.18) (1.19) The final approximate value is much higher than the early estimate of 3 in., but the other early estimate of 10 ft (120 in.) was roughly correct. How did the approximation measure up to your first guess? What can this exercise tell you in terms of rough “guesstimates” versus carefully calculated approximations? Check Your Understanding Using mental math and your understanding of fundamental units, approximate the area of a regulation basketball court. Describe the process you used to arrive at your final approximation. Solution An average male is about two meters tall. It would take approximately 15 men laid out end to end to cover the length, and about 7 to cover the width. That gives an approximate area of 420 m2 . This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 \| Introduction: The Nature of Science and Physics 29 Glossary accuracy: the degree to which a measured value agrees with correct value for that measurement approximation: an estimated value based on prior experience and reasoning classical physics: physics that was developed from the Renaissance to the end of the 19th century conversion factor: a ratio expressing how many of one unit are equal to another unit derived units: units that can be calculated using algebraic combinations of the fundamental units English units: system of measurement used in the United States; includes units of measurement such as feet, gallons, and pounds fundamental units: units that can only be expressed relative to the procedure used to measure them kilogram: the SI unit for mass, abbreviated (kg) law: a description, using concise language or a mathematical formula, a generalized pattern in nature that is supported by scientific evidence and repeated experiments meter: the SI unit for length, abbreviated (m) method of adding percents: the percent uncertainty in a quantity calculated by multiplication or division is the sum of the percent uncertainties in the items used to make the calculation metric system: a system in which values can be calculated in factors of 10 model: representation of something that is often too difficult (or impossible) to display directly modern physics: the study of relativity, quantum mechanics, or both order of magnitude: refers to the size of a quantity as it relates to a power of 10 percent uncertainty: the ratio of the uncertainty of a measurement to the measured value, expressed as a percentage physical quantity : a characteristic or property of an object that can be measured or calculated from other measurements physics: the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon precision: the degree to which repeated measurements agree with each other quantum mechanics: the study of objects smaller than can be seen with a microscope relativity: the study of objects moving at speeds greater than about 1% of the speed of light, or of objects being affected by a strong gravitational field scientific method: a method that typically begins with an observation and question that the scientist will research; next, the scientist typically performs some research about the topic and then devises a hypothesis; then, the scientist will test the hypothesis by performing an experiment; finally, the scientist analyzes the results of the experiment and draws a conclusion second: the SI unit for time, abbreviated (s) SI units : the international system of units that scientists in most countries have agreed to use; includes units such as meters, liters, and grams significant figures: express the precision of a measuring tool used to measure a value theory: an explanation for patterns in nature that is supported by scientific evidence and verified multiple times by various groups of researchers uncertainty: a quantitative measure of how much your measured values deviate from a standard or expected value units : a standard used for expressing and comparing measurements Section Summary 1.1 Physics: An Introduction • Science seeks to discover and describe the underlying order and simplicity in nature. 30 Chapter 1 \| Introduction: The Nature of Science and Physics • Physics is the most basic of the sciences, concerning itself with energy, matter, space and time, and their interactions. • Scientific laws and theories express the general truths of nature and the body of knowledge they encompass. These laws of nature are rules that all natural processes appear to follow. 1.2 Physical Quantities and Units • Physical quantities are a characteristic or property of an object that can be measured or calculated from other measurements. • Units are standards for expressing and comparing the measurement of physical quantities. All units can be expressed as combinations of four fundamental units. • The four fundamental units we will use in this text are the meter (for length), the kilogram (for mass), the second (for time), and the ampere (for electric current). These units are part of the metric system, which uses powers of 10 to relate quantities over the vast ranges encountered in nature. • The four fundamental units are abbreviated as follows: meter, m; kilogram, kg; second, s; and ampere, A. The metric system also uses a standard set of prefixes to denote each order of magnitude greater than or lesser than the fundamental unit itself. • Unit conversions involve changing a value expressed in one type of unit to another type of unit. This is done by using conversion factors, which are ratios relating equal quantities of different units. 1.3 Accuracy, Precision, and Significant Figures • Accuracy of a measured value refers to how close a measurement is to the correct value. The uncertainty in a measurement is an estimate of the amount by which the measurement result may differ from this value. • Precision of measured values refers to how close the agreement is between repeated measurements. • The precision of a measuring tool is related to the size of its measurement increments. The smaller the measurement increment, the more precise the tool. • Significant figures express the precision of a measuring tool. • When multiplying or dividing measured values, the final answer can contain only as many significant figures as the least precise value. • When adding or subtracting measured values, the final answer cannot contain more decimal places than the least precise value. 1.4 Approximation Scientists often approximate the values of quantities to perform calculations and analyze systems. Conceptual Questions 1.1 Physics: An Introduction 1. Models are particularly useful in relativity and quantum mechanics, where conditions are outside those
normally encountered by humans. What is a model? 2. How does a model differ from a theory? 3. If two different theories describe experimental observations equally well, can one be said to be more valid than the other (assuming both use accepted rules of logic)? 4. What determines the validity of a theory? 5. Certain criteria must be satisfied if a measurement or observation is to be believed. Will the criteria necessarily be as strict for an expected result as for an unexpected result? 6. Can the validity of a model be limited, or must it be universally valid? How does this compare to the required validity of a theory or a law? 7. Classical physics is a good approximation to modern physics under certain circumstances. What are they? 8. When is it necessary to use relativistic quantum mechanics? 9. Can classical physics be used to accurately describe a satellite moving at a speed of 7500 m/s? Explain why or why not. 1.2 Physical Quantities and Units 10. Identify some advantages of metric units. 1.3 Accuracy, Precision, and Significant Figures 11. What is the relationship between the accuracy and uncertainty of a measurement? 12. Prescriptions for vision correction are given in units called diopters (D). Determine the meaning of that unit. Obtain information (perhaps by calling an optometrist or performing an internet search) on the minimum uncertainty with which corrections in diopters are determined and the accuracy with which corrective lenses can be produced. Discuss the sources of uncertainties in both the prescription and accuracy in the manufacture of lenses. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 \| Introduction: The Nature of Science and Physics 31 Problems & Exercises 1.2 Physical Quantities and Units 1. The speed limit on some interstate highways is roughly 100 km/h. (a) What is this in meters per second? (b) How many miles per hour is this? 2. A car is traveling at a speed of 33 m/s . (a) What is its speed in kilometers per hour? (b) Is it exceeding the 90 km/h speed limit? 3. Show that 1.0 m/s = 3.6 km/h . Hint: Show the explicit steps involved in converting 1.0 m/s = 3.6 km/h. 4. American football is played on a 100-yd-long field, excluding the end zones. How long is the field in meters? (Assume that 1 meter equals 3.281 feet.) 5. Soccer fields vary in size. A large soccer field is 115 m long and 85 m wide. What are its dimensions in feet and inches? (Assume that 1 meter equals 3.281 feet.) 6. What is the height in meters of a person who is 6 ft 1.0 in. tall? (Assume that 1 meter equals 39.37 in.) 7. Mount Everest, at 29,028 feet, is the tallest mountain on the Earth. What is its height in kilometers? (Assume that 1 kilometer equals 3,281 feet.) 8. The speed of sound is measured to be 342 m/s on a certain day. What is this in km/h? 9. Tectonic plates are large segments of the Earth's crust that move slowly. Suppose that one such plate has an average speed of 4.0 cm/year. (a) What distance does it move in 1 s at this speed? (b) What is its speed in kilometers per million years? 10. (a) Refer to Table 1.3 to determine the average distance between the Earth and the Sun. Then calculate the average speed of the Earth in its orbit in kilometers per second. (b) What is this in meters per second? 1.3 Accuracy, Precision, and Significant Figures Express your answers to problems in this section to the correct number of significant figures and proper units. 11. Suppose that your bathroom scale reads your mass as 65 kg with a 3% uncertainty. What is the uncertainty in your mass (in kilograms)? 12. A good-quality measuring tape can be off by 0.50 cm over a distance of 20 m. What is its percent uncertainty? 13. (a) A car speedometer has a 5.0% uncertainty. What is the range of possible speeds when it reads 90 km/h ? (b) Convert this range to miles per hour. (1 km = 0.6214 mi) 14. An infant's pulse rate is measured to be 130 ± 5 beats/ min. What is the percent uncertainty in this measurement? 15. (a) Suppose that a person has an average heart rate of 72.0 beats/min. How many beats does he or she have in 2.0 y? (b) In 2.00 y? (c) In 2.000 y? (106.7)(98.2) / (46.210)(1.01) (b) (18.7)2 (c) 1.60×10−19 (3712) . 18. (a) How many significant figures are in the numbers 99 and 100? (b) If the uncertainty in each number is 1, what is the percent uncertainty in each? (c) Which is a more meaningful way to express the accuracy of these two numbers, significant figures or percent uncertainties? 19. (a) If your speedometer has an uncertainty of 2.0 km/h at a speed of 90 km/h , what is the percent uncertainty? (b) If it has the same percent uncertainty when it reads 60 km/h , what is the range of speeds you could be going? 20. (a) A person's blood pressure is measured to be 120 ± 2 mm Hg . What is its percent uncertainty? (b) Assuming the same percent uncertainty, what is the uncertainty in a blood pressure measurement of 80 mm Hg? 21. A person measures his or her heart rate by counting the number of beats in 30 s . If 40 ± 1 beats are counted in 30.0 ± 0.5 s , what is the heart rate and its uncertainty in beats per minute? 22. What is the area of a circle 3.102 cm in diameter? 23. If a marathon runner averages 9.5 mi/h, how long does it take him or her to run a 26.22 mi marathon? 24. A marathon runner completes a 42.188 km course in 2 h , 30 min, and 12 s . There is an uncertainty of 25 m in the distance traveled and an uncertainty of 1 s in the elapsed time. (a) Calculate the percent uncertainty in the distance. (b) Calculate the uncertainty in the elapsed time. (c) What is the average speed in meters per second? (d) What is the uncertainty in the average speed? 25. The sides of a small rectangular box are measured to be 1.80 ± 0.01 cm , 2.05 ± 0.02 cm, and 3.1 ± 0.1 cm long. Calculate its volume and uncertainty in cubic centimeters. 26. When non-metric units were used in the United Kingdom, a unit of mass called the pound-mass (lbm) was employed, where 1 lbm = 0.4539 kg . (a) If there is an uncertainty of 0.0001 kg in the pound-mass unit, what is its percent uncertainty? (b) Based on that percent uncertainty, what mass in pound-mass has an uncertainty of 1 kg when converted to kilograms? 27. The length and width of a rectangular room are measured to be 3.955 ± 0.005 m and 3.050 ± 0.005 m . Calculate the area of the room and its uncertainty in square meters. 28. A car engine moves a piston with a circular cross section of 7.500 ± 0.002 cm diameter a distance of 3.250 ± 0.001 cm to compress the gas in the cylinder. (a) By what amount is the gas decreased in volume in cubic centimeters? (b) Find the uncertainty in this volume. 16. A can contains 375 mL of soda. How much is left after 308 mL is removed? 1.4 Approximation 17. State how many significant figures are proper in the results of the following calculations: (a) 29. How many heartbeats are there in a lifetime? 32 Chapter 1 \| Introduction: The Nature of Science and Physics 30. A generation is about one-third of a lifetime. Approximately how many generations have passed since the year 0 AD? 31. How many times longer than the mean life of an extremely unstable atomic nucleus is the lifetime of a human? (Hint: The lifetime of an unstable atomic nucleus is on the order of 10−22 s .) 32. Calculate the approximate number of atoms in a bacterium. Assume that the average mass of an atom in the bacterium is ten times the mass of a hydrogen atom. (Hint: The mass of a hydrogen atom is on the order of 10−27 kg and the mass of a bacterium is on the order of 10−15 kg. ) Figure 1.28 This color-enhanced photo shows Salmonella typhimurium (red) attacking human cells. These bacteria are commonly known for causing foodborne illness. Can you estimate the number of atoms in each bacterium? (credit: Rocky Mountain Laboratories, NIAID, NIH) 33. Approximately how many atoms thick is a cell membrane, assuming all atoms there average about twice the size of a hydrogen atom? 34. (a) What fraction of Earth's diameter is the greatest ocean depth? (b) The greatest mountain height? 35. (a) Calculate the number of cells in a hummingbird assuming the mass of an average cell is ten times the mass of a bacterium. (b) Making the same assumption, how many cells are there in a human? 36. Assuming one nerve impulse must end before another can begin, what is the maximum firing rate of a nerve in impulses per second? This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 33 2 KINEMATICS Figure 2.1 The motion of an American kestrel through the air can be described by the bird's displacement, speed, velocity, and acceleration. When it flies in a straight line without any change in direction, its motion is said to be one dimensional. (credit: Vince Maidens, Wikimedia Commons) Chapter Outline 2.1. Displacement 2.2. Vectors, Scalars, and Coordinate Systems 2.3. Time, Velocity, and Speed 2.4. Acceleration 2.5. Motion Equations for Constant Acceleration in One Dimension 2.6. Problem-Solving Basics for One Dimensional Kinematics 2.7. Falling Objects 2.8. Graphical Analysis of One Dimensional Motion Connection for AP® Courses Objects are in motion everywhere we look. Everything from a tennis game to a space-probe flyby of the planet Neptune involves motion. When you are resting, your heart moves blood through your veins. Even in inanimate objects, there is a continuous motion in the vibrations of atoms and molecules. Questions about motion are interesting in and of themselves: How long will it take for a space probe to get to Mars? Where will a football land if it is thrown at a certain angle? Understanding motion will not only provide answers to these questions, but will be key to understanding more advanced concepts in physics. For example, the discussion of force in Chapter 4 will not fully make sense until you understand acceleration. This relationship between force and acceleration is also critical to understanding Big Idea 3. Ad
ditionally, this unit will explore the topic of reference frames, a critical component to quantifying how things move. If you have ever waved to a departing friend at a train station, you are likely familiar with this idea. While you see your friend move away from you at a considerable rate, those sitting with her will likely see her as not moving. The effect that the chosen reference frame has on your observations is substantial, and an understanding of this is needed to grasp both Enduring Understanding 3.A and Essential Knowledge 3.A.1. Our formal study of physics begins with kinematics, which is defined as the study of motion without considering its causes. In one- and two-dimensional kinematics we will study only the motion of a football, for example, without worrying about what forces cause or change its motion. In this chapter, we examine the simplest type of motion—namely, motion along a straight line, or one-dimensional motion. Later, in two-dimensional kinematics, we apply concepts developed here to study motion along curved paths (two- and three-dimensional motion), for example, that of a car rounding a curve. The content in this chapter supports: Big Idea 3 The interactions of an object with other objects can be described by forces. 34 Chapter 2 \| Kinematics Enduring Understanding 3.A All forces share certain common characteristics when considered by observers in inertial reference frames. Essential Knowledge 3.A.1 An observer in a particular reference frame can describe the motion of an object using such quantities as position, displacement, distance, velocity, speed, and acceleration. 2.1 Displacement Figure 2.2 These cyclists in Vietnam can be described by their position relative to buildings and a canal. Their motion can be described by their change in position, or displacement, in the frame of reference. (credit: Suzan Black, Fotopedia) Learning Objectives By the end of this section, you will be able to: • Define position, displacement, distance, and distance traveled in a particular frame of reference. • Explain the relationship between position and displacement. • Distinguish between displacement and distance traveled. • Calculate displacement and distance given initial position, final position, and the path between the two. The information presented in this section supports the following AP® learning objectives and science practices: • 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, and graphical representations. (S.P. 1.5, 2.1, 2.2) • 3.A.1.3 The student is able to analyze experimental data describing the motion of an object and is able to express the results of the analysis using narrative, mathematical, and graphical representations. (S.P. 5.1) Position In order to describe the motion of an object, you must first be able to describe its position—where it is at any particular time. More precisely, you need to specify its position relative to a convenient reference frame. Earth is often used as a reference frame, and we often describe the position of an object as it relates to stationary objects in that reference frame. For example, a rocket launch would be described in terms of the position of the rocket with respect to the Earth as a whole, while a professor's position could be described in terms of where she is in relation to the nearby white board. (See Figure 2.3.) In other cases, we use reference frames that are not stationary but are in motion relative to the Earth. To describe the position of a person in an airplane, for example, we use the airplane, not the Earth, as the reference frame. (See Figure 2.4.) Displacement If an object moves relative to a reference frame (for example, if a professor moves to the right relative to a white board or a passenger moves toward the rear of an airplane), then the object's position changes. This change in position is known as displacement. The word “displacement” implies that an object has moved, or has been displaced. Displacement Displacement is the change in position of an object: where Δ is displacement, f is the final position, and 0 is the initial position. Δ = f − 0, (2.1) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 35 In this text the upper case Greek letter Δ (delta) always means “change in” whatever quantity follows it; thus, Δ means change in position. Always solve for displacement by subtracting initial position 0 from final position f . Note that the SI unit for displacement is the meter (m) (see Physical Quantities and Units), but sometimes kilometers, miles, feet, and other units of length are used. Keep in mind that when units other than the meter are used in a problem, you may need to convert them into meters to complete the calculation. Figure 2.3 A professor paces left and right while lecturing. Her position relative to the blackboard is given by . The +2.0 m displacement of the professor relative to the blackboard is represented by an arrow pointing to the right. Figure 2.4 A passenger moves from his seat to the back of the plane. His location relative to the airplane is given by . The −4 m displacement of the passenger relative to the plane is represented by an arrow toward the rear of the plane. Notice that the arrow representing his displacement is twice as long as the arrow representing the displacement of the professor (he moves twice as far) in Figure 2.3. Note that displacement has a direction as well as a magnitude. The professor's displacement is 2.0 m to the right, and the airline passenger's displacement is 4.0 m toward the rear. In one-dimensional motion, direction can be specified with a plus or minus sign. When you begin a problem, you should select which direction is positive (usually that will be to the right or up, but you are free to select positive as being any direction). The professor's initial position is 0 = 1.5 m and her final position is f = 3.5 m . Thus her displacement is Δ = f −0 = 3.5 m − 1.5 m = + 2.0 m. (2.2) In this coordinate system, motion to the right is positive, whereas motion to the left is negative. Similarly, the airplane passenger's initial position is 0 = 6.0 m and his final position is f = 2.0 m , so his displacement is 36 Chapter 2 \| Kinematics His displacement is negative because his motion is toward the rear of the plane, or in the negative direction in our coordinate system. Δ = f −0 = 2.0 m − 6.0 m = −4.0 m. (2.3) Distance Although displacement is described in terms of direction, distance is not. Distance is defined to be the magnitude or size of displacement between two positions. Note that the distance between two positions is not the same as the distance traveled between them. Distance traveled is the total length of the path traveled between two positions. Distance has no direction and, thus, no sign. For example, the distance the professor walks is 2.0 m. The distance the airplane passenger walks is 4.0 m. Misconception Alert: Distance Traveled vs. Magnitude of Displacement It is important to note that the distance traveled, however, can be greater than the magnitude of the displacement (by magnitude, we mean just the size of the displacement without regard to its direction; that is, just a number with a unit). For example, the professor could pace back and forth many times, perhaps walking a distance of 150 m during a lecture, yet still end up only 2.0 m to the right of her starting point. In this case her displacement would be +2.0 m, the magnitude of her displacement would be 2.0 m, but the distance she traveled would be 150 m. In kinematics we nearly always deal with displacement and magnitude of displacement, and almost never with distance traveled. One way to think about this is to assume you marked the start of the motion and the end of the motion. The displacement is simply the difference in the position of the two marks and is independent of the path taken in traveling between the two marks. The distance traveled, however, is the total length of the path taken between the two marks. Check Your Understanding A cyclist rides 3 km west and then turns around and rides 2 km east. (a) What is her displacement? (b) What distance does she ride? (c) What is the magnitude of her displacement? Solution Figure 2.5 (a) The rider's displacement is Δ = f − 0 = −1 km . (The displacement is negative because we take east to be positive and west to be negative.) (b) The distance traveled is 3 km + 2 km = 5 km . (c) The magnitude of the displacement is 1 km . This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 37 2.2 Vectors, Scalars, and Coordinate Systems Figure 2.6 The motion of this Eclipse Concept jet can be described in terms of the distance it has traveled (a scalar quantity) or its displacement in a specific direction (a vector quantity). In order to specify the direction of motion, its displacement must be described based on a coordinate system. In this case, it may be convenient to choose motion toward the left as positive motion (it is the forward direction for the plane), although in many cases, the -coordinate runs from left to right, with motion to the right as positive and motion to the left as negative. (credit: Armchair Aviator, Flickr) Learning Objectives By the end of this section, you will be able to: • Define and distinguish between scalar and vector quantities. • Assign a coordinate system for a scenario involving one-dimensional motion. The information presented in this section supports the following AP® learning objectives and science practices: • 3.A.1.2 The student is able to design an experimental investigation of the motion of an object. What is the difference between distance and displacement? Whereas displacement is defined by both direction and magnitude, distance is defined only by magnitude. Displacement is an example of a vector quantity. Distance is an example of a scalar quantity. A vector is any quantit
y with both magnitude and direction. Other examples of vectors include a velocity of 90 km/h east and a force of 500 newtons straight down. The direction of a vector in one-dimensional motion is given simply by a plus ( + ) or minus ( − ) sign. Vectors are represented graphically by arrows. An arrow used to represent a vector has a length proportional to the vector's magnitude (e.g., the larger the magnitude, the longer the length of the vector) and points in the same direction as the vector. Some physical quantities, like distance, either have no direction or none is specified. A scalar is any quantity that has a magnitude, but no direction. For example, a 20ºC temperature, the 250 kilocalories (250 Calories) of energy in a candy bar, a 90 km/h speed limit, a person's 1.8 m height, and a distance of 2.0 m are all scalars—quantities with no specified direction. Note, however, that a scalar can be negative, such as a −20ºC temperature. In this case, the minus sign indicates a point on a scale rather than a direction. Scalars are never represented by arrows. Coordinate Systems for One-Dimensional Motion In order to describe the direction of a vector quantity, you must designate a coordinate system within the reference frame. For one-dimensional motion, this is a simple coordinate system consisting of a one-dimensional coordinate line. In general, when describing horizontal motion, motion to the right is usually considered positive, and motion to the left is considered negative. With vertical motion, motion up is usually positive and motion down is negative. In some cases, however, as with the jet in Figure 2.6, it can be more convenient to switch the positive and negative directions. For example, if you are analyzing the motion of falling objects, it can be useful to define downwards as the positive direction. If people in a race are running to the left, it is useful to define left as the positive direction. It does not matter as long as the system is clear and consistent. Once you assign a positive direction and start solving a problem, you cannot change it. 38 Chapter 2 \| Kinematics Figure 2.7 It is usually convenient to consider motion upward or to the right as positive ( + ) and motion downward or to the left as negative ( − ) . Check Your Understanding A person's speed can stay the same as he or she rounds a corner and changes direction. Given this information, is speed a scalar or a vector quantity? Explain. Solution Speed is a scalar quantity. It does not change at all with direction changes; therefore, it has magnitude only. If it were a vector quantity, it would change as direction changes (even if its magnitude remained constant). Switching Reference Frames A fundamental tenet of physics is that information about an event can be gathered from a variety of reference frames. For example, imagine that you are a passenger walking toward the front of a bus. As you walk, your motion is observed by a fellow bus passenger and by an observer standing on the sidewalk. Both the bus passenger and sidewalk observer will be able to collect information about you. They can determine how far you moved and how much time it took you to do so. However, while you moved at a consistent pace, both observers will get different results. To the passenger sitting on the bus, you moved forward at what one would consider a normal pace, something similar to how quickly you would walk outside on a sunny day. To the sidewalk observer though, you will have moved much quicker. Because the bus is also moving forward, the distance you move forward against the sidewalk each second increases, and the sidewalk observer must conclude that you are moving at a greater pace. To show that you understand this concept, you will need to create an event and think of a way to view this event from two different frames of reference. In order to ensure that the event is being observed simultaneously from both frames, you will need an assistant to help out. An example of a possible event is to have a friend ride on a skateboard while tossing a ball. How will your friend observe the ball toss, and how will those observations be different from your own? Your task is to describe your event and the observations of your event from both frames of reference. Answer the following questions below to demonstrate your understanding. For assistance, you can review the information given in the ‘Position' paragraph at the start of Section 2.1. 1. What is your event? What object are both you and your assistant observing? 2. What do you see as the event takes place? 3. What does your assistant see as the event takes place? 4. How do your reference frames cause you and your assistant to have two different sets of observations? This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 39 2.3 Time, Velocity, and Speed Figure 2.8 The motion of these racing snails can be described by their speeds and their velocities. (credit: tobitasflickr, Flickr) By the end of this section, you will be able to: Learning Objectives • Explain the relationships between instantaneous velocity, average velocity, instantaneous speed, average speed, displacement, and time. • Calculate velocity and speed given initial position, initial time, final position, and final time. • Derive a graph of velocity vs. time given a graph of position vs. time. • Interpret a graph of velocity vs. time. The information presented in this section supports the following AP® learning objectives and science practices: • 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, and graphical representations. (S.P. 1.5, 2.1, 2.2) • 3.A.1.3 The student is able to analyze experimental data describing the motion of an object and is able to express the results of the analysis using narrative, mathematical, and graphical representations. (S.P. 5.1) There is more to motion than distance and displacement. Questions such as, “How long does a foot race take?” and “What was the runner's speed?” cannot be answered without an understanding of other concepts. In this section we add definitions of time, velocity, and speed to expand our description of motion. Time As discussed in Physical Quantities and Units, the most fundamental physical quantities are defined by how they are measured. This is the case with time. Every measurement of time involves measuring a change in some physical quantity. It may be a number on a digital clock, a heartbeat, or the position of the Sun in the sky. In physics, the definition of time is simple— time is change, or the interval over which change occurs. It is impossible to know that time has passed unless something changes. The amount of time or change is calibrated by comparison with a standard. The SI unit for time is the second, abbreviated s. We might, for example, observe that a certain pendulum makes one full swing every 0.75 s. We could then use the pendulum to measure time by counting its swings or, of course, by connecting the pendulum to a clock mechanism that registers time on a dial. This allows us to not only measure the amount of time, but also to determine a sequence of events. How does time relate to motion? We are usually interested in elapsed time for a particular motion, such as how long it takes an airplane passenger to get from his seat to the back of the plane. To find elapsed time, we note the time at the beginning and end of the motion and subtract the two. For example, a lecture may start at 11:00 A.M. and end at 11:50 A.M., so that the elapsed time would be 50 min. Elapsed time Δt is the difference between the ending time and beginning time, Δ = f − 0, (2.4) where Δ is the change in time or elapsed time, f is the time at the end of the motion, and 0 is the time at the beginning of the motion. (As usual, the delta symbol, Δ , means the change in the quantity that follows it.) Life is simpler if the beginning time 0 is taken to be zero, as when we use a stopwatch. If we were using a stopwatch, it would simply read zero at the start of the lecture and 50 min at the end. If 0 = 0 , then Δ = f ≡ . In this text, for simplicity's sake, • motion starts at time equal to zero (0 = 0) • the symbol is used for elapsed time unless otherwise specified (Δ = f ≡ ) 40 Velocity Chapter 2 \| Kinematics Your notion of velocity is probably the same as its scientific definition. You know that if you have a large displacement in a small amount of time you have a large velocity, and that velocity has units of distance divided by time, such as miles per hour or kilometers per hour. Average Velocity Average velocity is displacement (change in position) divided by the time of travel, f − 0 f − 0 - is the average (indicated by the bar over the ) velocity, Δ is the change in position (or displacement), and f where and 0 are the final and beginning positions at times f and 0 , respectively. If the starting time 0 is taken to be zero, then the average velocity is simply - = Δ Δ = (2.5) - = Δ . (2.6) Notice that this definition indicates that velocity is a vector because displacement is a vector. It has both magnitude and direction. The SI unit for velocity is meters per second or m/s, but many other units, such as km/h, mi/h (also written as mph), and cm/s, are in common use. Suppose, for example, an airplane passenger took 5 seconds to move −4 m (the minus sign indicates that displacement is toward the back of the plane). His average velocity would be - = Δ = −4 m 5 s = − 0.8 m/s. (2.7) The minus sign indicates the average velocity is also toward the rear of the plane. The average velocity of an object does not tell us anything about what happens to it between the starting point and ending point, however. For example, we cannot tell from average velocity whether the airplane passenger stops momentarily or backs up before he goes to the back of the plane. To get more details, we must consider smaller segments of the trip over small
er time intervals. Figure 2.9 A more detailed record of an airplane passenger heading toward the back of the plane, showing smaller segments of his trip. The smaller the time intervals considered in a motion, the more detailed the information. When we carry this process to its logical conclusion, we are left with an infinitesimally small interval. Over such an interval, the average velocity becomes the instantaneous velocity or the velocity at a specific instant. A car's speedometer, for example, shows the magnitude (but not the direction) of the instantaneous velocity of the car. (Police give tickets based on instantaneous velocity, but when calculating how long it will take to get from one place to another on a road trip, you need to use average velocity.) Instantaneous velocity is the average velocity at a specific instant in time (or over an infinitesimally small time interval). Mathematically, finding instantaneous velocity, , at a precise instant can involve taking a limit, a calculus operation beyond the scope of this text. However, under many circumstances, we can find precise values for instantaneous velocity without calculus. Speed In everyday language, most people use the terms “speed” and “velocity” interchangeably. In physics, however, they do not have the same meaning and they are distinct concepts. One major difference is that speed has no direction. Thus speed is a scalar. Just as we need to distinguish between instantaneous velocity and average velocity, we also need to distinguish between instantaneous speed and average speed. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 41 Instantaneous speed is the magnitude of instantaneous velocity. For example, suppose the airplane passenger at one instant had an instantaneous velocity of −3.0 m/s (the minus meaning toward the rear of the plane). At that same time his instantaneous speed was 3.0 m/s. Or suppose that at one time during a shopping trip your instantaneous velocity is 40 km/h due north. Your instantaneous speed at that instant would be 40 km/h—the same magnitude but without a direction. Average speed, however, is very different from average velocity. Average speed is the distance traveled divided by elapsed time. We have noted that distance traveled can be greater than displacement. So average speed can be greater than average velocity, which is displacement divided by time. For example, if you drive to a store and return home in half an hour, and your car's odometer shows the total distance traveled was 6 km, then your average speed was 12 km/h. Your average velocity, however, was zero, because your displacement for the round trip is zero. (Displacement is change in position and, thus, is zero for a round trip.) Thus average speed is not simply the magnitude of average velocity. Figure 2.10 During a 30-minute round trip to the store, the total distance traveled is 6 km. The average speed is 12 km/h. The displacement for the round trip is zero, since there was no net change in position. Thus the average velocity is zero. Another way of visualizing the motion of an object is to use a graph. A plot of position or of velocity as a function of time can be very useful. For example, for this trip to the store, the position, velocity, and speed-vs.-time graphs are displayed in Figure 2.11. (Note that these graphs depict a very simplified model of the trip. We are assuming that speed is constant during the trip, which is unrealistic given that we'll probably stop at the store. But for simplicity's sake, we will model it with no stops or changes in speed. We are also assuming that the route between the store and the house is a perfectly straight line.) 42 Chapter 2 \| Kinematics Figure 2.11 Position vs. time, velocity vs. time, and speed vs. time on a trip. Note that the velocity for the return trip is negative. Making Connections: Take-Home Investigation—Getting a Sense of Speed If you have spent much time driving, you probably have a good sense of speeds between about 10 and 70 miles per hour. But what are these in meters per second? What do we mean when we say that something is moving at 10 m/s? To get a better sense of what these values really mean, do some observations and calculations on your own: • calculate typical car speeds in meters per second • estimate jogging and walking speed by timing yourself; convert the measurements into both m/s and mi/h • determine the speed of an ant, snail, or falling leaf This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 43 Check Your Understanding A commuter train travels from Baltimore to Washington, DC, and back in 1 hour and 45 minutes. The distance between the two stations is approximately 40 miles. What is (a) the average velocity of the train, and (b) the average speed of the train in m/s? Solution (a) The average velocity of the train is zero because f = 0 ; the train ends up at the same place it starts. (b) The average speed of the train is calculated below. Note that the train travels 40 miles one way and 40 miles back, for a total distance of 80 miles. = 80 miles 105 minutes distance time × 5280 feet 1 mile × 1 meter 3.28 feet × 1 minute 60 seconds = 20 m/s (2.8) (2.9) 80 miles 105 minutes 2.4 Acceleration Figure 2.12 A plane decelerates, or slows down, as it comes in for landing in St. Maarten. Its acceleration is opposite in direction to its velocity. (credit: Steve Conry, Flickr) Learning Objectives By the end of this section, you will be able to: • Define and distinguish between instantaneous acceleration and average acceleration. • Calculate acceleration given initial time, initial velocity, final time, and final velocity. The information presented in this section supports the following AP® learning objectives and science practices: • 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, and graphical representations. (S.P. 1.5, 2.1, 2.2) • 3.A.1.3 The student is able to analyze experimental data describing the motion of an object and is able to express the results of the analysis using narrative, mathematical, and graphical representations. (S.P. 5.1) In everyday conversation, to accelerate means to speed up. The accelerator in a car can in fact cause it to speed up. The greater the acceleration, the greater the change in velocity over a given time. The formal definition of acceleration is consistent with these notions, but more inclusive. Average Acceleration Average Acceleration is the rate at which velocity changes, where is average acceleration, is velocity, and is time. (The bar over the means average acceleration.) - = 2.10) Because acceleration is velocity in m/s divided by time in s, the SI units for acceleration are m/s2 , meters per second squared or meters per second per second, which literally means by how many meters per second the velocity changes every second. 44 Chapter 2 \| Kinematics Recall that velocity is a vector—it has both magnitude and direction. This means that a change in velocity can be a change in magnitude (or speed), but it can also be a change in direction. For example, if a car turns a corner at constant speed, it is accelerating because its direction is changing. The quicker you turn, the greater the acceleration. So there is an acceleration when velocity changes either in magnitude (an increase or decrease in speed) or in direction, or both. Acceleration as a Vector Acceleration is a vector in the same direction as the change in velocity, Δ . Since velocity is a vector, it can change either in magnitude or in direction. Acceleration is therefore a change in either speed or direction, or both. Keep in mind that although acceleration is in the direction of the change in velocity, it is not always in the direction of motion. When an object's acceleration is in the same direction of its motion, the object will speed up. However, when an object's acceleration is opposite to the direction of its motion, the object will slow down. Speeding up and slowing down should not be confused with a positive and negative acceleration. The next two examples should help to make this distinction clear. Figure 2.13 A subway train in Sao Paulo, Brazil, decelerates as it comes into a station. It is accelerating in a direction opposite to its direction of motion. (credit: Yusuke Kawasaki, Flickr) Making Connections: Car Motion Figure 2.14 Above are arrows representing the motion of five cars (A–E). In all five cases, the positive direction should be considered to the right of the page. Consider the acceleration and velocity of each car in terms of its direction of travel. Figure 2.15 Car A is speeding up. Because the positive direction is considered to the right of the paper, Car A is moving with a positive velocity. Because it is speeding up while moving with a positive velocity, its acceleration is also considered positive. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 45 Figure 2.16 Car B is slowing down. Because the positive direction is considered to the right of the paper, Car B is also moving with a positive velocity. However, because it is slowing down while moving with a positive velocity, its acceleration is considered negative. (This can be viewed in a mathematical manner as well. If the car was originally moving with a velocity of +25 m/s, it is finishing with a speed less than that, like +5 m/s. Because the change in velocity is negative, the acceleration will be as well.) Figure 2.17 Car C has a constant speed. Because the positive direction is considered to the right of the paper, Car C is moving with a positive velocity. Because all arrows are of the same length, this car is not changing its speed. As a result, its change in velocity is zero, and its acceleration must be zero as well. Figure 2.18 Car D is speeding up in the opposite direction of Cars A, B, C. Because the car is mo
ving opposite to the positive direction, Car D is moving with a negative velocity. Because it is speeding up while moving in a negative direction, its acceleration is negative as well. Figure 2.19 Car E is slowing down in the same direction as Car D and opposite of Cars A, B, C. Because it is moving opposite to the positive direction, Car E is moving with a negative velocity as well. However, because it is slowing down while moving in a negative direction, its acceleration is actually positive. As in example B, this may be more easily understood in a mathematical sense. The car is originally moving with a large negative velocity (−25 m/s) but slows to a final velocity that is less negative (−5 m/s). This change in velocity, from −25 m/s to −5 m/s, is actually a positive change ( − = − 5 m/s − − 25 m/s of 20 m/s. Because the change in velocity is positive, the acceleration must also be positive. Making Connection - Illustrative Example The three graphs below are labeled A, B, and C. Each one represents the position of a moving object plotted against time. 46 Chapter 2 \| Kinematics Figure 2.20 Three position and time graphs: A, B, and C. As we did in the previous example, let's consider the acceleration and velocity of each object in terms of its direction of travel. Figure 2.21 Graph A of Position (y axis) vs. Time (x axis). Object A is continually increasing its position in the positive direction. As a result, its velocity is considered positive. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 47 Figure 2.22 Breakdown of Graph A into two separate sections. During the first portion of time (shaded grey) the position of the object does not change much, resulting in a small positive velocity. During a later portion of time (shaded green) the position of the object changes more, resulting in a larger positive velocity. Because this positive velocity is increasing over time, the acceleration of the object is considered positive. Figure 2.23 Graph B of Position (y axis) vs. Time (x axis). As in case A, Object B is continually increasing its position in the positive direction. As a result, its velocity is considered positive. Figure 2.24 Breakdown of Graph B into two separate sections. During the first portion of time (shaded grey) the position of the object changes a large amount, resulting in a large positive velocity. During a later portion of time (shaded green) the position of the object does not change as much, resulting in a smaller positive velocity. Because this positive velocity is decreasing over time, the acceleration of the object is considered negative. 48 Chapter 2 \| Kinematics Figure 2.25 Graph C of Position (y axis) vs. Time (x axis). Object C is continually decreasing its position in the positive direction. As a result, its velocity is considered negative. Figure 2.26 Breakdown of Graph C into two separate sections. During the first portion of time (shaded grey) the position of the object does not change a large amount, resulting in a small negative velocity. During a later portion of time (shaded green) the position of the object changes a much larger amount, resulting in a larger negative velocity. Because the velocity of the object is becoming more negative during the time period, the change in velocity is negative. As a result, the object experiences a negative acceleration. Example 2.1 Calculating Acceleration: A Racehorse Leaves the Gate A racehorse coming out of the gate accelerates from rest to a velocity of 15.0 m/s due west in 1.80 s. What is its average acceleration? Figure 2.27 (credit: Jon Sullivan, PD Photo.org) Strategy This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 49 First we draw a sketch and assign a coordinate system to the problem. This is a simple problem, but it always helps to visualize it. Notice that we assign east as positive and west as negative. Thus, in this case, we have negative velocity. Figure 2.28 We can solve this problem by identifying Δ and Δ from the given information and then calculating the average f − 0 f − 0 acceleration directly from the equation - = Δ Δ = . Solution 1. Identify the knowns. 0 = 0 , f = −15.0 m/s (the minus sign indicates direction toward the west), Δ = 1.80 s . 2. Find the change in velocity. Since the horse is going from zero to − 15.0 m/s , its change in velocity equals its final velocity: Δ = f = −15.0 m/s . - . 3. Plug in the known values ( Δ and Δ ) and solve for the unknown - = Δ Δ = −15.0 m/s 1.80 s = −8.33 m/s2. (2.11) Discussion The minus sign for acceleration indicates that acceleration is toward the west. An acceleration of 8.33 m/s2 due west means that the horse increases its velocity by 8.33 m/s due west each second, that is, 8.33 meters per second per second, which we write as 8.33 m/s2 . This is truly an average acceleration, because the ride is not smooth. We shall see later that an acceleration of this magnitude would require the rider to hang on with a force nearly equal to his weight. Instantaneous Acceleration Instantaneous acceleration , or the acceleration at a specific instant in time, is obtained by the same process as discussed for instantaneous velocity in Time, Velocity, and Speed—that is, by considering an infinitesimally small interval of time. How do we find instantaneous acceleration using only algebra? The answer is that we choose an average acceleration that is representative of the motion. Figure 2.29 shows graphs of instantaneous acceleration versus time for two very different motions. In Figure 2.29(a), the acceleration varies slightly and the average over the entire interval is nearly the same as the instantaneous acceleration at any time. In this case, we should treat this motion as if it had a constant acceleration equal to the average (in this case about 1.8 m/s2 ). In Figure 2.29(b), the acceleration varies drastically over time. In such situations it is best to consider smaller time intervals and choose an average acceleration for each. For example, we could consider motion over the time intervals from 0 to 1.0 s and from 1.0 to 3.0 s as separate motions with accelerations of +3.0 m/s2 and –2.0 m/s2 , respectively. 50 Chapter 2 \| Kinematics Figure 2.29 Graphs of instantaneous acceleration versus time for two different one-dimensional motions. (a) Here acceleration varies only slightly and is always in the same direction, since it is positive. The average over the interval is nearly the same as the acceleration at any given time. (b) Here the acceleration varies greatly, perhaps representing a package on a post office conveyor belt that is accelerated forward and backward as it bumps along. It is necessary to consider small time intervals (such as from 0 to 1.0 s) with constant or nearly constant acceleration in such a situation. The next several examples consider the motion of the subway train shown in Figure 2.30. In (a) the shuttle moves to the right, and in (b) it moves to the left. The examples are designed to further illustrate aspects of motion and to illustrate some of the reasoning that goes into solving problems. Figure 2.30 One-dimensional motion of a subway train considered in Example 2.2, Example 2.3, Example 2.4, Example 2.5, Example 2.6, and Example 2.7. Here we have chosen the -axis so that + means to the right and − means to the left for displacements, velocities, and accelerations. (a) The subway train moves to the right from 0 to f . Its displacement Δ is +2.0 km. (b) The train moves to the left from ′0 to ′f . Its displacement Δ′ is −1.5 km . (Note that the prime symbol (′) is used simply to distinguish between displacement in the two different situations. The distances of travel and the size of the cars are on different scales to fit everything into the diagram.) Example 2.2 Calculating Displacement: A Subway Train What are the magnitude and sign of displacements for the motions of the subway train shown in parts (a) and (b) of Figure 2.30? Strategy This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 51 A drawing with a coordinate system is already provided, so we don't need to make a sketch, but we should analyze it to make sure we understand what it is showing. Pay particular attention to the coordinate system. To find displacement, we use the equation Δ = f − 0 . This is straightforward since the initial and final positions are given. Solution 1. Identify the knowns. In the figure we see that f = 6.70 km and 0 = 4.70 km for part (a), and ′f = 3.75 km and ′0 = 5.25 km for part (b). 2. Solve for displacement in part (a). Δ = f − 0 = 6.70 km − 4.70 km= +2.00 km 3. Solve for displacement in part (b). Δ′ = ′f − ′0 = 3.75 km − 5.25 km = − 1.50 km Discussion (2.12) (2.13) The direction of the motion in (a) is to the right and therefore its displacement has a positive sign, whereas motion in (b) is to the left and thus has a minus sign. Example 2.3 Comparing Distance Traveled with Displacement: A Subway Train What are the distances traveled for the motions shown in parts (a) and (b) of the subway train in Figure 2.30? Strategy To answer this question, think about the definitions of distance and distance traveled, and how they are related to displacement. Distance between two positions is defined to be the magnitude of displacement, which was found in Example 2.2. Distance traveled is the total length of the path traveled between the two positions. (See Displacement.) In the case of the subway train shown in Figure 2.30, the distance traveled is the same as the distance between the initial and final positions of the train. Solution 1. The displacement for part (a) was +2.00 km. Therefore, the distance between the initial and final positions was 2.00 km, and the distance traveled was 2.00 km. 2. The displacement for part (b) was −1.5 km. Therefore, the distance between the initial and final positions was 1.50 km, and the distance traveled w
as 1.50 km. Discussion Distance is a scalar. It has magnitude but no sign to indicate direction. Example 2.4 Calculating Acceleration: A Subway Train Speeding Up Suppose the train in Figure 2.30(a) accelerates from rest to 30.0 km/h in the first 20.0 s of its motion. What is its average acceleration during that time interval? Strategy It is worth it at this point to make a simple sketch: Figure 2.31 This problem involves three steps. First we must determine the change in velocity, then we must determine the change in time, and finally we use these values to calculate the acceleration. Solution 52 Chapter 2 \| Kinematics 1. Identify the knowns. 0 = 0 (the trains starts at rest), f = 30.0 km/h , and Δ = 20.0 s . 2. Calculate Δ . Since the train starts from rest, its change in velocity is Δ= +30.0 km/h , where the plus sign means velocity to the right. - . 3. Plug in known values and solve for the unknown, - = Δ Δ = +30.0 km/h 20.0 s 4. Since the units are mixed (we have both hours and seconds for time), we need to convert everything into SI units of meters and seconds. (See Physical Quantities and Units for more guidance.) - = +30 km/h 20.0 s 103 m 1 km 1 h 3600 s = 0.417 m/s2 (2.14) (2.15) Discussion The plus sign means that acceleration is to the right. This is reasonable because the train starts from rest and ends up with a velocity to the right (also positive). So acceleration is in the same direction as the change in velocity, as is always the case. Example 2.5 Calculate Acceleration: A Subway Train Slowing Down Now suppose that at the end of its trip, the train in Figure 2.30(a) slows to a stop from a speed of 30.0 km/h in 8.00 s. What is its average acceleration while stopping? Strategy Figure 2.32 In this case, the train is decelerating and its acceleration is negative because it is toward the left. As in the previous example, we must find the change in velocity and the change in time and then solve for acceleration. Solution 1. Identify the knowns. 0 = 30.0 km/h , f = 0 km/h (the train is stopped, so its velocity is 0), and Δ = 8.00 s . 2. Solve for the change in velocity − 30.0 km/h = −30.0 km/h - . 3. Plug in the knowns, Δ and Δ , and solve for - = Δ Δ = −30.0 km/h 8.00 s 4. Convert the units to meters and seconds. - = Δ Δ = −30.0 km/h 8.00 s 103 m 1 km 1 h 3600 s = −1.04 m/s2. (2.16) (2.17) (2.18) Discussion The minus sign indicates that acceleration is to the left. This sign is reasonable because the train initially has a positive velocity in this problem, and a negative acceleration would oppose the motion. Again, acceleration is in the same direction as the change in velocity, which is negative here. This acceleration can be called a deceleration because it has a direction opposite to the velocity. The graphs of position, velocity, and acceleration vs. time for the trains in Example 2.4 and Example 2.5 are displayed in Figure 2.33. (We have taken the velocity to remain constant from 20 to 40 s, after which the train decelerates.) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 53 Figure 2.33 (a) Position of the train over time. Notice that the train's position changes slowly at the beginning of the journey, then more and more quickly as it picks up speed. Its position then changes more slowly as it slows down at the end of the journey. In the middle of the journey, while the velocity remains constant, the position changes at a constant rate. (b) Velocity of the train over time. The train's velocity increases as it accelerates at the beginning of the journey. It remains the same in the middle of the journey (where there is no acceleration). It decreases as the train decelerates at the end of the journey. (c) The acceleration of the train over time. The train has positive acceleration as it speeds up at the beginning of the journey. It has no acceleration as it travels at constant velocity in the middle of the journey. Its acceleration is negative as it slows down at the end of the journey. Example 2.6 Calculating Average Velocity: The Subway Train What is the average velocity of the train in part b of Example 2.2, and shown again below, if it takes 5.00 min to make its trip? 54 Chapter 2 \| Kinematics Figure 2.34 Strategy Average velocity is displacement divided by time. It will be negative here, since the train moves to the left and has a negative displacement. Solution 1. Identify the knowns. ′f = 3.75 km , ′0 = 5.25 km , Δ = 5.00 min . 2. Determine displacement, Δ′ . We found Δ′ to be − 1.5 km in Example 2.2. 3. Solve for average velocity. 4. Convert units. - = Δ′ Δ = −1.50 km 5.00 min - = Δ′ Δ = −1.50 km 5.00 min 60 min 1 h = −18.0 km/h Discussion The negative velocity indicates motion to the left. Example 2.7 Calculating Deceleration: The Subway Train Finally, suppose the train in Figure 2.34 slows to a stop from a velocity of 20.0 km/h in 10.0 s. What is its average acceleration? Strategy Once again, let's draw a sketch: Figure 2.35 As before, we must find the change in velocity and the change in time to calculate average acceleration. Solution 1. Identify the knowns. 0 = −20 km/h , f = 0 km/h , Δ = 10.0 s . 2. Calculate Δ . The change in velocity here is actually positive, since Δ = f − 0 = 0 − (−20 km/h)=+20 km/h. - . 3. Solve for 4. Convert units. - = Δ Δ = +20.0 km/h 10.0 s This content is available for free at http://cnx.org/content/col11844/1.13 (2.19) (2.20) (2.21) (2.22) Chapter 2 \| Kinematics Discussion - = +20.0 km/h 10.0 s 103 m 1 km 1 h 3600 s = +0.556 m/s2 55 (2.23) The plus sign means that acceleration is to the right. This is reasonable because the train initially has a negative velocity (to the left) in this problem and a positive acceleration opposes the motion (and so it is to the right). Again, acceleration is in the same direction as the change in velocity, which is positive here. As in Example 2.5, this acceleration can be called a deceleration since it is in the direction opposite to the velocity. Sign and Direction Perhaps the most important thing to note about these examples is the signs of the answers. In our chosen coordinate system, plus means the quantity is to the right and minus means it is to the left. This is easy to imagine for displacement and velocity. But it is a little less obvious for acceleration. Most people interpret negative acceleration as the slowing of an object. This was not the case in Example 2.7, where a positive acceleration slowed a negative velocity. The crucial distinction was that the acceleration was in the opposite direction from the velocity. In fact, a negative acceleration will increase a negative velocity. For example, the train moving to the left in Figure 2.34 is sped up by an acceleration to the left. In that case, both and are negative. The plus and minus signs give the directions of the accelerations. If acceleration has the same sign as the velocity, the object is speeding up. If acceleration has the opposite sign as the velocity, the object is slowing down. Check Your Understanding An airplane lands on a runway traveling east. Describe its acceleration. Solution If we take east to be positive, then the airplane has negative acceleration, as it is accelerating toward the west. It is also decelerating: its acceleration is opposite in direction to its velocity. PhET Explorations: Moving Man Simulation Learn about position, velocity, and acceleration graphs. Move the little man back and forth with the mouse and plot his motion. Set the position, velocity, or acceleration and let the simulation move the man for you. Figure 2.36 Moving Man (http://cnx.org/content/m54772/1.3/moving-man_en.jar) 2.5 Motion Equations for Constant Acceleration in One Dimension Figure 2.37 Kinematic equations can help us describe and predict the motion of moving objects such as these kayaks racing in Newbury, England. (credit: Barry Skeates, Flickr) By the end of this section, you will be able to: Learning Objectives 56 Chapter 2 \| Kinematics • Calculate displacement of an object that is not accelerating, given initial position and velocity. • Calculate final velocity of an accelerating object, given initial velocity, acceleration, and time. • Calculate displacement and final position of an accelerating object, given initial position, initial velocity, time, and acceleration. The information presented in this section supports the following AP® learning objectives and science practices: • 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, or graphical representations. (S.P. 1.5, 2.1, 2.2) • 3.A.1.3 The student is able to analyze experimental data describing the motion of an object and is able to express the results of the analysis using narrative, mathematical, and graphical representations. (S.P. 5.1) We might know that the greater the acceleration of, say, a car moving away from a stop sign, the greater the displacement in a given time. But we have not developed a specific equation that relates acceleration and displacement. In this section, we develop some convenient equations for kinematic relationships, starting from the definitions of displacement, velocity, and acceleration already covered. Notation: t, x, v, a First, let us make some simplifications in notation. Taking the initial time to be zero, as if time is measured with a stopwatch, is a great simplification. Since elapsed time is Δ = f − 0 , taking 0 = 0 means that Δ = f , the final time on the stopwatch. When initial time is taken to be zero, we use the subscript 0 to denote initial values of position and velocity. That is, 0 is the initial position and 0 is the initial velocity. We put no subscripts on the final values. That is, is the final time, is the final position, and is the final velocity. This gives a simpler expression for elapsed time—now, Δ = . It also simplifies the expression for displacement, which is now Δ = − 0 . Also, it simplifies the expression for change in velocity, which is now
Δ = − 0 . To summarize, using the simplified notation, with the initial time taken to be zero2.24) where the subscript 0 denotes an initial value and the absence of a subscript denotes a final value in whatever motion is under consideration. We now make the important assumption that acceleration is constant. This assumption allows us to avoid using calculus to find instantaneous acceleration. Since acceleration is constant, the average and instantaneous accelerations are equal. That is, - = = constant, (2.25) so we use the symbol for acceleration at all times. Assuming acceleration to be constant does not seriously limit the situations we can study nor degrade the accuracy of our treatment. For one thing, acceleration is constant in a great number of situations. Furthermore, in many other situations we can accurately describe motion by assuming a constant acceleration equal to the average acceleration for that motion. Finally, in motions where acceleration changes drastically, such as a car accelerating to top speed and then braking to a stop, the motion can be considered in separate parts, each of which has its own constant acceleration. Solving for Displacement ( Δ ) and Final Position ( ) from Average Velocity when Acceleration ( ) is Constant To get our first two new equations, we start with the definition of average velocity: - = Δ Δ . Substituting the simplified notation for Δ and Δ yields - = − 0 . Solving for yields where the average velocity is = 0 + , - = 0 + 2 (constant ). This content is available for free at http://cnx.org/content/col11844/1.13 (2.26) (2.27) (2.28) (2.29) Chapter 2 \| Kinematics The equation - = 0 + 2 reflects the fact that, when acceleration is constant, is just the simple average of the initial and 57 final velocities. For example, if you steadily increase your velocity (that is, with constant acceleration) from 30 to 60 km/h, then your average velocity during this steady increase is 45 km/h. Using the equation - = 0 + 2 to check this, we see that (2.30) - = 0 + 2 = 30 km/h + 60 km/h 2 = 45 km/h, which seems logical. Example 2.8 Calculating Displacement: How Far does the Jogger Run? A jogger runs down a straight stretch of road with an average velocity of 4.00 m/s for 2.00 min. What is his final position, taking his initial position to be zero? Strategy Draw a sketch. Figure 2.38 The final position is given by the equation To find , we identify the values of 0 , , and from the statement of the problem and substitute them into the equation. Solution = 0 + . (2.31) 1. Identify the knowns. - = 4.00 m/s , Δ = 2.00 min , and 0 = 0 m . 2. Enter the known values into the equation. = 0 + = 0 + (4.00 m/s)(120 s) = 480 m (2.32) Discussion Velocity and final displacement are both positive, which means they are in the same direction. gives insight into the relationship between displacement, average velocity, and time. It shows, for The equation = 0 + example, that displacement is a linear function of average velocity. (By linear function, we mean that displacement depends on rather than on raised to some other power, such as . When graphed, linear functions look like straight lines with a constant slope.) On a car trip, for example, we will get twice as far in a given time if we average 90 km/h than if we average 45 km/h. 58 Chapter 2 \| Kinematics Figure 2.39 There is a linear relationship between displacement and average velocity. For a given time , an object moving twice as fast as another object will move twice as far as the other object. Solving for Final Velocity We can derive another useful equation by manipulating the definition of acceleration. Substituting the simplified notation for Δ and Δ gives us = Δ Δ Solving for yields = − 0 (constant ). = 0 + (constant ). (2.33) (2.34) (2.35) Example 2.9 Calculating Final Velocity: An Airplane Slowing Down after Landing An airplane lands with an initial velocity of 70.0 m/s and then decelerates at 1.50 m/s2 for 40.0 s. What is its final velocity? Strategy Draw a sketch. We draw the acceleration vector in the direction opposite the velocity vector because the plane is decelerating. Figure 2.40 Solution 1. Identify the knowns. 0 = 70.0 m/s , = −1.50 m/s2 , = 40.0 s . 2. Identify the unknown. In this case, it is final velocity, f . This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 3. Determine which equation to use. We can calculate the final velocity using the equation = 0 + . 4. Plug in the known values and solve. = 0 + = 70.0 m/s + −1.50 m/s2 (40.0 s) = 10.0 m/s 59 (2.36) Discussion The final velocity is much less than the initial velocity, as desired when slowing down, but still positive. With jet engines, reverse thrust could be maintained long enough to stop the plane and start moving it backward. That would be indicated by a negative final velocity, which is not the case here. Figure 2.41 The airplane lands with an initial velocity of 70.0 m/s and slows to a final velocity of 10.0 m/s before heading for the terminal. Note that the acceleration is negative because its direction is opposite to its velocity, which is positive. In addition to being useful in problem solving, the equation = 0 + gives us insight into the relationships among velocity, acceleration, and time. From it we can see, for example, that • • • final velocity depends on how large the acceleration is and how long it lasts if the acceleration is zero, then the final velocity equals the initial velocity ( = 0) , as expected (i.e., velocity is constant) if is negative, then the final velocity is less than the initial velocity (All of these observations fit our intuition, and it is always useful to examine basic equations in light of our intuition and experiences to check that they do indeed describe nature accurately.) Making Connections: Real-World Connection Figure 2.42 The Space Shuttle Endeavor blasts off from the Kennedy Space Center in February 2010. (credit: Matthew Simantov, Flickr) An intercontinental ballistic missile (ICBM) has a larger average acceleration than the Space Shuttle and achieves a greater velocity in the first minute or two of flight (actual ICBM burn times are classified—short-burn-time missiles are more difficult for an enemy to destroy). But the Space Shuttle obtains a greater final velocity, so that it can orbit the earth rather than come directly back down as an ICBM does. The Space Shuttle does this by accelerating for a longer time. Solving for Final Position When Velocity is Not Constant ( ≠ 0 ) We can combine the equations above to find a third equation that allows us to calculate the final position of an object experiencing constant acceleration. We start with Adding 0 to each side of this equation and dividing by 2 gives = . (2.37) (2.38) 60 Chapter 2 \| Kinematics Since 0 + 2 - for constant acceleration, then = Now we substitute this expression for - = 0 + 1 2. - into the equation for displacementconstant ). , yielding (2.39) (2.40) Example 2.10 Calculating Displacement of an Accelerating Object: Dragsters Dragsters can achieve average accelerations of 26.0 m/s2 . Suppose such a dragster accelerates from rest at this rate for 5.56 s. How far does it travel in this time? Figure 2.43 U.S. Army Top Fuel pilot Tony “The Sarge” Schumacher begins a race with a controlled burnout. (credit: Lt. Col. William Thurmond. Photo Courtesy of U.S. Army.) Strategy Draw a sketch. Figure 2.44 We are asked to find displacement, which is if we take 0 to be zero. (Think about it like the starting line of a race. It can be anywhere, but we call it 0 and measure all other positions relative to it.) We can use the equation = 0 + 0 + 1 2 2 once we identify 0 , , and from the statement of the problem. Solution 1. Identify the knowns. Starting from rest means that 0 = 0 , is given as 26.0 m/s2 and is given as 5.56 s. 2. Plug the known values into the equation to solve for the unknown : 2 2. = 0 + 0 + 1 Since the initial position and velocity are both zero, this simplifies to Substituting the identified values of and gives = 1 2 2. (2.41) (2.42) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics yielding Discussion = 1 2 26.0 m/s2 (5.56 s)2 = 402 m. 61 (2.43) (2.44) If we convert 402 m to miles, we find that the distance covered is very close to one quarter of a mile, the standard distance for drag racing. So the answer is reasonable. This is an impressive displacement in only 5.56 s, but top-notch dragsters can do a quarter mile in even less time than this. What else can we learn by examining the equation = 0 + 0 + 1 2 2 ? We see that: • displacement depends on the square of the elapsed time when acceleration is not zero. In Example 2.10, the dragster • covers only one fourth of the total distance in the first half of the elapsed time if acceleration is zero, then the initial velocity equals average velocity ( 0 = = 0 + 0 - ) and = 0 + 0 + 1 2 2 becomes Solving for Final Velocity when Velocity Is Not Constant ( ≠ 0 ) A fourth useful equation can be obtained from another algebraic manipulation of previous equations. If we solve = 0 + for , we get Substituting this and - = 0 + 2 into = 0 + , we get = − 0 . 2 = 0 2 + 2( − 0) (constant). Example 2.11 Calculating Final Velocity: Dragsters Calculate the final velocity of the dragster in Example 2.10 without using information about time. Strategy Draw a sketch. Figure 2.45 The equation 2 = 0 displacement, and no time information is required. 2 + 2( − 0) is ideally suited to this task because it relates velocities, acceleration, and Solution 1. Identify the known values. We know that 0 = 0 , since the dragster starts from rest. Then we note that − 0 = 402 m (this was the answer in Example 2.10). Finally, the average acceleration was given to be = 26.0 m/s2 . 2. Plug the knowns into the equation 2 = 0 2 + 2( − 0) and solve for . 2 = 0 + 2 26.0 m/s2 (402 m). (2.45) (2.46) (2.47) 62 Thus To get , we take the square root: Discussion 2
= 2.09×104 m2 /s2. = 2.09×104 m2 /s2 = 145 m/s. Chapter 2 \| Kinematics (2.48) (2.49) 145 m/s is about 522 km/h or about 324 mi/h, but even this breakneck speed is short of the record for the quarter mile. Also, note that a square root has two values; we took the positive value to indicate a velocity in the same direction as the acceleration. An examination of the equation 2 = 0 physical quantities: 2 + 2( − 0) can produce further insights into the general relationships among • The final velocity depends on how large the acceleration is and the distance over which it acts • For a fixed deceleration, a car that is going twice as fast doesn't simply stop in twice the distance—it takes much further to stop. (This is why we have reduced speed zones near schools.) Putting Equations Together In the following examples, we further explore one-dimensional motion, but in situations requiring slightly more algebraic manipulation. The examples also give insight into problem-solving techniques. The box below provides easy reference to the equations needed. Summary of Kinematic Equations (constant ) = ( − 0) (2.50) (2.51) (2.52) (2.53) (2.54) Example 2.12 Calculating Displacement: How Far Does a Car Go When Coming to a Halt? On dry concrete, a car can decelerate at a rate of 7.00 m/s2 , whereas on wet concrete it can decelerate at only 5.00 m/s2 . Find the distances necessary to stop a car moving at 30.0 m/s (about 110 km/h) (a) on dry concrete and (b) on wet concrete. (c) Repeat both calculations, finding the displacement from the point where the driver sees a traffic light turn red, taking into account his reaction time of 0.500 s to get his foot on the brake. Strategy Draw a sketch. Figure 2.46 This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 63 In order to determine which equations are best to use, we need to list all of the known values and identify exactly what we need to solve for. We shall do this explicitly in the next several examples, using tables to set them off. Solution for (a) 1. Identify the knowns and what we want to solve for. We know that 0 = 30.0 m/s ; = 0 ; = −7.00 m/s2 ( is negative because it is in a direction opposite to velocity). We take 0 to be 0. We are looking for displacement Δ , or − 0 . 2. Identify the equation that will help up solve the problem. The best equation to use is 2 = 0 2 + 2( − 0). (2.55) This equation is best because it includes only one unknown, . We know the values of all the other variables in this equation. (There are other equations that would allow us to solve for , but they require us to know the stopping time, , which we do not know. We could use them but it would entail additional calculations.) 3. Rearrange the equation to solve for . 4. Enter known values. Thus = 02 − (30.0 m/s)2 −7.00 m/s2 2 = 64.3 m on dry concrete. (2.56) (2.57) (2.58) Solution for (b) This part can be solved in exactly the same manner as Part A. The only difference is that the deceleration is – 5.00 m/s2 . The result is wet = 90.0 m on wet concrete. (2.59) Solution for (c) Once the driver reacts, the stopping distance is the same as it is in Parts A and B for dry and wet concrete. So to answer this question, we need to calculate how far the car travels during the reaction time, and then add that to the stopping time. It is reasonable to assume that the velocity remains constant during the driver's reaction time. 1. Identify the knowns and what we want to solve for. We know that We take 0 − reaction to be 0. We are looking for reaction . - = 30.0 m/s ; reaction = 0.500 s ; reaction = 0 . 2. Identify the best equation to use. = 0 + works well because the only unknown value is , which is what we want to solve for. 3. Plug in the knowns to solve the equation. = 0 + (30.0 m/s)(0.500 s) = 15.0 m. (2.60) This means the car travels 15.0 m while the driver reacts, making the total displacements in the two cases of dry and wet concrete 15.0 m greater than if he reacted instantly. 4. Add the displacement during the reaction time to the displacement when braking. braking + reaction = total (2.61) a. 64.3 m + 15.0 m = 79.3 m when dry b. 90.0 m + 15.0 m = 105 m when wet 64 Chapter 2 \| Kinematics Figure 2.47 The distance necessary to stop a car varies greatly, depending on road conditions and driver reaction time. Shown here are the braking distances for dry and wet pavement, as calculated in this example, for a car initially traveling at 30.0 m/s. Also shown are the total distances traveled from the point where the driver first sees a light turn red, assuming a 0.500 s reaction time. Discussion The displacements found in this example seem reasonable for stopping a fast-moving car. It should take longer to stop a car on wet rather than dry pavement. It is interesting that reaction time adds significantly to the displacements. But more important is the general approach to solving problems. We identify the knowns and the quantities to be determined and then find an appropriate equation. There is often more than one way to solve a problem. The various parts of this example can in fact be solved by other methods, but the solutions presented above are the shortest. Example 2.13 Calculating Time: A Car Merges into Traffic Suppose a car merges into freeway traffic on a 200-m-long ramp. If its initial velocity is 10.0 m/s and it accelerates at 2.00 m/s2 , how long does it take to travel the 200 m up the ramp? (Such information might be useful to a traffic engineer.) Strategy Draw a sketch. Figure 2.48 We are asked to solve for the time . As before, we identify the known quantities in order to choose a convenient physical relationship (that is, an equation with one unknown, ). Solution 1. Identify the knowns and what we want to solve for. We know that 0 = 10 m/s ; = 2.00 m/s2 ; and = 200 m . 2. We need to solve for . Choose the best equation. = 0 + 0 + 1 equation is the variable for which we need to solve. 2 2 works best because the only unknown in the 3. We will need to rearrange the equation to solve for . In this case, it will be easier to plug in the knowns first. 200 m = 0 m + (10.0 m/s) + 1 2 This content is available for free at http://cnx.org/content/col11844/1.13 2.00 m/s2 2 (2.62) Chapter 2 \| Kinematics 65 4. Simplify the equation. The units of meters (m) cancel because they are in each term. We can get the units of seconds (s) to cancel by taking = s , where is the magnitude of time and s is the unit. Doing so leaves 5. Use the quadratic formula to solve for . (a) Rearrange the equation to get 0 on one side of the equation. 200 = 10 + 2. This is a quadratic equation of the form 2 + 10 − 200 = 0 2 + + = 0, where the constants are = 1.00, = 10.0, and = −200 . (b) Its solutions are given by the quadratic formula: This yields two solutions for , which are = − ± 2 − 4 2 . = 10.0 and−20.0. In this case, then, the time is = in seconds, or = 10.0 s and − 20.0 s. (2.63) (2.64) (2.65) (2.66) (2.67) (2.68) A negative value for time is unreasonable, since it would mean that the event happened 20 s before the motion began. We can discard that solution. Thus, = 10.0 s. (2.69) Discussion Whenever an equation contains an unknown squared, there will be two solutions. In some problems both solutions are meaningful, but in others, such as the above, only one solution is reasonable. The 10.0 s answer seems reasonable for a typical freeway on-ramp. With the basics of kinematics established, we can go on to many other interesting examples and applications. In the process of developing kinematics, we have also glimpsed a general approach to problem solving that produces both correct answers and insights into physical relationships. Problem-Solving Basics discusses problem-solving basics and outlines an approach that will help you succeed in this invaluable task. Making Connections: Take-Home Experiment—Breaking News We have been using SI units of meters per second squared to describe some examples of acceleration or deceleration of cars, runners, and trains. To achieve a better feel for these numbers, one can measure the braking deceleration of a car - = Δ / Δ . While traveling in a car, slowly apply the doing a slow (and safe) stop. Recall that, for average acceleration, brakes as you come up to a stop sign. Have a passenger note the initial speed in miles per hour and the time taken (in seconds) to stop. From this, calculate the deceleration in miles per hour per second. Convert this to meters per second squared and compare with other decelerations mentioned in this chapter. Calculate the distance traveled in braking. Check Your Understanding A manned rocket accelerates at a rate of 20 m/s2 during launch. How long does it take the rocket reach a velocity of 400 m/s? Solution To answer this, choose an equation that allows you to solve for time , given only , 0 , and . Rearrange to solve for . = 0 + = − = 400 m/s − 0 m/s 20 m/s2 = 20 s (2.70) (2.71) 66 Chapter 2 \| Kinematics 2.6 Problem-Solving Basics for One Dimensional Kinematics Figure 2.49 Problem-solving skills are essential to your success in Physics. (credit: scui3asteveo, Flickr) By the end of this section, you will be able to: Learning Objectives • Apply problem-solving steps and strategies to solve problems of one-dimensional kinematics. • Apply strategies to determine whether or not the result of a problem is reasonable, and if not, determine the cause. Problem-solving skills are obviously essential to success in a quantitative course in physics. More importantly, the ability to apply broad physical principles, usually represented by equations, to specific situations is a very powerful form of knowledge. It is much more powerful than memorizing a list of facts. Analytical skills and problem-solving abilities can be applied to new situations, whereas a list of facts cannot be made long enough to contain every possible circumstance. Such analytical skills are useful both for solving problems in this text and for applying physi
cs in everyday and professional life. Problem-Solving Steps While there is no simple step-by-step method that works for every problem, the following general procedures facilitate problem solving and make it more meaningful. A certain amount of creativity and insight is required as well. Step 1 Examine the situation to determine which physical principles are involved. It often helps to draw a simple sketch at the outset. You will also need to decide which direction is positive and note that on your sketch. Once you have identified the physical principles, it is much easier to find and apply the equations representing those principles. Although finding the correct equation is essential, keep in mind that equations represent physical principles, laws of nature, and relationships among physical quantities. Without a conceptual understanding of a problem, a numerical solution is meaningless. Step 2 Make a list of what is given or can be inferred from the problem as stated (identify the knowns). Many problems are stated very succinctly and require some inspection to determine what is known. A sketch can also be very useful at this point. Formally identifying the knowns is of particular importance in applying physics to real-world situations. Remember, “stopped” means velocity is zero, and we often can take initial time and position as zero. Step 3 Identify exactly what needs to be determined in the problem (identify the unknowns). In complex problems, especially, it is not always obvious what needs to be found or in what sequence. Making a list can help. Step 4 Find an equation or set of equations that can help you solve the problem. Your list of knowns and unknowns can help here. It is easiest if you can find equations that contain only one unknown—that is, all of the other variables are known, so you can easily solve for the unknown. If the equation contains more than one unknown, then an additional equation is needed to solve the problem. In some problems, several unknowns must be determined to get at the one needed most. In such problems it is especially important to keep physical principles in mind to avoid going astray in a sea of equations. You may have to use two (or more) different equations to get the final answer. Step 5 Substitute the knowns along with their units into the appropriate equation, and obtain numerical solutions complete with units. This step produces the numerical answer; it also provides a check on units that can help you find errors. If the units of the answer are incorrect, then an error has been made. However, be warned that correct units do not guarantee that the numerical part of the answer is also correct. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics Step 6 67 Check the answer to see if it is reasonable: Does it make sense? This final step is extremely important—the goal of physics is to accurately describe nature. To see if the answer is reasonable, check both its magnitude and its sign, in addition to its units. Your judgment will improve as you solve more and more physics problems, and it will become possible for you to make finer and finer judgments regarding whether nature is adequately described by the answer to a problem. This step brings the problem back to its conceptual meaning. If you can judge whether the answer is reasonable, you have a deeper understanding of physics than just being able to mechanically solve a problem. When solving problems, we often perform these steps in different order, and we also tend to do several steps simultaneously. There is no rigid procedure that will work every time. Creativity and insight grow with experience, and the basics of problem solving become almost automatic. One way to get practice is to work out the text's examples for yourself as you read. Another is to work as many end-of-section problems as possible, starting with the easiest to build confidence and progressing to the more difficult. Once you become involved in physics, you will see it all around you, and you can begin to apply it to situations you encounter outside the classroom, just as is done in many of the applications in this text. Unreasonable Results Physics must describe nature accurately. Some problems have results that are unreasonable because one premise is unreasonable or because certain premises are inconsistent with one another. The physical principle applied correctly then produces an unreasonable result. For example, if a person starting a foot race accelerates at 0.40 m/s2 for 100 s, his final speed will be 40 m/s (about 150 km/h)—clearly unreasonable because the time of 100 s is an unreasonable premise. The physics is correct in a sense, but there is more to describing nature than just manipulating equations correctly. Checking the result of a problem to see if it is reasonable does more than help uncover errors in problem solving—it also builds intuition in judging whether nature is being accurately described. Use the following strategies to determine whether an answer is reasonable and, if it is not, to determine what is the cause. Step 1 Solve the problem using strategies as outlined and in the format followed in the worked examples in the text. In the example given in the preceding paragraph, you would identify the givens as the acceleration and time and use the equation below to find the unknown final velocity. That is, = 0 + = 0 + 0.40 m/s2 (100 s) = 40 m/s. (2.72) Step 2 Check to see if the answer is reasonable. Is it too large or too small, or does it have the wrong sign, improper units, …? In this case, you may need to convert meters per second into a more familiar unit, such as miles per hour. 40 m s 3.28 ft m 1 mi 5280 ft 60 s min 60 min 1 h = 89 mph (2.73) This velocity is about four times greater than a person can run—so it is too large. Step 3 If the answer is unreasonable, look for what specifically could cause the identified difficulty. In the example of the runner, there are only two assumptions that are suspect. The acceleration could be too great or the time too long. First look at the acceleration and think about what the number means. If someone accelerates at 0.40 m/s2 , their velocity is increasing by 0.4 m/s each second. Does this seem reasonable? If so, the time must be too long. It is not possible for someone to accelerate at a constant rate of 0.40 m/s2 for 100 s (almost two minutes). 2.7 Falling Objects Learning Objectives By the end of this section, you will be able to: • Describe the effects of gravity on objects in motion. • Describe the motion of objects that are in free fall. • Calculate the position and velocity of objects in free fall. The information presented in this section supports the following AP® learning objectives and science practices: • 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, or graphical representations. (S.P. 1.5, 2.1, 2.2) • 3.A.1.2 The student is able to design an experimental investigation of the motion of an object. (S.P. 4.2) • 3.A.1.3 The student is able to analyze experimental data describing the motion of an object and is able to express the results of the analysis using narrative, mathematical, and graphical representations. (S.P. 5.1) 68 Chapter 2 \| Kinematics Falling objects form an interesting class of motion problems. For example, we can estimate the depth of a vertical mine shaft by dropping a rock into it and listening for the rock to hit the bottom. By applying the kinematics developed so far to falling objects, we can examine some interesting situations and learn much about gravity in the process. Gravity The most remarkable and unexpected fact about falling objects is that, if air resistance and friction are negligible, then in a given location all objects fall toward the center of Earth with the same constant acceleration, independent of their mass. This experimentally determined fact is unexpected, because we are so accustomed to the effects of air resistance and friction that we expect light objects to fall slower than heavy ones. Figure 2.50 A hammer and a feather will fall with the same constant acceleration if air resistance is considered negligible. This is a general characteristic of gravity not unique to Earth, as astronaut David R. Scott demonstrated on the Moon in 1971, where the acceleration due to gravity is only 1.67 m/s2 . In the real world, air resistance can cause a lighter object to fall slower than a heavier object of the same size. A tennis ball will reach the ground after a hard baseball dropped at the same time. (It might be difficult to observe the difference if the height is not large.) Air resistance opposes the motion of an object through the air, while friction between objects—such as between clothes and a laundry chute or between a stone and a pool into which it is dropped—also opposes motion between them. For the ideal situations of these first few chapters, an object falling without air resistance or friction is defined to be in free-fall. The force of gravity causes objects to fall toward the center of Earth. The acceleration of free-falling objects is therefore called the acceleration due to gravity. The acceleration due to gravity is constant, which means we can apply the kinematics equations to any falling object where air resistance and friction are negligible. This opens a broad class of interesting situations to us. The acceleration due to gravity is so important that its magnitude is given its own symbol, . It is constant at any given location on Earth and has the average value = 9.80 m/s2. (2.74) Although varies from 9.78 m/s2 to 9.83 m/s2 , depending on latitude, altitude, underlying geological formations, and local topography, the average value of 9.80 m/s2 will be used in this text unless otherwise specified. The direction of the acceleration due to gravity is downward (towards the center of Earth). In fact, its direction defines what we call ve
rtical. Note that whether the acceleration in the kinematic equations has the value + or − depends on how we define our coordinate system. If we define the upward direction as positive, then = − = −9.80 m/s2 , and if we define the downward direction as positive, then = = 9.80 m/s2 . One-Dimensional Motion Involving Gravity The best way to see the basic features of motion involving gravity is to start with the simplest situations and then progress toward more complex ones. So we start by considering straight up and down motion with no air resistance or friction. These assumptions mean that the velocity (if there is any) is vertical. If the object is dropped, we know the initial velocity is zero. Once the object has left contact with whatever held or threw it, the object is in free-fall. Under these circumstances, the motion is onedimensional and has constant acceleration of magnitude . We will also represent vertical displacement with the symbol and use for horizontal displacement. Kinematic Equations for Objects in Free-Fall where Acceleration = -( − 0) (2.75) (2.76) (2.77) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 69 Example 2.14 Calculating Position and Velocity of a Falling Object: A Rock Thrown Upward A person standing on the edge of a high cliff throws a rock straight up with an initial velocity of 13.0 m/s. The rock misses the edge of the cliff as it falls back to Earth. Calculate the position and velocity of the rock 1.00 s, 2.00 s, and 3.00 s after it is thrown, neglecting the effects of air resistance. Strategy Draw a sketch. Figure 2.51 We are asked to determine the position at various times. It is reasonable to take the initial position 0 to be zero. This problem involves one-dimensional motion in the vertical direction. We use plus and minus signs to indicate direction, with up being positive and down negative. Since up is positive, and the rock is thrown upward, the initial velocity must be positive too. The acceleration due to gravity is downward, so is negative. It is crucial that the initial velocity and the acceleration due to gravity have opposite signs. Opposite signs indicate that the acceleration due to gravity opposes the initial motion and will slow and eventually reverse it. Since we are asked for values of position and velocity at three times, we will refer to these as 1 and 1 ; and 2 ; and 3 and 3 . Solution for Position 1 1. Identify the knowns. We know that 0 = 0 ; 0 = 13.0 m/s ; = − = −9.80 m/s2 ; and = 1.00 s . 2. Identify the best equation to use. We will use = 0 + 0 + 1 here), which is the value we want to find. 3. Plug in the known values and solve for 1 . 2 2 because it includes only one unknown, (or 1 , = 0 + (13.0 m/s)(1.00 s) + 1 2 −9.80 m/s2 (1.00 s)2 = 8.10 m (2.78) Discussion The rock is 8.10 m above its starting point at = 1.00 s, since 1 > 0 . It could be moving up or down; the only way to tell is to calculate 1 and find out if it is positive or negative. Solution for Velocity 1 1. Identify the knowns. We know that 0 = 0 ; 0 = 13.0 m/s ; = − = −9.80 m/s2 ; and = 1.00 s . We also know from the solution above that 1 = 8.10 m . 2. Identify the best equation to use. The most straightforward is = 0 − (from = 0 + , where = gravitational acceleration = − ). 3. Plug in the knowns and solve. 1 = 0 − = 13.0 m/s − 9.80 m/s2 (1.00 s) = 3.20 m/s (2.79) Discussion The positive value for 1 means that the rock is still heading upward at = 1.00 s . However, it has slowed from its original 13.0 m/s, as expected. Solution for Remaining Times The procedures for calculating the position and velocity at = 2.00 s and 3.00 s are the same as those above. The results are summarized in Table 2.1 and illustrated in Figure 2.52. 70 Chapter 2 \| Kinematics Table 2.1 Results Time, t Position, y Velocity, v Acceleration, a 1.00 s 8.10 m 3.20 m/s 2.00 s 6.40 m −6.60 m/s 3.00 s −5.10 m −16.4 m/s −9.80 m/s2 −9.80 m/s2 −9.80 m/s2 Graphing the data helps us understand it more clearly. Figure 2.52 Vertical position, vertical velocity, and vertical acceleration vs. time for a rock thrown vertically up at the edge of a cliff. Notice that velocity changes linearly with time and that acceleration is constant. Misconception Alert! Notice that the position vs. time graph shows vertical position only. It is easy to get the impression that the graph shows some horizontal motion—the shape of the graph looks like the path of a projectile. But this is not the case; the horizontal axis is time, not space. The actual path of the rock in space is straight up, and straight down. Discussion The interpretation of these results is important. At 1.00 s the rock is above its starting point and heading upward, since 1 and 1 are both positive. At 2.00 s, the rock is still above its starting point, but the negative velocity means it is moving downward. At 3.00 s, both 3 and 3 are negative, meaning the rock is below its starting point and continuing to move This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 71 downward. Notice that when the rock is at its highest point (at 1.5 s), its velocity is zero, but its acceleration is still −9.80 m/s2 . Its acceleration is −9.80 m/s2 for the whole trip—while it is moving up and while it is moving down. Note that the values for are the positions (or displacements) of the rock, not the total distances traveled. Finally, note that freefall applies to upward motion as well as downward. Both have the same acceleration—the acceleration due to gravity, which remains constant the entire time. Astronauts training in the famous Vomit Comet, for example, experience free-fall while arcing up as well as down, as we will discuss in more detail later. Making Connections: Take-Home Experiment—Reaction Time A simple experiment can be done to determine your reaction time. Have a friend hold a ruler between your thumb and index finger, separated by about 1 cm. Note the mark on the ruler that is right between your fingers. Have your friend drop the ruler unexpectedly, and try to catch it between your two fingers. Note the new reading on the ruler. Assuming acceleration is that due to gravity, calculate your reaction time. How far would you travel in a car (moving at 30 m/s) if the time it took your foot to go from the gas pedal to the brake was twice this reaction time? Example 2.15 Calculating Velocity of a Falling Object: A Rock Thrown Down What happens if the person on the cliff throws the rock straight down, instead of straight up? To explore this question, calculate the velocity of the rock when it is 5.10 m below the starting point, and has been thrown downward with an initial speed of 13.0 m/s. Strategy Draw a sketch. Figure 2.53 Since up is positive, the final position of the rock will be negative because it finishes below the starting point at 0 = 0 . Similarly, the initial velocity is downward and therefore negative, as is the acceleration due to gravity. We expect the final velocity to be negative since the rock will continue to move downward. Solution 1. Identify the knowns. 0 = 0 ; 1 = − 5.10 m ; 0 = −13.0 m/s ; = − = −9.80 m/s2 . 2. Choose the kinematic equation that makes it easiest to solve the problem. The equation 2 = 0 well because the only unknown in it is . (We will plug 1 in for .) 2 + 2( − 0) works 3. Enter the known values 2 = (−13.0 m/s)2 + 2 −9.80 m/s2 (−5.10 m − 0 m) = 268.96 m2 /s2, where we have retained extra significant figures because this is an intermediate result. Taking the square root, and noting that a square root can be positive or negative, gives = ±16.4 m/s. The negative root is chosen to indicate that the rock is still heading down. Thus, = −16.4 m/s. Discussion (2.80) (2.81) (2.82) Note that this is exactly the same velocity the rock had at this position when it was thrown straight upward with the same initial speed. (See Example 2.14 and Figure 2.54(a).) This is not a coincidental result. Because we only consider the acceleration due to gravity in this problem, the speed of a falling object depends only on its initial speed and its vertical position relative to the starting point. For example, if the velocity of the rock is calculated at a height of 8.10 m above the starting point (using the method from Example 2.14) when the initial velocity is 13.0 m/s straight up, a result of ±3.20 m/s 72 Chapter 2 \| Kinematics is obtained. Here both signs are meaningful; the positive value occurs when the rock is at 8.10 m and heading up, and the negative value occurs when the rock is at 8.10 m and heading back down. It has the same speed but the opposite direction. Figure 2.54 (a) A person throws a rock straight up, as explored in Example 2.14. The arrows are velocity vectors at 0, 1.00, 2.00, and 3.00 s. (b) A person throws a rock straight down from a cliff with the same initial speed as before, as in Example 2.15. Note that at the same distance below the point of release, the rock has the same velocity in both cases. Another way to look at it is this: In Example 2.14, the rock is thrown up with an initial velocity of 13.0 m/s . It rises and then falls back down. When its position is = 0 on its way back down, its velocity is −13.0 m/s . That is, it has the same speed on its way down as on its way up. We would then expect its velocity at a position of = −5.10 m to be the same whether we have thrown it upwards at +13.0 m/s or thrown it downwards at −13.0 m/s . The velocity of the rock on its way down from = 0 is the same whether we have thrown it up or down to start with, as long as the speed with which it was initially thrown is the same. Example 2.16 Find g from Data on a Falling Object The acceleration due to gravity on Earth differs slightly from place to place, depending on topography (e.g., whether you are on a hill or in a valley) and subsurface geology (whether there is dense rock like iron ore as opposed to light rock like salt beneath you.) The precise acceleration du
e to gravity can be calculated from data taken in an introductory physics This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 73 laboratory course. An object, usually a metal ball for which air resistance is negligible, is dropped and the time it takes to fall a known distance is measured. See, for example, Figure 2.55. Very precise results can be produced with this method if sufficient care is taken in measuring the distance fallen and the elapsed time. Figure 2.55 Positions and velocities of a metal ball released from rest when air resistance is negligible. Velocity is seen to increase linearly with time while displacement increases with time squared. Acceleration is a constant and is equal to gravitational acceleration. Suppose the ball falls 1.0000 m in 0.45173 s. Assuming the ball is not affected by air resistance, what is the precise acceleration due to gravity at this location? Strategy Draw a sketch. 74 Chapter 2 \| Kinematics Figure 2.56 We need to solve for acceleration . Note that in this case, displacement is downward and therefore negative, as is acceleration. Solution 1. Identify the knowns. 0 = 0 ; = –1.0000 m ; = 0.45173 ; 0 = 0 . 2. Choose the equation that allows you to solve for using the known values. Substitute 0 for 0 and rearrange the equation to solve for . Substituting 0 for 0 yields Solving for gives 4. Substitute known values yields = 0 + 1 2 2. = 2( − 0) 2 . = 2( − 1.0000 m – 0) (0.45173 s)2 = −9.8010 m/s2 so, because = − with the directions we have chosen, = 9.8010 m/s2. Discussion (2.83) (2.84) (2.85) (2.86) (2.87) The negative value for indicates that the gravitational acceleration is downward, as expected. We expect the value to be somewhere around the average value of 9.80 m/s2 , so 9.8010 m/s2 makes sense. Since the data going into the calculation are relatively precise, this value for is more precise than the average value of 9.80 m/s2 ; it represents the local value for the acceleration due to gravity. Applying the Science Practices: Finding Acceleration Due to Gravity While it is well established that the acceleration due to gravity is quite nearly 9.8 m/s2 at all locations on Earth, you can verify this for yourself with some basic materials. Your task is to find the acceleration due to gravity at your location. Achieving an acceleration of precisely 9.8 m/s2 will be difficult. However, with good preparation and attention to detail, you should be able to get close. Before you begin working, consider the following questions. What measurements will you need to take in order to find the acceleration due to gravity? What relationships and equations found in this chapter may be useful in calculating the acceleration? What variables will you need to hold constant? What materials will you use to record your measurements? Upon completing these four questions, record your procedure. Once recorded, you may carry out the experiment. If you find that your experiment cannot be carried out, you may revise your procedure. Once you have found your experimental acceleration, compare it to the assumed value of 9.8 m/s2. If error exists, what were the likely sources of this error? How could you change your procedure in order to improve the accuracy of your findings? This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 75 Check Your Understanding A chunk of ice breaks off a glacier and falls 30.0 meters before it hits the water. Assuming it falls freely (there is no air resistance), how long does it take to hit the water? Solution We know that initial position 0 = 0 , final position = −30.0 m , and = − = −9.80 m/s2 . We can then use the equation = 0 + 0 + 1 (2.88) 2 2 to solve for . Inserting = − , we obtain = ( − 30.0 m) −9.80 m/s2 = ± 6.12 s2 = 2.47 s ≈ 2.5 s where we take the positive value as the physically relevant answer. Thus, it takes about 2.5 seconds for the piece of ice to hit the water. PhET Explorations: Equation Grapher Learn about graphing polynomials. The shape of the curve changes as the constants are adjusted. View the curves for the individual terms (e.g. = ) to see how they add to generate the polynomial curve. Figure 2.57 Equation Grapher (http://cnx.org/content/m54775/1.5/equation-grapher_en.jar) 2.8 Graphical Analysis of One Dimensional Motion By the end of this section, you will be able to: Learning Objectives • Describe a straight-line graph in terms of its slope and y-intercept. • Determine average velocity or instantaneous velocity from a graph of position vs. time. • Determine average or instantaneous acceleration from a graph of velocity vs. time. • Derive a graph of velocity vs. time from a graph of position vs. time. • Derive a graph of acceleration vs. time from a graph of velocity vs. time. A graph, like a picture, is worth a thousand words. Graphs not only contain numerical information; they also reveal relationships between physical quantities. This section uses graphs of displacement, velocity, and acceleration versus time to illustrate onedimensional kinematics. Slopes and General Relationships First note that graphs in this text have perpendicular axes, one horizontal and the other vertical. When two physical quantities are plotted against one another in such a graph, the horizontal axis is usually considered to be an independent variable and the vertical axis a dependent variable. If we call the horizontal axis the -axis and the vertical axis the -axis, as in Figure 2.58, a straight-line graph has the general form = + . (2.89) Here is the slope, defined to be the rise divided by the run (as seen in the figure) of the straight line. The letter is used for the y-intercept, which is the point at which the line crosses the vertical axis. 76 Chapter 2 \| Kinematics Figure 2.58 A straight-line graph. The equation for a straight line is = + . Graph of Displacement vs. Time (a = 0, so v is constant) Time is usually an independent variable that other quantities, such as displacement, depend upon. A graph of displacement versus time would, thus, have on the vertical axis and on the horizontal axis. Figure 2.59 is just such a straight-line graph. It shows a graph of displacement versus time for a jet-powered car on a very flat dry lake bed in Nevada. Figure 2.59 Graph of displacement versus time for a jet-powered car on the Bonneville Salt Flats. Using the relationship between dependent and independent variables, we see that the slope in the graph above is average velocity - and the intercept is displacement at time zero—that is, 0 . Substituting these symbols into = + gives + 0 = (2.90) or Thus a graph of displacement versus time gives a general relationship among displacement, velocity, and time, as well as giving detailed numerical information about a specific situation. = 0 + . (2.91) The Slope of x vs. t The slope of the graph of displacement vs. time is velocity . Notice that this equation is the same as that derived algebraically from other motion equations in Motion Equations for Constant Acceleration in One Dimension. slope = Δ Δ = (2.92) From the figure we can see that the car has a displacement of 400 m at time 0.650 m at = 1.0 s, and so on. Its displacement at times other than those listed in the table can be read from the graph; furthermore, information about its velocity and acceleration can also be obtained from the graph. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 77 Example 2.17 Determining Average Velocity from a Graph of Displacement versus Time: Jet Car Find the average velocity of the car whose position is graphed in Figure 2.59. Strategy The slope of a graph of vs. is average velocity, since slope equals rise over run. In this case, rise = change in displacement and run = change in time, so that slope = Δ Δ - . = (2.93) Since the slope is constant here, any two points on the graph can be used to find the slope. (Generally speaking, it is most accurate to use two widely separated points on the straight line. This is because any error in reading data from the graph is proportionally smaller if the interval is larger.) Solution 1. Choose two points on the line. In this case, we choose the points labeled on the graph: (6.4 s, 2000 m) and (0.50 s, 525 m). (Note, however, that you could choose any two points.) 2. Substitute the and values of the chosen points into the equation. Remember in calculating change (Δ) we always use final value minus initial value. yielding Discussion - = Δ Δ = 2000 m − 525 m 6.4 s − 0.50 s , - = 250 m/s. (2.94) (2.95) This is an impressively large land speed (900 km/h, or about 560 mi/h): much greater than the typical highway speed limit of 60 mi/h (27 m/s or 96 km/h), but considerably shy of the record of 343 m/s (1234 km/h or 766 mi/h) set in 1997. Graphs of Motion when is constant but ≠ 0 The graphs in Figure 2.60 below represent the motion of the jet-powered car as it accelerates toward its top speed, but only during the time when its acceleration is constant. Time starts at zero for this motion (as if measured with a stopwatch), and the displacement and velocity are initially 200 m and 15 m/s, respectively. 78 Chapter 2 \| Kinematics Figure 2.60 Graphs of motion of a jet-powered car during the time span when its acceleration is constant. (a) The slope of an vs. graph is velocity. This is shown at two points, and the instantaneous velocities obtained are plotted in the next graph. Instantaneous velocity at any point is the slope of the tangent at that point. (b) The slope of the vs. graph is constant for this part of the motion, indicating constant acceleration. (c) Acceleration has the constant value of 5.0 m/s2 over the time interval plotted. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 79 Figure 2.61 A U.S. Air Force jet car speeds down a track. (credit: Matt Trostle, Flickr) The graph of displacement versus time in Figure
2.60(a) is a curve rather than a straight line. The slope of the curve becomes steeper as time progresses, showing that the velocity is increasing over time. The slope at any point on a displacement-versustime graph is the instantaneous velocity at that point. It is found by drawing a straight line tangent to the curve at the point of interest and taking the slope of this straight line. Tangent lines are shown for two points in Figure 2.60(a). If this is done at every point on the curve and the values are plotted against time, then the graph of velocity versus time shown in Figure 2.60(b) is obtained. Furthermore, the slope of the graph of velocity versus time is acceleration, which is shown in Figure 2.60(c). Example 2.18 Determining Instantaneous Velocity from the Slope at a Point: Jet Car Calculate the velocity of the jet car at a time of 25 s by finding the slope of the vs. graph in the graph below. Figure 2.62 The slope of an vs. graph is velocity. This is shown at two points. Instantaneous velocity at any point is the slope of the tangent at that point. Strategy The slope of a curve at a point is equal to the slope of a straight line tangent to the curve at that point. This principle is illustrated in Figure 2.62, where Q is the point at = 25 s . Solution 1. Find the tangent line to the curve at = 25 s . 2. Determine the endpoints of the tangent. These correspond to a position of 1300 m at time 19 s and a position of 3120 m at time 32 s. 3. Plug these endpoints into the equation to solve for the slope, . slope = Q = ΔQ Δ Q = (3120 m − 1300 m) (32 s − 19 s) Q = 1820 m 13 s = 140 m/s. (2.96) (2.97) Thus, Discussion 80 Chapter 2 \| Kinematics This is the value given in this figure's table for at = 25 s . The value of 140 m/s for Q is plotted in Figure 2.62. The entire graph of vs. can be obtained in this fashion. Carrying this one step further, we note that the slope of a velocity versus time graph is acceleration. Slope is rise divided by run; on a vs. graph, rise = change in velocity Δ and run = change in time Δ . The Slope of v vs. t The slope of a graph of velocity vs. time is acceleration . slope = Δ Δ = (2.98) Since the velocity versus time graph in Figure 2.60(b) is a straight line, its slope is the same everywhere, implying that acceleration is constant. Acceleration versus time is graphed in Figure 2.60(c). Additional general information can be obtained from Figure 2.62 and the expression for a straight line, = + . In this case, the vertical axis is , the intercept is 0 , the slope is , and the horizontal axis is . Substituting these symbols yields = 0 + . (2.99) A general relationship for velocity, acceleration, and time has again been obtained from a graph. Notice that this equation was also derived algebraically from other motion equations in Motion Equations for Constant Acceleration in One Dimension. It is not accidental that the same equations are obtained by graphical analysis as by algebraic techniques. In fact, an important way to discover physical relationships is to measure various physical quantities and then make graphs of one quantity against another to see if they are correlated in any way. Correlations imply physical relationships and might be shown by smooth graphs such as those above. From such graphs, mathematical relationships can sometimes be postulated. Further experiments are then performed to determine the validity of the hypothesized relationships. Graphs of Motion Where Acceleration is Not Constant Now consider the motion of the jet car as it goes from 165 m/s to its top velocity of 250 m/s, graphed in Figure 2.63. Time again starts at zero, and the initial displacement and velocity are 2900 m and 165 m/s, respectively. (These were the final displacement and velocity of the car in the motion graphed in Figure 2.60.) Acceleration gradually decreases from 5.0 m/s2 to zero when the car hits 250 m/s. The slope of the vs. graph increases until = 55 s , after which time the slope is constant. Similarly, velocity increases until 55 s and then becomes constant, since acceleration decreases to zero at 55 s and remains zero afterward. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 81 Figure 2.63 Graphs of motion of a jet-powered car as it reaches its top velocity. This motion begins where the motion in Figure 2.60 ends. (a) The slope of this graph is velocity; it is plotted in the next graph. (b) The velocity gradually approaches its top value. The slope of this graph is acceleration; it is plotted in the final graph. (c) Acceleration gradually declines to zero when velocity becomes constant. Example 2.19 Calculating Acceleration from a Graph of Velocity versus Time Calculate the acceleration of the jet car at a time of 25 s by finding the slope of the vs. graph in Figure 2.63(b). Strategy The slope of the curve at = 25 s is equal to the slope of the line tangent at that point, as illustrated in Figure 2.63(b). Solution Determine endpoints of the tangent line from the figure, and then plug them into the equation to solve for slope, . slope = Δ Δ = (260 m/s − 210 m/s) (51 s − 1.0 s) (2.100) 82 Discussion = 50 m/s 50 s = 1.0 m/s2. Chapter 2 \| Kinematics (2.101) Note that this value for is consistent with the value plotted in Figure 2.63(c) at = 25 s . A graph of displacement versus time can be used to generate a graph of velocity versus time, and a graph of velocity versus time can be used to generate a graph of acceleration versus time. We do this by finding the slope of the graphs at every point. If the graph is linear (i.e., a line with a constant slope), it is easy to find the slope at any point and you have the slope for every point. Graphical analysis of motion can be used to describe both specific and general characteristics of kinematics. Graphs can also be used for other topics in physics. An important aspect of exploring physical relationships is to graph them and look for underlying relationships. Check Your Understanding A graph of velocity vs. time of a ship coming into a harbor is shown below. (a) Describe the motion of the ship based on the graph. (b)What would a graph of the ship's acceleration look like? Figure 2.64 Solution (a) The ship moves at constant velocity and then begins to decelerate at a constant rate. At some point, its deceleration rate decreases. It maintains this lower deceleration rate until it stops moving. (b) A graph of acceleration vs. time would show zero acceleration in the first leg, large and constant negative acceleration in the second leg, and constant negative acceleration. Figure 2.65 Glossary acceleration: the rate of change in velocity; the change in velocity over time acceleration due to gravity: acceleration of an object as a result of gravity average acceleration: the change in velocity divided by the time over which it changes average speed: distance traveled divided by time during which motion occurs average velocity: displacement divided by time over which displacement occurs deceleration: acceleration in the direction opposite to velocity; acceleration that results in a decrease in velocity dependent variable: the variable that is being measured; usually plotted along the -axis displacement: the change in position of an object distance: the magnitude of displacement between two positions distance traveled: the total length of the path traveled between two positions This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 83 elapsed time: the difference between the ending time and beginning time free-fall: the state of movement that results from gravitational force only independent variable: the variable that the dependent variable is measured with respect to; usually plotted along the -axis instantaneous acceleration: acceleration at a specific point in time instantaneous speed: magnitude of the instantaneous velocity instantaneous velocity: velocity at a specific instant, or the average velocity over an infinitesimal time interval kinematics: the study of motion without considering its causes model: simplified description that contains only those elements necessary to describe the physics of a physical situation position: the location of an object at a particular time scalar: a quantity that is described by magnitude, but not direction slope: the difference in -value (the rise) divided by the difference in -value (the run) of two points on a straight line time: change, or the interval over which change occurs vector: a quantity that is described by both magnitude and direction y-intercept: the - value when = 0, or when the graph crosses the -axis Section Summary 2.1 Displacement • Kinematics is the study of motion without considering its causes. In this chapter, it is limited to motion along a straight line, called one-dimensional motion. • Displacement is the change in position of an object. In symbols, displacement Δ is defined to be • Δ = f − 0, where 0 is the initial position and f is the final position. In this text, the Greek letter Δ (delta) always means “change in” whatever quantity follows it. The SI unit for displacement is the meter (m). Displacement has a direction as well as a magnitude. • When you start a problem, assign which direction will be positive. • Distance is the magnitude of displacement between two positions. • Distance traveled is the total length of the path traveled between two positions. 2.2 Vectors, Scalars, and Coordinate Systems • A vector is any quantity that has magnitude and direction. • A scalar is any quantity that has magnitude but no direction. • Displacement and velocity are vectors, whereas distance and speed are scalars. • In one-dimensional motion, direction is specified by a plus or minus sign to signify left or right, up or down, and the like. 2.3 Time, Velocity, and Speed • Time is measured in terms of change, and its SI unit is the second (s). Elapsed time for an event is Δ = f − 0, where f is
the final time and 0 is the initial time. The initial time is often taken to be zero, as if measured with a stopwatch; the elapsed time is then just . - is defined as displacement divided by the travel time. In symbols, average velocity is • Average velocity - = Δ Δ = f − 0 f − 0 . • The SI unit for velocity is m/s. • Velocity is a vector and thus has a direction. • Instantaneous velocity is the velocity at a specific instant or the average velocity for an infinitesimal interval. Instantaneous speed is the magnitude of the instantaneous velocity. Instantaneous speed is a scalar quantity, as it has no direction specified. • • 84 Chapter 2 \| Kinematics • Average speed is the total distance traveled divided by the elapsed time. (Average speed is not the magnitude of the average velocity.) Speed is a scalar quantity; it has no direction associated with it. 2.4 Acceleration • Acceleration is the rate at which velocity changes. In symbols, average acceleration - is - = Δ Δ = f − 0 f − 0 . • The SI unit for acceleration is m/s2 . • Acceleration is a vector, and thus has a both a magnitude and direction. • Acceleration can be caused by either a change in the magnitude or the direction of the velocity. • Instantaneous acceleration is the acceleration at a specific instant in time. • Deceleration is an acceleration with a direction opposite to that of the velocity. 2.5 Motion Equations for Constant Acceleration in One Dimension • To simplify calculations we take acceleration to be constant, so that • We also take initial time to be zero. • Initial position and velocity are given a subscript 0; final values have no subscript. Thus, - = at all times. • The following kinematic equations for motion with constant are useful( − 0) • In vertical motion, is substituted for . 2.6 Problem-Solving Basics for One Dimensional Kinematics • The six basic problem solving steps for physics are: Step 1. Examine the situation to determine which physical principles are involved. Step 2. Make a list of what is given or can be inferred from the problem as stated (identify the knowns). Step 3. Identify exactly what needs to be determined in the problem (identify the unknowns). Step 4. Find an equation or set of equations that can help you solve the problem. Step 5. Substitute the knowns along with their units into the appropriate equation, and obtain numerical solutions complete with units. Step 6. Check the answer to see if it is reasonable: Does it make sense? 2.7 Falling Objects • An object in free-fall experiences constant acceleration if air resistance is negligible. • On Earth, all free-falling objects have an acceleration due to gravity , which averages • Whether the acceleration a should be taken as + or − is determined by your choice of coordinate system. If you = 9.80 m/s2. choose the upward direction as positive, = − = −9.80 m/s2 is negative. In the opposite case, = +g = 9.80 m/s2 is positive. Since acceleration is constant, the kinematic equations above can be applied with the appropriate + or − substituted for . • For objects in free-fall, up is normally taken as positive for displacement, velocity, and acceleration. 2.8 Graphical Analysis of One Dimensional Motion • Graphs of motion can be used to analyze motion. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 85 • Graphical solutions yield identical solutions to mathematical methods for deriving motion equations. • The slope of a graph of displacement vs. time is velocity . • The slope of a graph of velocity vs. time graph is acceleration . • Average velocity, instantaneous velocity, and acceleration can all be obtained by analyzing graphs. Conceptual Questions 2.1 Displacement 1. Give an example in which there are clear distinctions among distance traveled, displacement, and magnitude of displacement. Specifically identify each quantity in your example. 2. Under what circumstances does distance traveled equal magnitude of displacement? What is the only case in which magnitude of displacement and displacement are exactly the same? 3. Bacteria move back and forth by using their flagella (structures that look like little tails). Speeds of up to 50 μm/s have been observed. The total distance traveled by a bacterium is large for its size, while its 50×10−6 m/s displacement is small. Why is this? 2.2 Vectors, Scalars, and Coordinate Systems 4. A student writes, “A bird that is diving for prey has a speed of − / .” What is wrong with the student's statement? What has the student actually described? Explain. 5. What is the speed of the bird in Exercise 2.4? 6. Acceleration is the change in velocity over time. Given this information, is acceleration a vector or a scalar quantity? Explain. 7. A weather forecast states that the temperature is predicted to be −5ºC the following day. Is this temperature a vector or a scalar quantity? Explain. 2.3 Time, Velocity, and Speed 8. Give an example (but not one from the text) of a device used to measure time and identify what change in that device indicates a change in time. 9. There is a distinction between average speed and the magnitude of average velocity. Give an example that illustrates the difference between these two quantities. 10. Does a car's odometer measure position or displacement? Does its speedometer measure speed or velocity? 11. If you divide the total distance traveled on a car trip (as determined by the odometer) by the time for the trip, are you calculating the average speed or the magnitude of the average velocity? Under what circumstances are these two quantities the same? 12. How are instantaneous velocity and instantaneous speed related to one another? How do they differ? 2.4 Acceleration 13. Is it possible for speed to be constant while acceleration is not zero? Give an example of such a situation. 14. Is it possible for velocity to be constant while acceleration is not zero? Explain. 15. Give an example in which velocity is zero yet acceleration is not. 16. If a subway train is moving to the left (has a negative velocity) and then comes to a stop, what is the direction of its acceleration? Is the acceleration positive or negative? 17. Plus and minus signs are used in one-dimensional motion to indicate direction. What is the sign of an acceleration that reduces the magnitude of a negative velocity? Of a positive velocity? 2.6 Problem-Solving Basics for One Dimensional Kinematics 18. What information do you need in order to choose which equation or equations to use to solve a problem? Explain. 19. What is the last thing you should do when solving a problem? Explain. 2.7 Falling Objects 20. What is the acceleration of a rock thrown straight upward on the way up? At the top of its flight? On the way down? 21. An object that is thrown straight up falls back to Earth. This is one-dimensional motion. (a) When is its velocity zero? (b) Does its velocity change direction? (c) Does the acceleration due to gravity have the same sign on the way up as on the way down? 22. Suppose you throw a rock nearly straight up at a coconut in a palm tree, and the rock misses on the way up but hits the coconut on the way down. Neglecting air resistance, how does the speed of the rock when it hits the coconut on the way down compare with what it would have been if it had hit the coconut on the way up? Is it more likely to dislodge the coconut on the way up or down? Explain. 86 Chapter 2 \| Kinematics 23. If an object is thrown straight up and air resistance is negligible, then its speed when it returns to the starting point is the same as when it was released. If air resistance were not negligible, how would its speed upon return compare with its initial speed? How would the maximum height to which it rises be affected? 24. The severity of a fall depends on your speed when you strike the ground. All factors but the acceleration due to gravity being the same, how many times higher could a safe fall on the Moon be than on Earth (gravitational acceleration on the Moon is about 1/6 that of the Earth)? 25. How many times higher could an astronaut jump on the Moon than on Earth if his takeoff speed is the same in both locations (gravitational acceleration on the Moon is about 1/6 of on Earth)? 2.8 Graphical Analysis of One Dimensional Motion 26. (a) Explain how you can use the graph of position versus time in Figure 2.66 to describe the change in velocity over time. Identify (b) the time ( a , b , c , d , or e ) at which the instantaneous velocity is greatest, (c) the time at which it is zero, and (d) the time at which it is negative. Figure 2.66 27. (a) Sketch a graph of velocity versus time corresponding to the graph of displacement versus time given in Figure 2.67. (b) Identify the time or times ( a , b , c , etc.) at which the instantaneous velocity is greatest. (c) At which times is it zero? (d) At which times is it negative? Figure 2.67 28. (a) Explain how you can determine the acceleration over time from a velocity versus time graph such as the one in Figure 2.68. (b) Based on the graph, how does acceleration change over time? This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 \| Kinematics 87 Figure 2.68 29. (a) Sketch a graph of acceleration versus time corresponding to the graph of velocity versus time given in Figure 2.69. (b) Identify the time or times ( a , b , c , etc.) at which the acceleration is greatest. (c) At which times is it zero? (d) At which times is it negative? Figure 2.69 30. Consider the velocity vs. time graph of a person in an elevator shown in Figure 2.70. Suppose the elevator is initially at rest. It then accelerates for 3 seconds, maintains that velocity for 15 seconds, then decelerates for 5 seconds until it stops. The acceleration for the entire trip is not constant so we cannot use the equations of motion from Motion Equations for Constant Acceleration in One Dimension for the complete trip. (We could, however, use them in the three in
dividual sections where acceleration is a constant.) Sketch graphs of (a) position vs. time and (b) acceleration vs. time for this trip. Figure 2.70 31. A cylinder is given a push and then rolls up an inclined plane. If the origin is the starting point, sketch the position, velocity, and acceleration of the cylinder vs. time as it goes up and then down the plane. 88 Chapter 2 \| Kinematics Problems & Exercises 2.1 Displacement Figure 2.71 1. Find the following for path A in Figure 2.71: (a) The distance traveled. (b) The magnitude of the displacement from start to finish. (c) The displacement from start to finish. 2. Find the following for path B in Figure 2.71: (a) The distance traveled. (b) The magnitude of the displacement from start to finish. (c) The displacement from start to finish. 3. Find the following for path C in Figure 2.71: (a) The distance traveled. (b) The magnitude of the displacement from start to finish. (c) The displacement from start to finish. 4. Find the following for path D in Figure 2.71: (a) The distance traveled. (b) The magnitude of the displacement from start to finish. (c) The displacement from start to finish. 2.3 Time, Velocity, and Speed 5. (a) Calculate Earth's average speed relative to the Sun. (b) What is its average velocity over a period of one year? 6. A helicopter blade spins at exactly 100 revolutions per minute. Its tip is 5.00 m from the center of rotation. (a) Calculate the average speed of the blade tip in the helicopter's frame of reference. (b) What is its average velocity over one revolution? 7. The North American and European continents are moving apart at a rate of about 3 cm/y. At this rate how long will it take them to drift 500 km farther apart than they are at present? 8. Land west of the San Andreas fault in southern California is moving at an average velocity of about 6 cm/y northwest relative to land east of the fault. Los Angeles is west of the fault and may thus someday be at the same latitude as San Francisco, which is east of the fault. How far in the future will this occur if the displacement to be made is 590 km northwest, assuming the motion remains constant? 9. On May 26, 1934, a streamlined, stainless steel diesel train called the Zephyr set the world's nonstop long-distance speed record for trains. Its run from Denver to Chicago took 13 hours, 4 minutes, 58 seconds, and was witnessed by more than a million people along the route. The total distance traveled was 1633.8 km. What was its average speed in km/h and m/s? 10. Tidal friction is slowing the rotation of the Earth. As a result, the orbit of the Moon is increasing in radius at a rate of approximately 4 cm/year. Assuming this to be a constant rate, how many years will pass before the radius of the Moon's orbit increases by 3.84×106 m (1%)? This content is available for free at http://cnx.org/content/col11844/1.13 11. A student drove to the university from her home and noted that the odometer reading of her car increased by 12.0 km. The trip took 18.0 min. (a) What was her average speed? (b) If the straight-line distance from her home to the university is 10.3 km in a direction 25.0º south of east, what was her average velocity? (c) If she returned home by the same path 7 h 30 min after she left, what were her average speed and velocity for the entire trip? 12. The speed of propagation of the action potential (an electrical signal) in a nerve cell depends (inversely) on the diameter of the axon (nerve fiber). If the nerve cell connecting the spinal cord to your feet is 1.1 m long, and the nerve impulse speed is 18 m/s, how long does it take for the nerve signal to travel this distance? 13. Conversations with astronauts on the lunar surface were characterized by a kind of echo in which the earthbound person's voice was so loud in the astronaut's space helmet that it was picked up by the astronaut's microphone and transmitted back to Earth. It is reasonable to assume that the echo time equals the time necessary for the radio wave to travel from the Earth to the Moon and back (that is, neglecting any time delays in the electronic equipment). Calculate the distance from Earth to the Moon given that the echo time was 2.56 s and that radio waves travel at the speed of light (3.00×108 m/s) . 14. A football quarterback runs 15.0 m straight down the playing field in 2.50 s. He is then hit and pushed 3.00 m straight backward in 1.75 s. He breaks the tackle and runs straight forward another 21.0 m in 5.20 s. Calculate his average velocity (a) for each of the three intervals and (b) for the entire motion. 15. The planetary model of the atom pictures electrons orbiting the atomic nucleus much as planets orbit the Sun. In this model you can view hydrogen, the simplest atom, as having a single electron in a circular orbit 1.06×10−10 m in diameter. (a) If the average speed of the electron in this orbit is known to be 2.20×106 m/s , calculate the number of revolutions per second it makes about the nucleus. (b) What is the electron's average velocity? 2.4 Acceleration 16. A cheetah can accelerate from rest to a speed of 30.0 m/s in 7.00 s. What is its acceleration? 17. Professional Application Dr. John Paul Stapp was U.S. Air Force officer who studied the effects of extreme deceleration on the human body. On December 10, 1954, Stapp rode a rocket sled, accelerating from rest to a top speed of 282 m/s (1015 km/h) in 5.00 s, and was brought jarringly back to rest in only 1.40 s! Calculate his (a) acceleration and (b) deceleration. Express each in multiples of (9.80 m/s2) by taking its ratio to the acceleration of gravity. 18. A commuter backs her car out of her garage with an acceleration of 1.40 m/s2 . (a) How long does it take her to reach a speed of 2.00 m/s? (b) If she then brakes to a stop in 0.800 s, what is her deceleration? 19. Assume that an intercontinental ballistic missile goes from rest to a suborbital speed of 6.50 km/s in 60.0 s (the actual speed and time are classified). What is its average acceleration in m/s2 and in multiples of (9.80 m/s2) ? Chapter 2 \| Kinematics 89 2.5 Motion Equations for Constant Acceleration in One Dimension 20. An Olympic-class sprinter starts a race with an acceleration of 4.50 m/s2 . (a) What is her speed 2.40 s later? (b) Sketch a graph of her position vs. time for this period. 21. A well-thrown ball is caught in a well-padded mitt. If the deceleration of the ball is 2.10×104 m/s2 , and 1.85 ms (1 ms = 10−3 s) elapses from the time the ball first touches the mitt until it stops, what was the initial velocity of the ball? 22. A bullet in a gun is accelerated from the firing chamber to the end of the barrel at an average rate of 6.20×105 m/s2 for 8.10×10−4 s . What is its muzzle velocity (that is, its final velocity)? 23. (a) A light-rail commuter train accelerates at a rate of 1.35 m/s2 . How long does it take to reach its top speed of 80.0 km/h, starting from rest? (b) The same train ordinarily decelerates at a rate of 1.65 m/s2 . How long does it take to come to a stop from its top speed? (c) In emergencies the train can decelerate more rapidly, coming to rest from 80.0 km/h in 8.30 s. What is its emergency deceleration in m/s2 ? 24. While entering a freeway, a car accelerates from rest at a rate of 2.40 m/s2 for 12.0 s. (a) Draw a sketch of the situation. (b) List the knowns in this problem. (c) How far does the car travel in those 12.0 s? To solve this part, first identify the unknown, and then discuss how you chose the appropriate equation to solve for it. After choosing the equation, show your steps in solving for the unknown, check your units, and discuss whether the answer is reasonable. (d) What is the car's final velocity? Solve for this unknown in the same manner as in part (c), showing all steps explicitly. 25. At the end of a race, a runner decelerates from a velocity of 9.00 m/s at a rate of 2.00 m/s2 . (a) How far does she travel in the next 5.00 s? (b) What is her final velocity? (c) Evaluate the result. Does it make sense? 26. Professional Application: Blood is accelerated from rest to 30.0 cm/s in a distance of 1.80 cm by the left ventricle of the heart. (a) Make a sketch of the situation. (b) List the knowns in this problem. (c) How long does the acceleration take? To solve this part, first identify the unknown, and then discuss how you chose the appropriate equation to solve for it. After choosing the equation, show your steps in solving for the unknown, checking your units. (d) Is the answer reasonable when compared with the time for a heartbeat? 27. In a slap shot, a hockey player accelerates the puck from a velocity of 8.00 m/s to 40.0 m/s in the same direction. If this shot takes 3.33×10−2 s , calculate the distance over which the puck accelerates. 28. A powerful motorcycle can accelerate from rest to 26.8 m/ s (100 km/h) in only 3.90 s. (a) What is its average acceleration? (b) How far does it travel in that time? 29. Freight trains can produce only relatively small accelerations and decelerations. (a) What is the final velocity of a freight train that accelerates at a rate of 0.0500 m/s2 for 8.00 min, starting with an initial velocity of 4.00 m/s? (b) If the train can slow down at a rate of 0.550 m/s2 , how long will it take to come to a stop from this velocity? (c) How far will it travel in each case? 30. A fireworks shell is accelerated from rest to a velocity of 65.0 m/s over a distance of 0.250 m. (a) How long did the acceleration last? (b) Calculate the acceleration. 31. A swan on a lake gets airborne by flapping its wings and running on top of the water. (a) If the swan must reach a velocity of 6.00 m/s to take off and it accelerates from rest at an average rate of 0.350 m/s2 , how far will it travel before becoming airborne? (b) How long does this take? 32. Professional Application: A woodpecker's brain is specially protected from large decelerations by tendon-like attachments inside the skull. While pecking on a tree, the woodpecker's
head comes to a stop from an initial velocity of 0.600 m/s in a distance of only 2.00 mm. (a) Find the acceleration in m/s2 and in multiples of . (b) Calculate the stopping time. (c) The tendons cradling the brain stretch, making its stopping distance 4.50 mm (greater than the head and, hence, less deceleration of the brain). What is the brain's deceleration, expressed in multiples of ? = 9.80 m/s2 33. An unwary football player collides with a padded goalpost while running at a velocity of 7.50 m/s and comes to a full stop after compressing the padding and his body 0.350 m. (a) What is his deceleration? (b) How long does the collision last? 34. In World War II, there were several reported cases of airmen who jumped from their flaming airplanes with no parachute to escape certain death. Some fell about 20,000 feet (6000 m), and some of them survived, with few lifethreatening injuries. For these lucky pilots, the tree branches and snow drifts on the ground allowed their deceleration to be relatively small. If we assume that a pilot's speed upon impact was 123 mph (54 m/s), then what was his deceleration? Assume that the trees and snow stopped him over a distance of 3.0 m. 35. Consider a grey squirrel falling out of a tree to the ground. (a) If we ignore air resistance in this case (only for the sake of this problem), determine a squirrel's velocity just before hitting the ground, assuming it fell from a height of 3.0 m. (b) If the squirrel stops in a distance of 2.0 cm through bending its limbs, compare its deceleration with that of the airman in the previous problem. 36. An express train passes through a station. It enters with an initial velocity of 22.0 m/s and decelerates at a rate of 0.150 m/s2 as it goes through. The station is 210 m long. (a) How long is the nose of the train in the station? (b) How fast is it going when the nose leaves the station? (c) If the train is 130 m long, when does the end of the train leave the station? (d) What is the velocity of the end of the train as it leaves? 37. Dragsters can actually reach a top speed of 145 m/s in only 4.45 s—considerably less time than given in Example 2.10 and Example 2.11. (a) Calculate the average acceleration for such a dragster. (b) Find the final velocity of this dragster starting from rest and accelerating at the rate found in (a) for 402 m (a quarter mile) without using any 90 Chapter 2 \| Kinematics information on time. (c) Why is the final velocity greater than that used to find the average acceleration? Hint: Consider whether the assumption of constant acceleration is valid for a dragster. If not, discuss whether the acceleration would be greater at the beginning or end of the run and what effect that would have on the final velocity. known and identify its value. Then identify the unknown, and discuss how you chose the appropriate equation to solve for it. After choosing the equation, show your steps in solving for the unknown, checking units, and discuss whether the answer is reasonable. (c) How long is the dolphin in the air? Neglect any effects due to his size or orientation. 38. A bicycle racer sprints at the end of a race to clinch a victory. The racer has an initial velocity of 11.5 m/s and accelerates at the rate of 0.500 m/s2 for 7.00 s. (a) What is his final velocity? (b) The racer continues at this velocity to the finish line. If he was 300 m from the finish line when he started to accelerate, how much time did he save? (c) One other racer was 5.00 m ahead when the winner started to accelerate, but he was unable to accelerate, and traveled at 11.8 m/s until the finish line. How far ahead of him (in meters and in seconds) did the winner finish? 39. In 1967, New Zealander Burt Munro set the world record for an Indian motorcycle, on the Bonneville Salt Flats in Utah, with a maximum speed of 183.58 mi/h. The one-way course was 5.00 mi long. Acceleration rates are often described by the time it takes to reach 60.0 mi/h from rest. If this time was 4.00 s, and Burt accelerated at this rate until he reached his maximum speed, how long did it take Burt to complete the course? 40. (a) A world record was set for the men's 100-m dash in the 2008 Olympic Games in Beijing by Usain Bolt of Jamaica. Bolt “coasted” across the finish line with a time of 9.69 s. If we assume that Bolt accelerated for 3.00 s to reach his maximum speed, and maintained that speed for the rest of the race, calculate his maximum speed and his acceleration. (b) During the same Olympics, Bolt also set the world record in the 200-m dash with a time of 19.30 s. Using the same assumptions as for the 100-m dash, what was his maximum speed for this race? 2.7 Falling Objects Assume air resistance is negligible unless otherwise stated. 41. Calculate the displacement and velocity at times of (a) 0.500, (b) 1.00, (c) 1.50, and (d) 2.00 s for a ball thrown straight up with an initial velocity of 15.0 m/s. Take the point of release to be 0 = 0 . 42. Calculate the displacement and velocity at times of (a) 0.500, (b) 1.00, (c) 1.50, (d) 2.00, and (e) 2.50 s for a rock thrown straight down with an initial velocity of 14.0 m/s from the Verrazano Narrows Bridge in New York City. The roadway of this bridge is 70.0 m above the water. 43. A basketball referee tosses the ball straight up for the starting tip-off. At what velocity must a basketball player leave the ground to rise 1.25 m above the floor in an attempt to get the ball? 44. A rescue helicopter is hovering over a person whose boat has sunk. One of the rescuers throws a life preserver straight down to the victim with an initial velocity of 1.40 m/s and observes that it takes 1.8 s to reach the water. (a) List the knowns in this problem. (b) How high above the water was the preserver released? Note that the downdraft of the helicopter reduces the effects of air resistance on the falling life preserver, so that an acceleration equal to that of gravity is reasonable. 45. A dolphin in an aquatic show jumps straight up out of the water at a velocity of 13.0 m/s. (a) List the knowns in this problem. (b) How high does his body rise above the water? To solve this part, first note that the final velocity is now a This content is available for free at http://cnx.org/content/col11844/1.13 46. A swimmer bounces straight up from a diving board and falls feet first into a pool. She starts with a velocity of 4.00 m/ s, and her takeoff point is 1.80 m above the pool. (a) How long are her feet in the air? (b) What is her highest point above the board? (c) What is her velocity when her feet hit the water? 47. (a) Calculate the height of a cliff if it takes 2.35 s for a rock to hit the ground when it is thrown straight up from the cliff with an initial velocity of 8.00 m/s. (b) How long would it take to reach the ground if it is thrown straight down with the same speed? 48. A very strong, but inept, shot putter puts the shot straight up vertically with an initial velocity of 11.0 m/s. How long does he have to get out of the way if the shot was released at a height of 2.20 m, and he is 1.80 m tall? 49. You throw a ball straight up with an initial velocity of 15.0 m/s. It passes a tree branch on the way up at a height of 7.00 m. How much additional time will pass before the ball passes the tree branch on the way back down? 50. A kangaroo can jump over an object 2.50 m high. (a) Calculate its vertical speed when it leaves the ground. (b) How long is it in the air? 51. Standing at the base of one of the cliffs of Mt. Arapiles in Victoria, Australia, a hiker hears a rock break loose from a height of 105 m. He can't see the rock right away but then does, 1.50 s later. (a) How far above the hiker is the rock when he can see it? (b) How much time does he have to move before the rock hits his head? 52. An object is dropped from a height of 75.0 m above ground level. (a) Determine the distance traveled during the first second. (b) Determine the final velocity at which the object hits the ground. (c) Determine the distance traveled during the last second of motion before hitting the ground. 53. There is a 250-m-high cliff at Half Dome in Yosemite National Park in California. Suppose a boulder breaks loose from the top of this cliff. (a) How fast will it be going when it strikes the ground? (b) Assuming a reaction time of 0.300 s, how long will a tourist at the bottom have to get out of the way after hearing the sound of the rock breaking loose (neglecting the height of the tourist, which would become negligible anyway if hit)? The speed of sound is 335 m/s on this day. 54. A ball is thrown straight up. It passes a 2.00-m-high window 7.50 m off the ground on its path up and takes 1.30 s to go past the window. What was the ball's initial velocity? 55. Suppose you drop a rock into a dark well and, using precision equipment, you measure the time for the sound of a splash to return. (a) Neglecting the time required for sound to travel up the well, calculate the distance to the water if the sound returns in 2.0000 s. (b) Now calculate the distance taking into account the time for sound to travel up the well. The speed of sound is 332.00 m/s in this well. 56. A steel ball is dropped onto a hard floor from a height of 1.50 m and rebounds to a height of 1.45 m. (a) Calculate its velocity just before it strikes the floor. (b) Calculate its velocity just after it leaves the floor on its way back up. (c) Calculate its acceleration during contact with the floor if that contact lasts 0.0800 ms (8.00×10−5 s) . (d) How much did the ball Chapter 2 \| Kinematics 91 compress during its collision with the floor, assuming the floor is absolutely rigid? 57. A coin is dropped from a hot-air balloon that is 300 m above the ground and rising at 10.0 m/s upward. For the coin, find (a) the maximum height reached, (b) its position and velocity 4.00 s after being released, and (c) the time before it hits the ground. 58. A soft tennis ball is dropped onto a hard floor from a height of 1.50 m
and rebounds to a height of 1.10 m. (a) Calculate its velocity just before it strikes the floor. (b) Calculate its velocity just after it leaves the floor on its way back up. (c) Calculate its acceleration during contact with the floor if that contact lasts 3.50 ms (3.50×10−3 s) . (d) How much did the ball compress during its collision with the floor, assuming the floor is absolutely rigid? 2.8 Graphical Analysis of One Dimensional Motion Note: There is always uncertainty in numbers taken from graphs. If your answers differ from expected values, examine them to see if they are within data extraction uncertainties estimated by you. 59. (a) By taking the slope of the curve in Figure 2.72, verify that the velocity of the jet car is 115 m/s at = 20 s . (b) By taking the slope of the curve at any point in Figure 2.73, verify that the jet car's acceleration is 5.0 m/s2 . Figure 2.74 61. Using approximate values, calculate the slope of the curve in Figure 2.74 to verify that the velocity at = 30.0 s is 0.238 m/s. Assume all values are known to 3 significant figures. 62. By taking the slope of the curve in Figure 2.75, verify that the acceleration is 3.2 m/s2 at = 10 s . Figure 2.72 Figure 2.75 63. Construct the displacement graph for the subway shuttle train as shown in Figure 2.30(a). Your graph should show the position of the train, in kilometers, from t = 0 to 20 s. You will need to use the information on acceleration and velocity given in the examples for this figure. 64. (a) Take the slope of the curve in Figure 2.76 to find the jogger's velocity at = 2.5 s . (b) Repeat at 7.5 s. These values must be consistent with the graph in Figure 2.77. Figure 2.73 60. Using approximate values, calculate the slope of the curve in Figure 2.74 to verify that the velocity at = 10.0 s is 0.208 m/s. Assume all values are known to 3 significant figures. Figure 2.76 92 Chapter 2 \| Kinematics Figure 2.77 Figure 2.80 Figure 2.78 65. A graph of () is shown for a world-class track sprinter in a 100-m race. (See Figure 2.79). (a) What is his average velocity for the first 4 s? (b) What is his instantaneous velocity at = 5 s ? (c) What is his average acceleration between 0 and 4 s? (d) What is his time for the race? Figure 2.79 66. Figure 2.80 shows the displacement graph for a particle for 5 s. Draw the corresponding velocity and acceleration graphs. Test Prep for AP® Courses 2.1 Displacement 1. Which of the following statements comparing position, distance, and displacement is correct? This content is available for free at http://cnx.org/content/col11844/1.13 a. An object may record a distance of zero while recording a non-zero displacement. b. An object may record a non-zero distance while recording a displacement of zero. c. An object may record a non-zero distance while maintaining a position of zero. Chapter 2 \| Kinematics 93 d. An object may record a non-zero displacement while maintaining a position of zero. velocity v as a function of time t is shown in the graph. The five labeled points divide the graph into four sections. 2.2 Vectors, Scalars, and Coordinate Systems 2. A student is trying to determine the acceleration of a feather as she drops it to the ground. If the student is looking to achieve a positive velocity and positive acceleration, what is the most sensible way to set up her coordinate system? a. Her hand should be a coordinate of zero and the upward direction should be considered positive. b. Her hand should be a coordinate of zero and the downward direction should be considered positive. c. The floor should be a coordinate of zero and the upward direction should be considered positive. d. The floor should be a coordinate of zero and the downward direction should be considered positive. 2.3 Time, Velocity, and Speed 3. A group of students has two carts, A and B, with wheels that turn with negligible friction. The two carts travel along a straight horizontal track and eventually collide. Before the collision, cart A travels to the right and cart B is initially at rest. After the collision, the carts stick together. a. Describe an experimental procedure to determine the velocities of the carts before and after the collision, including all the additional equipment you would need. You may include a labeled diagram of your setup to help in your description. Indicate what measurements you would take and how you would take them. Include enough detail so that another student could carry out your procedure. b. There will be sources of error in the measurements taken in the experiment both before and after the collision. Which velocity will be more greatly affected by this error: the velocity prior to the collision or the velocity after the collision? Or will both sets of data be affected equally? Justify your answer. 2.4 Acceleration 4. Figure 2.81 Graph showing Velocity vs. Time of a cart. A cart is constrained to move along a straight line. A varying net force along the direction of motion is exerted on the cart. The cart's Which of the following correctly ranks the magnitude of the average acceleration of the cart during the four sections of the graph? a. aCD > aAB > aBC > aDE b. aBC > aAB > aCD > aDE c. aAB > aBC > aDE > aCD d. aCD > aAB > aDE > aBC 5. Push a book across a table and observe it slow to a stop. Draw graphs showing the book's position vs. time and velocity vs. time if the direction of its motion is considered positive. Draw graphs showing the book's position vs. time and velocity vs. time if the direction of its motion is considered negative. 2.5 Motion Equations for Constant Acceleration in One Dimension 6. A group of students is attempting to determine the average acceleration of a marble released from the top of a long ramp. Below is a set of data representing the marble's position with respect to time. Position (cm) Time (s) 0.0 0.3 1.25 2.8 5.0 7.75 11.3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Use the data table above to construct a graph determining the acceleration of the marble. Select a set of data points from the table and plot those points on the graph. Fill in the blank column in the table for any quantities you graph other than the given data. Label the axes and indicate the scale for each. Draw a best-fit line or curve through your data points. Using the best-fit line, determine the value of the marble's acceleration. 2.7 Falling Objects 7. Observing a spacecraft land on a distant asteroid, scientists notice that the craft is falling at a rate of 5 m/s. When it is 100 m closer to the surface of the asteroid, the craft reports a velocity of 8 m/s. According to their data, what is the approximate gravitational acceleration on this asteroid? a. 0 m/s2 b. 0.03 m/s2 c. 0.20 m/s2 d. 0.65 m/s2 e. 33 m/s2 94 Chapter 2 \| Kinematics This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 3 \| Two-Dimensional Kinematics 95 3 TWO-DIMENSIONAL KINEMATICS Figure 3.1 Everyday motion that we experience is, thankfully, rarely as tortuous as a rollercoaster ride like this—the Dragon Khan in Spain's Universal Port Aventura Amusement Park. However, most motion is in curved, rather than straight-line, paths. Motion along a curved path is two- or threedimensional motion, and can be described in a similar fashion to one-dimensional motion. (credit: Boris23/Wikimedia Commons) Chapter Outline 3.1. Kinematics in Two Dimensions: An Introduction 3.2. Vector Addition and Subtraction: Graphical Methods 3.3. Vector Addition and Subtraction: Analytical Methods 3.4. Projectile Motion 3.5. Addition of Velocities Connection for AP® Courses Most instances of motion in everyday life involve changes in displacement and velocity that occur in more than one direction. For example, when you take a long road trip, you drive on different roads in different directions for different amounts of time at different speeds. How can these motions all be combined to determine information about the trip such as the total displacement and average velocity? If you kick a ball from ground level at some angle above the horizontal, how can you describe its motion? To what maximum height does the object rise above the ground? How long is the object in the air? How much horizontal distance is covered before the ball lands? To answer questions such as these, we need to describe motion in two dimensions. Examining two-dimensional motion requires an understanding of both the scalar and the vector quantities associated with the motion. You will learn how to combine vectors to incorporate both the magnitude and direction of vectors into your analysis. You will learn strategies for simplifying the calculations involved by choosing the appropriate reference frame and by treating each dimension of the motion separately as a one-dimensional problem, but you will also see that the motion itself occurs in the same way regardless of your chosen reference frame (Essential Knowledge 3.A.1). 96 Chapter 3 \| Two-Dimensional Kinematics This chapter lays a necessary foundation for examining interactions of objects described by forces (Big Idea 3). Changes in direction result from acceleration, which necessitates force on an object. In this chapter, you will concentrate on describing motion that involves changes in direction. In later chapters, you will apply this understanding as you learn about how forces cause these motions (Enduring Understanding 3.A). The concepts in this chapter support: Big Idea 3 The interactions of an object with other objects can be described by forces. Enduring Understanding 3.A All forces share certain common characteristics when considered by observers in inertial reference frames. Essential Knowledge 3.A.1 An observer in a particular reference frame can describe the motion of an object using such quantities as position, displacement, distance, velocity, speed, and acceleration. 3.1 Kinematics in Two Dimensions: An Introduction Learning Objectives By the end of this section, you will be able to: • Observe that motion in two
dimensions consists of horizontal and vertical components. • Understand the independence of horizontal and vertical vectors in two-dimensional motion. The information presented in this section supports the following AP® learning objectives and science practices: • 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, and graphical representations. (S.P. 1.5, 2.1, 2.2) • 3.A.1.2 The student is able to design an experimental investigation of the motion of an object. (S.P. 4.2) • 3.A.1.3 The student is able to analyze experimental data describing the motion of an object and is able to express the results of the analysis using narrative, mathematical, and graphical representations. (S.P. 5.1) Figure 3.2 Walkers and drivers in a city like New York are rarely able to travel in straight lines to reach their destinations. Instead, they must follow roads and sidewalks, making two-dimensional, zigzagged paths. (credit: Margaret W. Carruthers) Two-Dimensional Motion: Walking in a City Suppose you want to walk from one point to another in a city with uniform square blocks, as pictured in Figure 3.3. Figure 3.3 A pedestrian walks a two-dimensional path between two points in a city. In this scene, all blocks are square and are the same size. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 3 \| Two-Dimensional Kinematics 97 The straight-line path that a helicopter might fly is blocked to you as a pedestrian, and so you are forced to take a twodimensional path, such as the one shown. You walk 14 blocks in all, 9 east followed by 5 north. What is the straight-line distance? An old adage states that the shortest distance between two points is a straight line. The two legs of the trip and the straight-line path form a right triangle, and so the Pythagorean theorem, 2 + 2 = 2 , can be used to find the straight-line distance. Figure 3.4 The Pythagorean theorem relates the length of the legs of a right triangle, labeled and , with the hypotenuse, labeled . The . This can be rewritten, solving for : = 2 + 2 relationship is given by: 2 + 2 = 2 . The hypotenuse of the triangle is the straight-line path, and so in this case its length in units of city blocks is (9 blocks)2+ (5 blocks)2= 10.3 blocks , considerably shorter than the 14 blocks you walked. (Note that we are using three significant figures in the answer. Although it appears that “9” and “5” have only one significant digit, they are discrete numbers. In this case “9 blocks” is the same as “9.0 or 9.00 blocks.” We have decided to use three significant figures in the answer in order to show the result more precisely.) Figure 3.5 The straight-line path followed by a helicopter between the two points is shorter than the 14 blocks walked by the pedestrian. All blocks are square and the same size. The fact that the straight-line distance (10.3 blocks) in Figure 3.5 is less than the total distance walked (14 blocks) is one example of a general characteristic of vectors. (Recall that vectors are quantities that have both magnitude and direction.) As for one-dimensional kinematics, we use arrows to represent vectors. The length of the arrow is proportional to the vector's magnitude. The arrow's length is indicated by hash marks in Figure 3.3 and Figure 3.5. The arrow points in the same direction as the vector. For two-dimensional motion, the path of an object can be represented with three vectors: one vector shows the straight-line path between the initial and final points of the motion, one vector shows the horizontal component of the motion, and one vector shows the vertical component of the motion. The horizontal and vertical components of the motion add together to give the straight-line path. For example, observe the three vectors in Figure 3.5. The first represents a 9-block displacement east. The second represents a 5-block displacement north. These vectors are added to give the third vector, with a 10.3-block total displacement. The third vector is the straight-line path between the two points. Note that in this example, the vectors that we are adding are perpendicular to each other and thus form a right triangle. This means that we can use the Pythagorean theorem to calculate the magnitude of the total displacement. (Note that we cannot use the Pythagorean theorem to add vectors that are not perpendicular. We will develop techniques for adding vectors having any direction, not just those perpendicular to one another, in Vector Addition and Subtraction: Graphical Methods and Vector Addition and Subtraction: Analytical Methods.) The Independence of Perpendicular Motions The person taking the path shown in Figure 3.5 walks east and then north (two perpendicular directions). How far he or she walks east is only affected by his or her motion eastward. Similarly, how far he or she walks north is only affected by his or her motion northward. Independence of Motion The horizontal and vertical components of two-dimensional motion are independent of each other. Any motion in the horizontal direction does not affect motion in the vertical direction, and vice versa. 98 Chapter 3 \| Two-Dimensional Kinematics This is true in a simple scenario like that of walking in one direction first, followed by another. It is also true of more complicated motion involving movement in two directions at once. For example, let's compare the motions of two baseballs. One baseball is dropped from rest. At the same instant, another is thrown horizontally from the same height and follows a curved path. A stroboscope has captured the positions of the balls at fixed time intervals as they fall. Figure 3.6 This shows the motions of two identical balls—one falls from rest, the other has an initial horizontal velocity. Each subsequent position is an equal time interval. Arrows represent horizontal and vertical velocities at each position. The ball on the right has an initial horizontal velocity, while the ball on the left has no horizontal velocity. Despite the difference in horizontal velocities, the vertical velocities and positions are identical for both balls. This shows that the vertical and horizontal motions are independent. Applying the Science Practices: Independence of Horizontal and Vertical Motion or Maximum Height and Flight Time Choose one of the following experiments to design: Design an experiment to confirm what is shown in Figure 3.6, that the vertical motion of the two balls is independent of the horizontal motion. As you think about your experiment, consider the following questions: • How will you measure the horizontal and vertical positions of each ball over time? What equipment will this require? • How will you measure the time interval between each of your position measurements? What equipment will this require? If you were to create separate graphs of the horizontal velocity for each ball versus time, what do you predict it would look like? Explain. If you were to compare graphs of the vertical velocity for each ball versus time, what do you predict it would look like? Explain. If there is a significant amount of air resistance, how will that affect each of your graphs? • • • Design a two-dimensional ballistic motion experiment that demonstrates the relationship between the maximum height reached by an object and the object's time of flight. As you think about your experiment, consider the following questions: • How will you measure the maximum height reached by your object? • How can you take advantage of the symmetry of an object in ballistic motion launched from ground level, reaching maximum height, and returning to ground level? • Will it make a difference if your object has no horizontal component to its velocity? Explain. • Will you need to measure the time at multiple different positions? Why or why not? • Predict what a graph of travel time versus maximum height will look like. Will it be linear? Parabolic? Horizontal? Explain the shape of your predicted graph qualitatively or quantitatively. If there is a significant amount of air resistance, how will that affect your measurements and your results? • It is remarkable that for each flash of the strobe, the vertical positions of the two balls are the same. This similarity implies that the vertical motion is independent of whether or not the ball is moving horizontally. (Assuming no air resistance, the vertical motion of a falling object is influenced by gravity only, and not by any horizontal forces.) Careful examination of the ball thrown horizontally shows that it travels the same horizontal distance between flashes. This is due to the fact that there are no additional forces on the ball in the horizontal direction after it is thrown. This result means that the horizontal velocity is constant, and affected neither by vertical motion nor by gravity (which is vertical). Note that this case is true only for ideal conditions. In the real world, air resistance will affect the speed of the balls in both directions. The two-dimensional curved path of the horizontally thrown ball is composed of two independent one-dimensional motions (horizontal and vertical). The key to analyzing such motion, called projectile motion, is to resolve (break) it into motions along perpendicular directions. Resolving two-dimensional motion into perpendicular components is possible because the components are independent. We shall see how to resolve vectors in Vector Addition and Subtraction: Graphical Methods and Vector Addition and Subtraction: Analytical Methods. We will find such techniques to be useful in many areas of physics. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 3 \| Two-Dimensional Kinematics 99 PhET Explorations: Ladybug Motion 2D Learn about position, velocity and acceleration vectors. Move the ladybug by setting the position, velocity or acceleration, and see how the vectors change. Choose linear, circular or elliptical motion, an
d record and playback the motion to analyze the behavior. Figure 3.7 Ladybug Motion 2D (http://cnx.org/content/m54779/1.2/ladybug-motion-2d_en.jar) 3.2 Vector Addition and Subtraction: Graphical Methods By the end of this section, you will be able to: Learning Objectives • Understand the rules of vector addition, subtraction, and multiplication. • Apply graphical methods of vector addition and subtraction to determine the displacement of moving objects. The information presented in this section supports the following AP® learning objectives and science practices: • 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, and graphical representations. (S.P. 1.5, 2.1, 2.2) • 3.A.1.3 The student is able to analyze experimental data describing the motion of an object and is able to express the results of the analysis using narrative, mathematical, and graphical representations. (S.P. 5.1) Figure 3.8 Displacement can be determined graphically using a scale map, such as this one of the Hawaiian Islands. A journey from Hawai'i to Moloka'i has a number of legs, or journey segments. These segments can be added graphically with a ruler to determine the total two-dimensional displacement of the journey. (credit: US Geological Survey) Vectors in Two Dimensions A vector is a quantity that has magnitude and direction. Displacement, velocity, acceleration, and force, for example, are all vectors. In one-dimensional, or straight-line, motion, the direction of a vector can be given simply by a plus or minus sign. In two dimensions (2-d), however, we specify the direction of a vector relative to some reference frame (i.e., coordinate system), using an arrow having length proportional to the vector's magnitude and pointing in the direction of the vector. Figure 3.9 shows such a graphical representation of a vector, using as an example the total displacement for the person walking in a city considered in Kinematics in Two Dimensions: An Introduction. We shall use the notation that a boldface symbol, such as D , stands for a vector. Its magnitude is represented by the symbol in italics, , and its direction by . 100 Chapter 3 \| Two-Dimensional Kinematics Vectors in this Text In this text, we will represent a vector with a boldface variable. For example, we will represent the quantity force with the vector F , which has both magnitude and direction. The magnitude of the vector will be represented by a variable in italics, such as , and the direction of the variable will be given by an angle . Figure 3.9 A person walks 9 blocks east and 5 blocks north. The displacement is 10.3 blocks at an angle 29.1° north of east. Figure 3.10 To describe the resultant vector for the person walking in a city considered in Figure 3.9 graphically, draw an arrow to represent the total displacement vector D . Using a protractor, draw a line at an angle relative to the east-west axis. The length of the arrow is proportional to the vector's magnitude and is measured along the line with a ruler. In this example, the magnitude of the vector is 10.3 units, and the direction is 29.1° north of east. Vector Addition: Head-to-Tail Method The head-to-tail method is a graphical way to add vectors, described in Figure 3.11 below and in the steps following. The tail of the vector is the starting point of the vector, and the head (or tip) of a vector is the final, pointed end of the arrow. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 3 \| Two-Dimensional Kinematics 101 Figure 3.11 Head-to-Tail Method: The head-to-tail method of graphically adding vectors is illustrated for the two displacements of the person walking in a city considered in Figure 3.9. (a) Draw a vector representing the displacement to the east. (b) Draw a vector representing the displacement to the north. The tail of this vector should originate from the head of the first, east-pointing vector. (c) Draw a line from the tail of the east-pointing vector to the head of the north-pointing vector to form the sum or resultant vector D . The length of the arrow D is proportional to the vector's magnitude and is measured to be 10.3 units . Its direction, described as the angle with respect to the east (or horizontal axis) is measured with a protractor to be 29.1° . Step 1. Draw an arrow to represent the first vector (9 blocks to the east) using a ruler and protractor. Figure 3.12 Step 2. Now draw an arrow to represent the second vector (5 blocks to the north). Place the tail of the second vector at the head of the first vector. Figure 3.13 Step 3. If there are more than two vectors, continue this process for each vector to be added. Note that in our example, we have only two vectors, so we have finished placing arrows tip to tail. Step 4. Draw an arrow from the tail of the first vector to the head of the last vector. This is the resultant, or the sum, of the other vectors. 102 Chapter 3 \| Two-Dimensional Kinematics Figure 3.14 Step 5. To get the magnitude of the resultant, measure its length with a ruler. (Note that in most calculations, we will use the Pythagorean theorem to determine this length.) Step 6. To get the direction of the resultant, measure the angle it makes with the reference frame using a protractor. (Note that in most calculations, we will use trigonometric relationships to determine this angle.) The graphical addition of vectors is limited in accuracy only by the precision with which the drawings can be made and the precision of the measuring tools. It is valid for any number of vectors. Example 3.1 Adding Vectors Graphically Using the Head-to-Tail Method: A Woman Takes a Walk Use the graphical technique for adding vectors to find the total displacement of a person who walks the following three paths (displacements) on a flat field. First, she walks 25.0 m in a direction 49.0° north of east. Then, she walks 23.0 m heading 15.0° north of east. Finally, she turns and walks 32.0 m in a direction 68.0° south of east. Strategy Represent each displacement vector graphically with an arrow, labeling the first A , the second B , and the third C , making the lengths proportional to the distance and the directions as specified relative to an east-west line. The head-to-tail method outlined above will give a way to determine the magnitude and direction of the resultant displacement, denoted R . Solution (1) Draw the three displacement vectors. Figure 3.15 (2) Place the vectors head to tail retaining both their initial magnitude and direction. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 3 \| Two-Dimensional Kinematics 103 Figure 3.16 (3) Draw the resultant vector, R . Figure 3.17 (4) Use a ruler to measure the magnitude of R , and a protractor to measure the direction of R . While the direction of the vector can be specified in many ways, the easiest way is to measure the angle between the vector and the nearest horizontal or vertical axis. Since the resultant vector is south of the eastward pointing axis, we flip the protractor upside down and measure the angle between the eastward axis and the vector. Figure 3.18 In this case, the total displacement R is seen to have a magnitude of 50.0 m and to lie in a direction 7.0° south of east. By using its magnitude and direction, this vector can be expressed as = 50.0 m and = 7.0° south of east. Discussion The head-to-tail graphical method of vector addition works for any number of vectors. It is also important to note that the resultant is independent of the order in which the vectors are added. Therefore, we could add the vectors in any order as illustrated in Figure 3.19 and we will still get the same solution. 104 Chapter 3 \| Two-Dimensional Kinematics Figure 3.19 Here, we see that when the same vectors are added in a different order, the result is the same. This characteristic is true in every case and is an important characteristic of vectors. Vector addition is commutative. Vectors can be added in any order. A + B = B + A. (3.1) (This is true for the addition of ordinary numbers as well—you get the same result whether you add 2 + 3 or 3 + 2 , for example). Vector Subtraction Vector subtraction is a straightforward extension of vector addition. To define subtraction (say we want to subtract B from A , written A – B , we must first define what we mean by subtraction. The negative of a vector B is defined to be –B ; that is, graphically the negative of any vector has the same magnitude but the opposite direction, as shown in Figure 3.20. In other words, B has the same length as –B , but points in the opposite direction. Essentially, we just flip the vector so it points in the opposite direction. Figure 3.20 The negative of a vector is just another vector of the same magnitude but pointing in the opposite direction. So B is the negative of –B ; it has the same length but opposite direction. The subtraction of vector B from vector A is then simply defined to be the addition of –B to A . Note that vector subtraction is the addition of a negative vector. The order of subtraction does not affect the results. A – B = A + (–B). (3.2) This is analogous to the subtraction of scalars (where, for example, 5 – 2 = 5 + (–2) ). Again, the result is independent of the order in which the subtraction is made. When vectors are subtracted graphically, the techniques outlined above are used, as the following example illustrates. Example 3.2 Subtracting Vectors Graphically: A Woman Sailing a Boat A woman sailing a boat at night is following directions to a dock. The instructions read to first sail 27.5 m in a direction 66.0° north of east from her current location, and then travel 30.0 m in a direction 112° north of east (or 22.0° west of This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 3 \| Two-Dimensional Kinematics 105 north). If the woman makes a mistake and travels in the opposite direction for the second leg of the trip, where
will she end up? Compare this location with the location of the dock. Figure 3.21 Strategy We can represent the first leg of the trip with a vector A , and the second leg of the trip with a vector B . The dock is located at a location A + B . If the woman mistakenly travels in the opposite direction for the second leg of the journey, she will travel a distance (30.0 m) in the direction 180° – 112° = 68° south of east. We represent this as –B , as shown below. The vector –B has the same magnitude as B but is in the opposite direction. Thus, she will end up at a location A + (–B) , or A – B . Figure 3.22 We will perform vector addition to compare the location of the dock, A + B , with the location at which the woman mistakenly arrives, A + (–B) . Solution (1) To determine the location at which the woman arrives by accident, draw vectors A and –B . (2) Place the vectors head to tail. (3) Draw the resultant vector R . (4) Use a ruler and protractor to measure the magnitude and direction of R . Figure 3.23 In this case, = 23.0 m and = 7.5° south of east. 106 Chapter 3 \| Two-Dimensional Kinematics (5) To determine the location of the dock, we repeat this method to add vectors A and B . We obtain the resultant vector R ' : Figure 3.24 In this case = 52.9 m and = 90.1° north of east. We can see that the woman will end up a significant distance from the dock if she travels in the opposite direction for the second leg of the trip. Discussion Because subtraction of a vector is the same as addition of a vector with the opposite direction, the graphical method of subtracting vectors works the same as for addition. Multiplication of Vectors and Scalars If we decided to walk three times as far on the first leg of the trip considered in the preceding example, then we would walk 3 × 27.5 m , or 82.5 m, in a direction 66.0° north of east. This is an example of multiplying a vector by a positive scalar. Notice that the magnitude changes, but the direction stays the same. If the scalar is negative, then multiplying a vector by it changes the vector's magnitude and gives the new vector the opposite direction. For example, if you multiply by –2, the magnitude doubles but the direction changes. We can summarize these rules in the following way: When vector A is multiplied by a scalar , • • • the magnitude of the vector becomes the absolute value of , if is positive, the direction of the vector does not change, if is negative, the direction is reversed. In our case, = 3 and = 27.5 m . Vectors are multiplied by scalars in many situations. Note that division is the inverse of multiplication. For example, dividing by 2 is the same as multiplying by the value (1/2). The rules for multiplication of vectors by scalars are the same for division; simply treat the divisor as a scalar between 0 and 1. Resolving a Vector into Components In the examples above, we have been adding vectors to determine the resultant vector. In many cases, however, we will need to do the opposite. We will need to take a single vector and find what other vectors added together produce it. In most cases, this involves determining the perpendicular components of a single vector, for example the x- and y-components, or the north-south and east-west components. For example, we may know that the total displacement of a person walking in a city is 10.3 blocks in a direction 29.0° north of east and want to find out how many blocks east and north had to be walked. This method is called finding the components (or parts) of the displacement in the east and north directions, and it is the inverse of the process followed to find the total displacement. It is one example of finding the components of a vector. There are many applications in physics where this is a useful thing to do. We will see this soon in Projectile Motion, and much more when we cover forces in Dynamics: Newton's Laws of Motion. Most of these involve finding components along perpendicular axes (such as north and east), so that right triangles are involved. The analytical techniques presented in Vector Addition and Subtraction: Analytical Methods are ideal for finding vector components. PhET Explorations: Maze Game Learn about position, velocity, and acceleration in the "Arena of Pain". Use the green arrow to move the ball. Add more walls to the arena to make the game more difficult. Try to make a goal as fast as you can. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 3 \| Two-Dimensional Kinematics 107 Figure 3.25 Maze Game (http://cnx.org/content/m54781/1.2/maze-game_en.jar) 3.3 Vector Addition and Subtraction: Analytical Methods Learning Objectives By the end of this section, you will be able to: • Understand the rules of vector addition and subtraction using analytical methods. • Apply analytical methods to determine vertical and horizontal component vectors. • Apply analytical methods to determine the magnitude and direction of a resultant vector. The information presented in this section supports the following AP® learning objectives and science practices: • 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, and graphical representations. (S.P. 1.5, 2.1, 2.2) Analytical methods of vector addition and subtraction employ geometry and simple trigonometry rather than the ruler and protractor of graphical methods. Part of the graphical technique is retained, because vectors are still represented by arrows for easy visualization. However, analytical methods are more concise, accurate, and precise than graphical methods, which are limited by the accuracy with which a drawing can be made. Analytical methods are limited only by the accuracy and precision with which physical quantities are known. Resolving a Vector into Perpendicular Components Analytical techniques and right triangles go hand-in-hand in physics because (among other things) motions along perpendicular directions are independent. We very often need to separate a vector into perpendicular components. For example, given a vector like A in Figure 3.26, we may wish to find which two perpendicular vectors, A and A , add to produce it. Figure 3.26 The vector A , with its tail at the origin of an x, y-coordinate system, is shown together with its x- and y-components, A and A . These vectors form a right triangle. The analytical relationships among these vectors are summarized below. A and A are defined to be the components of A along the x- and y-axes. The three vectors A , A , and A form a right triangle: A + Ay = A. (3.3) Note that this relationship between vector components and the resultant vector holds only for vector quantities (which include both magnitude and direction). The relationship does not apply for the magnitudes alone. For example, if A = 3 m east, A = 4 m north, and A = 5 m north-east, then it is true that the vectors A + Ay = A . However, it is not true that the sum of the magnitudes of the vectors is also equal. That is3.4) Thus, 108 Chapter 3 \| Two-Dimensional Kinematics If the vector A is known, then its magnitude (its length) and its angle (its direction) are known. To find and , its xand y-components, we use the following relationships for a right triangle. + ≠ (3.5) and = cos = sin . (3.6) (3.7) Figure 3.27 The magnitudes of the vector components A and A can be related to the resultant vector A and the angle with trigonometric identities. Here we see that = cos and = sin . Suppose, for example, that A is the vector representing the total displacement of the person walking in a city considered in Kinematics in Two Dimensions: An Introduction and Vector Addition and Subtraction: Graphical Methods. Figure 3.28 We can use the relationships = cos and = sin to determine the magnitude of the horizontal and vertical component vectors in this example. Then = 10.3 blocks and = 29.1º , so that = cos = = sin = 10.3 blocks cos 29.1º sin 29.1º 10.3 blocks = 9.0 blocks = 5.0 blocks. (3.8) (3.9) Calculating a Resultant Vector If the perpendicular components A and A of a vector A are known, then A can also be found analytically. To find the magnitude and direction of a vector from its perpendicular components A and A , we use the following relationships: = 2 + 2 = tan−1( / ). (3.10) (3.11) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 3 \| Two-Dimensional Kinematics 109 Figure 3.29 The magnitude and direction of the resultant vector can be determined once the horizontal and vertical components and have been determined. Note that the equation = hypotenuse. For example, if and are 9 and 5 blocks, respectively, then = 92 +52=10.3 blocks, again consistent with the example of the person walking in a city. Finally, the direction is = tan–1(5/9)=29.1º , as before. 2 is just the Pythagorean theorem relating the legs of a right triangle to the length of the 2 + Determining Vectors and Vector Components with Analytical Methods Equations = cos and = sin are used to find the perpendicular components of a vector—that is, to go 2 and = tan–1( / ) are used to find a vector from its from and to and . Equations = perpendicular components—that is, to go from and to and . Both processes are crucial to analytical methods of vector addition and subtraction. 2 + Adding Vectors Using Analytical Methods To see how to add vectors using perpendicular components, consider Figure 3.30, in which the vectors A and B are added to produce the resultant R . Figure 3.30 Vectors A and B are two legs of a walk, and R is the resultant or total displacement. You can use analytical methods to determine the magnitude and direction of R . If A and B represent two legs of a walk (two displacements), then R is the total displacement. The person taking the walk ends up at the tip of R. There are many ways to arrive at the same point. In particular, the person could have walked first in the x-direction and then in the y-direction. Those paths are the x- and y-components of the r
esultant, R and R . If we know R 2 and = tan–1( / ) . When you use the analytical and R , we can find and using the equations = method of vector addition, you can determine the components or the magnitude and direction of a vector. 2 + Step 1. Identify the x- and y-axes that will be used in the problem. Then, find the components of each vector to be added along the chosen perpendicular axes. Use the equations = cos and = sin to find the components. In Figure 3.31, 110 Chapter 3 \| Two-Dimensional Kinematics these components are , , , and . The angles that vectors A and B make with the x-axis are A and B , respectively. Figure 3.31 To add vectors A and B , first determine the horizontal and vertical components of each vector. These are the dotted vectors A , A , B and B shown in the image. Step 2. Find the components of the resultant along each axis by adding the components of the individual vectors along that axis. That is, as shown in Figure 3.32, and = + = + . (3.12) (3.13) Figure 3.32 The magnitude of the vectors A and B add to give the magnitude of the resultant vector in the horizontal direction. Similarly, the magnitudes of the vectors A and B add to give the magnitude of the resultant vector in the vertical direction. Components along the same axis, say the x-axis, are vectors along the same line and, thus, can be added to one another like ordinary numbers. The same is true for components along the y-axis. (For example, a 9-block eastward walk could be taken in two legs, the first 3 blocks east and the second 6 blocks east, for a total of 9, because they are along the same direction.) So resolving vectors into components along common axes makes it easier to add them. Now that the components of R are known, its magnitude and direction can be found. Step 3. To get the magnitude of the resultant, use the Pythagorean theorem: Step 4. To get the direction of the resultant: = 2. 2 + = tan−1( / ). The following example illustrates this technique for adding vectors using perpendicular components. (3.14) (3.15) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 3 \| Two-Dimensional Kinematics 111 Example 3.3 Adding Vectors Using Analytical Methods Add the vector A to the vector B shown in Figure 3.33, using perpendicular components along the x- and y-axes. The xand y-axes are along the east–west and north–south directions, respectively. Vector A represents the first leg of a walk in which a person walks 53.0 m in a direction 20.0º north of east. Vector B represents the second leg, a displacement of 34.0 m in a direction 63.0º north of east. Figure 3.33 Vector A has magnitude 53.0 m and direction 20.0 º north of the x-axis. Vector B has magnitude 34.0 m and direction 63.0º north of the x-axis. You can use analytical methods to determine the magnitude and direction of R . Strategy The components of A and B along the x- and y-axes represent walking due east and due north to get to the same ending point. Once found, they are combined to produce the resultant. Solution Following the method outlined above, we first find the components of A and B along the x- and y-axes. Note that = 53.0 m , A = 20.0º , = 34.0 m , and B = 63.0º . We find the x-components by using = cos , which gives and = cos A = (53.0 m)(cos 20.0º) = (53.0 m)(0.940) = 49.8 m = cos B = (34.0 m)(cos 63.0º) = (34.0 m)(0.454) = 15.4 m. Similarly, the y-components are found using = sin A : and = sin A = (53.0 m)(sin 20.0º) = (53.0 m)(0.342) = 18.1 m = sin B = (34.0 m)(sin 63.0 º ) = (34.0 m)(0.891) = 30.3 m. The x- and y-components of the resultant are thus and = + = 49.8 m + 15.4 m = 65.2 m = + = 18.1 m+30.3 m = 48.4 m. Now we can find the magnitude of the resultant by using the Pythagorean theorem: = 2 + 2 = (65.2)2 + (48.4)2 m so that = 81.2 m. (3.16) (3.17) (3.18) (3.19) (3.20) (3.21) (3.22) (3.23) 112 Chapter 3 \| Two-Dimensional Kinematics Finally, we find the direction of the resultant: Thus, = tan−1( / )=+tan−1(48.4 / 65.2). = tan−1(0.742) = 36.6 º . (3.24) (3.25) Figure 3.34 Using analytical methods, we see that the magnitude of R is 81.2 m and its direction is 36.6º north of east. Discussion This example illustrates the addition of vectors using perpendicular components. Vector subtraction using perpendicular components is very similar—it is just the addition of a negative vector. Subtraction of vectors is accomplished by the addition of a negative vector. That is, A − B ≡ A + (–B) . Thus, the method for the subtraction of vectors using perpendicular components is identical to that for addition. The components of –B are the negatives of the components of B . The x- and y-components of the resultant A − B = R are thus and = + – = + – and the rest of the method outlined above is identical to that for addition. (See Figure 3.35.) (3.26) (3.27) Analyzing vectors using perpendicular components is very useful in many areas of physics, because perpendicular quantities are often independent of one another. The next module, Projectile Motion, is one of many in which using perpendicular components helps make the picture clear and simplifies the physics. Figure 3.35 The subtraction of the two vectors shown in Figure 3.30. The components of –B are the negatives of the components of B . The method of subtraction is the same as that for addition. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 3 \| Two-Dimensional Kinematics 113 PhET Explorations: Vector Addition Learn how to add vectors. Drag vectors onto a graph, change their length and angle, and sum them together. The magnitude, angle, and components of each vector can be displayed in several formats. Figure 3.36 Vector Addition (http://cnx.org/content/m54783/1.2/vector-addition_en.jar) 3.4 Projectile Motion By the end of this section, you will be able to: Learning Objectives • Identify and explain the properties of a projectile, such as acceleration due to gravity, range, maximum height, and trajectory. • Determine the location and velocity of a projectile at different points in its trajectory. • Apply the principle of independence of motion to solve projectile motion problems. The information presented in this section supports the following AP® learning objectives: • 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, and graphical representations. (S.P. 1.5, 2.1, 2.2) • 3.A.1.3 The student is able to analyze experimental data describing the motion of an object and is able to express the results of the analysis using narrative, mathematical, and graphical representations. (S.P. 5.1) Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. The object is called a projectile, and its path is called its trajectory. The motion of falling objects, as covered in Problem-Solving Basics for One-Dimensional Kinematics, is a simple one-dimensional type of projectile motion in which there is no horizontal movement. In this section, we consider two-dimensional projectile motion, such as that of a football or other object for which air resistance is negligible. The most important fact to remember here is that motions along perpendicular axes are independent and thus can be analyzed separately. This fact was discussed in Kinematics in Two Dimensions: An Introduction, where vertical and horizontal motions were seen to be independent. The key to analyzing two-dimensional projectile motion is to break it into two motions, one along the horizontal axis and the other along the vertical. (This choice of axes is the most sensible, because acceleration due to gravity is vertical—thus, there will be no acceleration along the horizontal axis when air resistance is negligible.) As is customary, we call the horizontal axis the x-axis and the vertical axis the y-axis. Figure 3.37 illustrates the notation for displacement, where s is defined to be the total displacement and x and y are its components along the horizontal and vertical axes, respectively. The magnitudes of these vectors are s, x, and y. (Note that in the last section we used the notation A to represent a vector with components A and A . If we continued this format, we would call displacement s with components s and s . However, to simplify the notation, we will simply represent the component vectors as x and y .) Of course, to describe motion we must deal with velocity and acceleration, as well as with displacement. We must find their components along the x- and y-axes, too. We will assume all forces except gravity (such as air resistance and friction, for example) are negligible. The components of acceleration are then very simple: = – = – 9.80 m/s2 . (Note that this definition assumes that the upwards direction is defined as the positive direction. If you arrange the coordinate system instead such that the downwards direction is positive, then acceleration due to gravity takes a positive value.) Because gravity is vertical, = 0 . Both accelerations are constant, so the kinematic equations can be used. Review of Kinematic Equations (constant ) = ( − 0). (3.28) (3.29) (3.30) (3.31) (3.32) 114 Chapter 3 \| Two-Dimensional Kinematics Figure 3.37 The total displacement s of a soccer ball at a point along its path. The vector s has components x and y along the horizontal and vertical axes. Its magnitude is , and it makes an angle with the horizontal. Given these assumptions, the following steps are then used to analyze projectile motion: Step 1. Resolve or break the motion into horizontal and vertical components along the x- and y-axes. These axes are perpendicular, so = cos and = sin are used. The magnitude of the components of displacement s along these axes are and The magnitudes of the components of the velocity v are = cos and = sin θ, where is the magnitude of the velocity and is its direction, as shown in Figure 3.38. Initial values are denoted with a subscript 0, as usual. Step 2. Treat the moti
on as two independent one-dimensional motions, one horizontal and the other vertical. The kinematic equations for horizontal and vertical motion take the following forms: Horizontal Motion( = 0) = 0 + = 0 = = velocity is a constant. Vertical Motion(assuming positive is up = − = −9.80m/s2) (( − 0). (3.33) (3.34) (3.35) (3.36) (3.37) (3.38) (3.39) (3.40) Step 3. Solve for the unknowns in the two separate motions—one horizontal and one vertical. Note that the only common variable between the motions is time . The problem solving procedures here are the same as for one-dimensional kinematics and are illustrated in the solved examples below. Step 4. Recombine the two motions to find the total displacement s and velocity v . Because the x - and y -motions are perpendicular, we determine these vectors by using the techniques outlined in the Vector Addition and Subtraction: Analytical Methods and employing = displacement s and is the direction of the velocity v : 2 and = tan−1( / ) in the following form, where is the direction of the 2 + Total displacement and velocity = 2 + 2 = tan−1( / ) 2 2 + = = tan−1( / ). (3.41) (3.42) (3.43) (3.44) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 3 \| Two-Dimensional Kinematics 115 Figure 3.38 (a) We analyze two-dimensional projectile motion by breaking it into two independent one-dimensional motions along the vertical and horizontal axes. (b) The horizontal motion is simple, because = 0 and is thus constant. (c) The velocity in the vertical direction begins to decrease as the object rises; at its highest point, the vertical velocity is zero. As the object falls towards the Earth again, the vertical velocity increases again in magnitude but points in the opposite direction to the initial vertical velocity. (d) The x - and y -motions are recombined to give the total velocity at any given point on the trajectory. Example 3.4 A Fireworks Projectile Explodes High and Away During a fireworks display, a shell is shot into the air with an initial speed of 70.0 m/s at an angle of 75.0° above the horizontal, as illustrated in Figure 3.39. The fuse is timed to ignite the shell just as it reaches its highest point above the ground. (a) Calculate the height at which the shell explodes. (b) How much time passed between the launch of the shell and the explosion? (c) What is the horizontal displacement of the shell when it explodes? Strategy Because air resistance is negligible for the unexploded shell, the analysis method outlined above can be used. The motion can be broken into horizontal and vertical motions in which = 0 and = – . We can then define 0 and 0 to be zero and solve for the desired quantities. Solution for (a) 116 Chapter 3 \| Two-Dimensional Kinematics By “height” we mean the altitude or vertical position above the starting point. The highest point in any trajectory, called the apex, is reached when = 0 . Since we know the initial and final velocities as well as the initial position, we use the following equation to find : 2 = 0 2 − 2( − 0). (3.45) Figure 3.39 The trajectory of a fireworks shell. The fuse is set to explode the shell at the highest point in its trajectory, which is found to be at a height of 233 m and 125 m away horizontally. Because 0 and are both zero, the equation simplifies to Solving for gives 3.46) (3.47) Now we must find 0 , the component of the initial velocity in the y-direction. It is given by 0 = 0 sin , where 0 is the initial velocity of 70.0 m/s, and 0 = 75.0° is the initial angle. Thus, 0 = 0 sin 0 = (70.0 m/s)(sin 75°) = 67.6 m/s. and is so that Discussion for (a) = (67.6 m/s)2 2(9.80 m/s2) , = 233m. (3.48) (3.49) (3.50) Note that because up is positive, the initial velocity is positive, as is the maximum height, but the acceleration due to gravity is negative. Note also that the maximum height depends only on the vertical component of the initial velocity, so that any projectile with a 67.6 m/s initial vertical component of velocity will reach a maximum height of 233 m (neglecting air resistance). The numbers in this example are reasonable for large fireworks displays, the shells of which do reach such heights before exploding. In practice, air resistance is not completely negligible, and so the initial velocity would have to be somewhat larger than that given to reach the same height. Solution for (b) As in many physics problems, there is more than one way to solve for the time to the highest point. In this case, the easiest method is to use = 0 + 1 (0 + ) . Because 0 is zero, this equation reduces to simply 2 This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 3 \| Two-Dimensional Kinematics Note that the final vertical velocity, , at the highest point is zero. Thus, = 1 2 (0 + ). = 2 (0y + ) = 2(233 m) (67.6 m/s) = 6.90 s. 117 (3.51) (3.52) Discussion for (b) This time is also reasonable for large fireworks. When you are able to see the launch of fireworks, you will notice several seconds pass before the shell explodes. (Another way of finding the time is by using = 0 + 0 − 1 the quadratic equation for .) 2 2 , and solving Solution for (c) Because air resistance is negligible, = 0 and the horizontal velocity is constant, as discussed above. The horizontal displacement is horizontal velocity multiplied by time as given by = 0 + , where 0 is equal to zero: = , where is the x-component of the velocity, which is given by = 0 cos 0 . Now, = 0 cos 0 = (70.0 m/s)(cos 75.0°) = 18.1 m/s. The time for both motions is the same, and so is = (18.1 m/s)(6.90 s) = 125 m. Discussion for (c) (3.53) (3.54) (3.55) The horizontal motion is a constant velocity in the absence of air resistance. The horizontal displacement found here could be useful in keeping the fireworks fragments from falling on spectators. Once the shell explodes, air resistance has a major effect, and many fragments will land directly below. In solving part (a) of the preceding example, the expression we found for is valid for any projectile motion where air resistance is negligible. Call the maximum height = ; then, = 2 0 2 . (3.56) This equation defines the maximum height of a projectile and depends only on the vertical component of the initial velocity. Defining a Coordinate System It is important to set up a coordinate system when analyzing projectile motion. One part of defining the coordinate system is to define an origin for the and positions. Often, it is convenient to choose the initial position of the object as the origin such that 0 = 0 and 0 = 0 . It is also important to define the positive and negative directions in the and directions. Typically, we define the positive vertical direction as upwards, and the positive horizontal direction is usually the direction of the object's motion. When this is the case, the vertical acceleration, , takes a negative value (since it is directed downwards towards the Earth). However, it is occasionally useful to define the coordinates differently. For example, if you are analyzing the motion of a ball thrown downwards from the top of a cliff, it may make sense to define the positive direction downwards since the motion of the ball is solely in the downwards direction. If this is the case, takes a positive value. Example 3.5 Calculating Projectile Motion: Hot Rock Projectile Kilauea in Hawaii is the world's most continuously active volcano. Very active volcanoes characteristically eject red-hot rocks and lava rather than smoke and ash. Suppose a large rock is ejected from the volcano with a speed of 25.0 m/s and at an angle 35.0° above the horizontal, as shown in Figure 3.40. The rock strikes the side of the volcano at an altitude 20.0 m lower than its starting point. (a) Calculate the time it takes the rock to follow this path. (b) What are the magnitude and direction of the rock's velocity at impact? 118 Chapter 3 \| Two-Dimensional Kinematics Figure 3.40 The trajectory of a rock ejected from the Kilauea volcano. Strategy Again, resolving this two-dimensional motion into two independent one-dimensional motions will allow us to solve for the desired quantities. The time a projectile is in the air is governed by its vertical motion alone. We will solve for first. While the rock is rising and falling vertically, the horizontal motion continues at a constant velocity. This example asks for the final velocity. Thus, the vertical and horizontal results will be recombined to obtain and at the final time determined in the first part of the example. Solution for (a) While the rock is in the air, it rises and then falls to a final position 20.0 m lower than its starting altitude. We can find the time for this by using = 0 + 0 − 1 2 2. (3.57) If we take the initial position 0 to be zero, then the final position is = −20.0 m. Now the initial vertical velocity is the vertical component of the initial velocity, found from 0 = 0 sin 0 = ( 25.0 m/s )( sin 35.0° ) = 14.3 m/s . Substituting known values yields Rearranging terms gives a quadratic equation in : −20.0 m = (14.3 m/s) − 4.90 m/s2 2. 4.90 m/s2 2 − (14.3 m/s) − (20.0 m) = 0. (3.58) (3.59) This expression is a quadratic equation of the form 2 + + = 0 , where the constants are = 4.90 , = – 14.3 , and = – 20.0. Its solutions are given by the quadratic formula3.60) This equation yields two solutions: = 3.96 and = – 1.03 . (It is left as an exercise for the reader to verify these solutions.) The time is = 3.96 s or – 1.03 s . The negative value of time implies an event before the start of motion, and so we discard it. Thus, = 3.96 s. (3.61) Discussion for (a) The time for projectile motion is completely determined by the vertical motion. So any projectile that has an initial vertical velocity of 14.3 m/s and lands 20.0 m below its starting altitude will spend 3.96 s in the air. Solution for (b) From the information now in hand, we can find the final horizontal and vertical velocities and and combine them to find the total velocity and the angle 0
it makes with the horizontal. Of course, is constant so we can solve for it at any horizontal location. In this case, we chose the starting point since we know both the initial velocity and initial angle. Therefore: = 0 cos 0 = (25.0 m/s)(cos 35°) = 20.5 m/s. The final vertical velocity is given by the following equation: = 0 − (3.62) (3.63) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 3 \| Two-Dimensional Kinematics 119 where 0y was found in part (a) to be 14.3 m/s . Thus, so that = 14.3 m/s − (9.80 m/s2)(3.96 s) = −24.5 m/s. To find the magnitude of the final velocity we combine its perpendicular components, using the following equation: = 2 + 2 = (20.5 m/s)2 + ( − 24.5 m/s)2, which gives The direction is found from the equation: = 31.9 m/s. = tan−1( / ) so that Thus, Discussion for (b) = tan−1( − 24.5 / 20.5) = tan−1( − 1.19). = −50.1 ° . (3.64) (3.65) (3.66) (3.67) (3.68) (3.69) (3.70) The negative angle means that the velocity is 50.1° below the horizontal. This result is consistent with the fact that the final vertical velocity is negative and hence downward—as you would expect because the final altitude is 20.0 m lower than the initial altitude. (See Figure 3.40.) One of the most important things illustrated by projectile motion is that vertical and horizontal motions are independent of each other. Galileo was the first person to fully comprehend this characteristic. He used it to predict the range of a projectile. On level ground, we define range to be the horizontal distance traveled by a projectile. Galileo and many others were interested in the range of projectiles primarily for military purposes—such as aiming cannons. However, investigating the range of projectiles can shed light on other interesting phenomena, such as the orbits of satellites around the Earth. Let us consider projectile range further. Figure 3.41 Trajectories of projectiles on level ground. (a) The greater the initial speed 0 , the greater the range for a given initial angle. (b) The effect of initial angle 0 on the range of a projectile with a given initial speed. Note that the range is the same for 15° and 75° , although the maximum heights of those paths are different. 120 Chapter 3 \| Two-Dimensional Kinematics How does the initial velocity of a projectile affect its range? Obviously, the greater the initial speed 0 , the greater the range, as shown in Figure 3.41(a). The initial angle 0 also has a dramatic effect on the range, as illustrated in Figure 3.41(b). For a fixed initial speed, such as might be produced by a cannon, the maximum range is obtained with 0 = 45° . This is true only for conditions neglecting air resistance. If air resistance is considered, the maximum angle is approximately 38° . Interestingly, for every initial angle except 45° , there are two angles that give the same range—the sum of those angles is 90° . The range also depends on the value of the acceleration of gravity . The lunar astronaut Alan Shepherd was able to drive a golf ball a great distance on the Moon because gravity is weaker there. The range of a projectile on level ground for which air resistance is negligible is given by = 2 sin 2 0 0 , (3.71) where 0 is the initial speed and 0 is the initial angle relative to the horizontal. The proof of this equation is left as an end-ofchapter problem (hints are given), but it does fit the major features of projectile range as described. When we speak of the range of a projectile on level ground, we assume that is very small compared with the circumference of the Earth. If, however, the range is large, the Earth curves away below the projectile and acceleration of gravity changes direction along the path. The range is larger than predicted by the range equation given above because the projectile has farther to fall than it would on level ground. (See Figure 3.42.) If the initial speed is great enough, the projectile goes into orbit. This possibility was recognized centuries before it could be accomplished. When an object is in orbit, the Earth curves away from underneath the object at the same rate as it falls. The object thus falls continuously but never hits the surface. These and other aspects of orbital motion, such as the rotation of the Earth, will be covered analytically and in greater depth later in this text. Once again we see that thinking about one topic, such as the range of a projectile, can lead us to others, such as the Earth orbits. In Addition of Velocities, we will examine the addition of velocities, which is another important aspect of two-dimensional kinematics and will also yield insights beyond the immediate topic. Figure 3.42 Projectile to satellite. In each case shown here, a projectile is launched from a very high tower to avoid air resistance. With increasing initial speed, the range increases and becomes longer than it would be on level ground because the Earth curves away underneath its path. With a large enough initial speed, orbit is achieved. PhET Explorations: Projectile Motion Blast a Buick out of a cannon! Learn about projectile motion by firing various objects. Set the angle, initial speed, and mass. Add air resistance. Make a game out of this simulation by trying to hit a target. Figure 3.43 Projectile Motion (http://cnx.org/content/m54787/1.2/projectile-motion_en.jar) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 3 \| Two-Dimensional Kinematics 121 3.5 Addition of Velocities Learning Objectives By the end of this section, you will be able to: • Apply principles of vector addition to determine relative velocity. • Explain the significance of the observer in the measurement of velocity. The information presented in this section supports the following AP® learning objectives and science practices: • 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, and graphical representations. (S.P. 1.5, 2.1, 2.2) • 3.A.1.3 The student is able to analyze experimental data describing the motion of an object and is able to express the results of the analysis using narrative, mathematical, and graphical representations. (S.P. 5.1) Relative Velocity If a person rows a boat across a rapidly flowing river and tries to head directly for the other shore, the boat instead moves diagonally relative to the shore, as in Figure 3.44. The boat does not move in the direction in which it is pointed. The reason, of course, is that the river carries the boat downstream. Similarly, if a small airplane flies overhead in a strong crosswind, you can sometimes see that the plane is not moving in the direction in which it is pointed, as illustrated in Figure 3.45. The plane is moving straight ahead relative to the air, but the movement of the air mass relative to the ground carries it sideways. Figure 3.44 A boat trying to head straight across a river will actually move diagonally relative to the shore as shown. Its total velocity (solid arrow) relative to the shore is the sum of its velocity relative to the river plus the velocity of the river relative to the shore. 122 Chapter 3 \| Two-Dimensional Kinematics Figure 3.45 An airplane heading straight north is instead carried to the west and slowed down by wind. The plane does not move relative to the ground in the direction it points; rather, it moves in the direction of its total velocity (solid arrow). In each of these situations, an object has a velocity relative to a medium (such as a river) and that medium has a velocity relative to an observer on solid ground. The velocity of the object relative to the observer is the sum of these velocity vectors, as indicated in Figure 3.44 and Figure 3.45. These situations are only two of many in which it is useful to add velocities. In this module, we first re-examine how to add velocities and then consider certain aspects of what relative velocity means. How do we add velocities? Velocity is a vector (it has both magnitude and direction); the rules of vector addition discussed in Vector Addition and Subtraction: Graphical Methods and Vector Addition and Subtraction: Analytical Methods apply to the addition of velocities, just as they do for any other vectors. In one-dimensional motion, the addition of velocities is simple—they add like ordinary numbers. For example, if a field hockey player is moving at 5 m/s straight toward the goal and drives the ball in the same direction with a velocity of 30 m/s relative to her body, then the velocity of the ball is 35 m/s relative to the stationary, profusely sweating goalkeeper standing in front of the goal. In two-dimensional motion, either graphical or analytical techniques can be used to add velocities. We will concentrate on analytical techniques. The following equations give the relationships between the magnitude and direction of velocity ( and ) and its components ( and ) along the x- and y-axes of an appropriately chosen coordinate system: = cos = sin 2 2 + = = tan−1( / ). (3.72) (3.73) (3.74) (3.75) Figure 3.46 The velocity, , of an object traveling at an angle to the horizontal axis is the sum of component vectors v and v . These equations are valid for any vectors and are adapted specifically for velocity. The first two equations are used to find the components of a velocity when its magnitude and direction are known. The last two are used to find the magnitude and direction of velocity when its components are known. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 3 \| Two-Dimensional Kinematics 123 Take-Home Experiment: Relative Velocity of a Boat Fill a bathtub half-full of water. Take a toy boat or some other object that floats in water. Unplug the drain so water starts to drain. Try pushing the boat from one side of the tub to the other and perpendicular to the flow of water. Which way do you need to push the boat so that it ends up immediately opposite? Compare the directions of the flow of water, heading
of the boat, and actual velocity of the boat. Example 3.6 Adding Velocities: A Boat on a River Figure 3.47 A boat attempts to travel straight across a river at a speed 0.75 m/s. The current in the river, however, flows at a speed of 1.20 m/s to the right. What is the total displacement of the boat relative to the shore? Refer to Figure 3.47, which shows a boat trying to go straight across the river. Let us calculate the magnitude and direction of the boat's velocity relative to an observer on the shore, vtot . The velocity of the boat, vboat , is 0.75 m/s in the direction relative to the river and the velocity of the river, vriver , is 1.20 m/s to the right. Strategy We start by choosing a coordinate system with its -axis parallel to the velocity of the river, as shown in Figure 3.47. Because the boat is directed straight toward the other shore, its velocity relative to the water is parallel to the -axis and perpendicular to the velocity of the river. Thus, we can add the two velocities by using the equations tot = = tan−1( / ) directly. 2 + 2 and Solution The magnitude of the total velocity is where and Thus, yielding tot = 2, 2 + = river = 1.20 m/s = boat = 0.750 m/s. tot = (1.20 m/s)2 + (0.750 m/s)2 tot = 1.42 m/s. The direction of the total velocity is given by: (3.76) (3.77) (3.78) (3.79) (3.80) 124 Chapter 3 \| Two-Dimensional Kinematics This equation gives Discussion = tan−1( / ) = tan−1(0.750 / 1.20). = 32.0º. (3.81) (3.82) Both the magnitude and the direction of the total velocity are consistent with Figure 3.47. Note that because the velocity of the river is large compared with the velocity of the boat, it is swept rapidly downstream. This result is evidenced by the small angle (only 32.0º ) the total velocity has relative to the riverbank. Example 3.7 Calculating Velocity: Wind Velocity Causes an Airplane to Drift Calculate the wind velocity for the situation shown in Figure 3.48. The plane is known to be moving at 45.0 m/s due north relative to the air mass, while its velocity relative to the ground (its total velocity) is 38.0 m/s in a direction 20.0º west of north. Figure 3.48 An airplane is known to be heading north at 45.0 m/s, though its velocity relative to the ground is 38.0 m/s at an angle west of north. What is the speed and direction of the wind? Strategy In this problem, somewhat different from the previous example, we know the total velocity vtot and that it is the sum of two other velocities, vw (the wind) and vp (the plane relative to the air mass). The quantity vp is known, and we are asked to find vw . None of the velocities are perpendicular, but it is possible to find their components along a common set of perpendicular axes. If we can find the components of vw , then we can combine them to solve for its magnitude and direction. As shown in Figure 3.48, we choose a coordinate system with its x-axis due east and its y-axis due north (parallel to vp ). (You may wish to look back at the discussion of the addition of vectors using perpendicular components in Vector Addition and Subtraction: Analytical Methods.) Solution Because vtot is the vector sum of the vw and vp , its x- and y-components are the sums of the x- and y-components of the wind and plane velocities. Note that the plane only has vertical component of velocity so p = 0 and p = p . That is, tot = w (3.83) and This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 3 \| Two-Dimensional Kinematics 125 tot = w + p. We can use the first of these two equations to find w : Because tot = 38.0 m / s and cos 110º = – 0.342 we have w = tot = totcos 110º. w = (38.0 m/s)(–0.342)=–13.0 m/s. The minus sign indicates motion west which is consistent with the diagram. Now, to find w we note that tot = w + p Here tot = totsin 110º ; thus, w = (38.0 m/s)(0.940) − 45.0 m/s = −9.29 m/s. (3.84) (3.85) (3.86) (3.87) (3.88) This minus sign indicates motion south which is consistent with the diagram. Now that the perpendicular components of the wind velocity w and w are known, we can find the magnitude and direction of vw . First, the magnitude is w = w 2 2 + w = ( − 13.0 m/s)2 + ( − 9.29 m/s)2 w = 16.0 m/s. = tan−1(w / w) = tan−1( − 9.29 / −13.0) = 35.6º. so that The direction is: giving Discussion (3.89) (3.90) (3.91) (3.92) The wind's speed and direction are consistent with the significant effect the wind has on the total velocity of the plane, as seen in Figure 3.48. Because the plane is fighting a strong combination of crosswind and head-wind, it ends up with a total velocity significantly less than its velocity relative to the air mass as well as heading in a different direction. Note that in both of the last two examples, we were able to make the mathematics easier by choosing a coordinate system with one axis parallel to one of the velocities. We will repeatedly find that choosing an appropriate coordinate system makes problem solving easier. For example, in projectile motion we always use a coordinate system with one axis parallel to gravity. Relative Velocities and Classical Relativity When adding velocities, we have been careful to specify that the velocity is relative to some reference frame. These velocities are called relative velocities. For example, the velocity of an airplane relative to an air mass is different from its velocity relative to the ground. Both are quite different from the velocity of an airplane relative to its passengers (which should be close to zero). Relative velocities are one aspect of relativity, which is defined to be the study of how different observers moving relative to each other measure the same phenomenon. Nearly everyone has heard of relativity and immediately associates it with Albert Einstein (1879–1955), the greatest physicist of the 20th century. Einstein revolutionized our view of nature with his modern theory of relativity, which we shall study in later chapters. The relative velocities in this section are actually aspects of classical relativity, first discussed correctly by Galileo and Isaac Newton. Classical relativity is limited to situations where speeds are less than about 1% of the speed of light—that is, less than 3,000 km/s . Most things we encounter in daily life move slower than this speed. Let us consider an example of what two different observers see in a situation analyzed long ago by Galileo. Suppose a sailor at the top of a mast on a moving ship drops his binoculars. Where will it hit the deck? Will it hit at the base of the mast, or will it hit behind the mast because the ship is moving forward? The answer is that if air resistance is negligible, the binoculars will hit at the base of the mast at a point directly below its point of release. Now let us consider what two different observers see when the binoculars drop. One observer is on the ship and the other on shore. The binoculars have no horizontal velocity relative to the observer on the ship, and so he sees them fall straight down the mast. (See Figure 3.49.) To the observer on shore, the 126 Chapter 3 \| Two-Dimensional Kinematics binoculars and the ship have the same horizontal velocity, so both move the same distance forward while the binoculars are falling. This observer sees the curved path shown in Figure 3.49. Although the paths look different to the different observers, each sees the same result—the binoculars hit at the base of the mast and not behind it. To get the correct description, it is crucial to correctly specify the velocities relative to the observer. Figure 3.49 Classical relativity. The same motion as viewed by two different observers. An observer on the moving ship sees the binoculars dropped from the top of its mast fall straight down. An observer on shore sees the binoculars take the curved path, moving forward with the ship. Both observers see the binoculars strike the deck at the base of the mast. The initial horizontal velocity is different relative to the two observers. (The ship is shown moving rather fast to emphasize the effect.) Example 3.8 Calculating Relative Velocity: An Airline Passenger Drops a Coin An airline passenger drops a coin while the plane is moving at 260 m/s. What is the velocity of the coin when it strikes the floor 1.50 m below its point of release: (a) Measured relative to the plane? (b) Measured relative to the Earth? Figure 3.50 The motion of a coin dropped inside an airplane as viewed by two different observers. (a) An observer in the plane sees the coin fall straight down. (b) An observer on the ground sees the coin move almost horizontally. Strategy Both problems can be solved with the techniques for falling objects and projectiles. In part (a), the initial velocity of the coin is zero relative to the plane, so the motion is that of a falling object (one-dimensional). In part (b), the initial velocity is 260 m/ This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 3 \| Two-Dimensional Kinematics 127 s horizontal relative to the Earth and gravity is vertical, so this motion is a projectile motion. In both parts, it is best to use a coordinate system with vertical and horizontal axes. Solution for (a) Using the given information, we note that the initial velocity and position are zero, and the final position is 1.50 m. The final velocity can be found using the equation: 2 = 0 2 − 2( − 0). Substituting known values into the equation, we get 2 = 02 − 2(9.80 m/s2)( − 1.50 m − 0 m) = 29.4 m2 /s2 yielding = −5.42 m/s. (3.93) (3.94) (3.95) We know that the square root of 29.4 has two roots: 5.42 and -5.42. We choose the negative root because we know that the velocity is directed downwards, and we have defined the positive direction to be upwards. There is no initial horizontal velocity relative to the plane and no horizontal acceleration, and so the motion is straight down relative to the plane. Solution for (b) Because the initial vertical velocity is zero relative to the ground and vertical motion is independent of hori
zontal motion, the final vertical velocity for the coin relative to the ground is = − 5.42 m/s , the same as found in part (a). In contrast to part (a), there now is a horizontal component of the velocity. However, since there is no horizontal acceleration, the initial and final horizontal velocities are the same and = 260 m/s . The x- and y-components of velocity can be combined to find the magnitude of the final velocity: Thus, yielding The direction is given by: so that Discussion = 2 + 2. = (260 m/s)2 + ( − 5.42 m/s)2 = 260.06 m/s. = tan−1( / ) = tan−1( − 5.42 / 260) = tan−1( − 0.0208) = −1.19º. (3.96) (3.97) (3.98) (3.99) (3.100) In part (a), the final velocity relative to the plane is the same as it would be if the coin were dropped from rest on the Earth and fell 1.50 m. This result fits our experience; objects in a plane fall the same way when the plane is flying horizontally as when it is at rest on the ground. This result is also true in moving cars. In part (b), an observer on the ground sees a much different motion for the coin. The plane is moving so fast horizontally to begin with that its final velocity is barely greater than the initial velocity. Once again, we see that in two dimensions, vectors do not add like ordinary numbers—the final velocity v in part (b) is not (260 – 5.42) m/s ; rather, it is 260.06 m/s . The velocity's magnitude had to be calculated to five digits to see any difference from that of the airplane. The motions as seen by different observers (one in the plane and one on the ground) in this example are analogous to those discussed for the binoculars dropped from the mast of a moving ship, except that the velocity of the plane is much larger, so that the two observers see very different paths. (See Figure 3.50.) In addition, both observers see the coin fall 1.50 m vertically, but the one on the ground also sees it move forward 144 m (this calculation is left for the reader). Thus, one observer sees a vertical path, the other a nearly horizontal path. Making Connections: Relativity and Einstein Because Einstein was able to clearly define how measurements are made (some involve light) and because the speed of light is the same for all observers, the outcomes are spectacularly unexpected. Time varies with observer, energy is stored as increased mass, and more surprises await. 128 Chapter 3 \| Two-Dimensional Kinematics PhET Explorations: Motion in 2D Try the new "Ladybug Motion 2D" simulation for the latest updated version. Learn about position, velocity, and acceleration vectors. Move the ball with the mouse or let the simulation move the ball in four types of motion (2 types of linear, simple harmonic, circle). Figure 3.51 Motion in 2D (http://cnx.org/content/m54798/1.2/motion-2d_en.jar) Glossary air resistance: a frictional force that slows the motion of objects as they travel through the air; when solving basic physics problems, air resistance is assumed to be zero analytical method: the method of determining the magnitude and direction of a resultant vector using the Pythagorean theorem and trigonometric identities classical relativity: the study of relative velocities in situations where speeds are less than about 1% of the speed of light—that is, less than 3000 km/s commutative: refers to the interchangeability of order in a function; vector addition is commutative because the order in which vectors are added together does not affect the final sum component (of a 2-d vector): a piece of a vector that points in either the vertical or the horizontal direction; every 2-d vector can be expressed as a sum of two vertical and horizontal vector components direction (of a vector): the orientation of a vector in space head (of a vector): the end point of a vector; the location of the tip of the vector's arrowhead; also referred to as the “tip” head-to-tail method: a method of adding vectors in which the tail of each vector is placed at the head of the previous vector kinematics: the study of motion without regard to mass or force magnitude (of a vector): the length or size of a vector; magnitude is a scalar quantity motion: displacement of an object as a function of time projectile: an object that travels through the air and experiences only acceleration due to gravity projectile motion: the motion of an object that is subject only to the acceleration of gravity range: the maximum horizontal distance that a projectile travels relative velocity: the velocity of an object as observed from a particular reference frame relativity: the study of how different observers moving relative to each other measure the same phenomenon resultant: the sum of two or more vectors resultant vector: the vector sum of two or more vectors scalar: a quantity with magnitude but no direction tail: the start point of a vector; opposite to the head or tip of the arrow trajectory: the path of a projectile through the air vector: a quantity that has both magnitude and direction; an arrow used to represent quantities with both magnitude and direction vector addition: the rules that apply to adding vectors together velocity: speed in a given direction This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 3 \| Two-Dimensional Kinematics 129 Section Summary 3.1 Kinematics in Two Dimensions: An Introduction • The shortest path between any two points is a straight line. In two dimensions, this path can be represented by a vector with horizontal and vertical components. • The horizontal and vertical components of a vector are independent of one another. Motion in the horizontal direction does not affect motion in the vertical direction, and vice versa. 3.2 Vector Addition and Subtraction: Graphical Methods • The graphical method of adding vectors A and B involves drawing vectors on a graph and adding them using the head-to-tail method. The resultant vector R is defined such that A + B = R . The magnitude and direction of R are then determined with a ruler and protractor, respectively. • The graphical method of subtracting vector B from A involves adding the opposite of vector B , which is defined as −B . In this case, A – B = A + (–B) = R . Then, the head-to-tail method of addition is followed in the usual way to obtain the resultant vector R . • Addition of vectors is commutative such that A + B = B + A . • The head-to-tail method of adding vectors involves drawing the first vector on a graph and then placing the tail of each subsequent vector at the head of the previous vector. The resultant vector is then drawn from the tail of the first vector to the head of the final vector. If a vector A is multiplied by a scalar quantity , the magnitude of the product is given by . If is positive, the direction of the product points in the same direction as A ; if is negative, the direction of the product points in the opposite direction as A . • 3.3 Vector Addition and Subtraction: Analytical Methods • The analytical method of vector addition and subtraction involves using the Pythagorean theorem and trigonometric identities to determine the magnitude and direction of a resultant vector. • The steps to add vectors A and B using the analytical method are as follows: Step 1: Determine the coordinate system for the vectors. Then, determine the horizontal and vertical components of each vector using the equations and = cos = cos = sin = sin . Step 2: Add the horizontal and vertical components of each vector to determine the components and of the resultant vector, R : and = + = + . Step 3: Use the Pythagorean theorem to determine the magnitude, , of the resultant vector R : Step 4: Use a trigonometric identity to determine the direction, , of R : = tan−1( / ). = 2. 2 + 3.4 Projectile Motion • Projectile motion is the motion of an object through the air that is subject only to the acceleration of gravity. • To solve projectile motion problems, perform the following steps: 1. Determine a coordinate system. Then, resolve the position and/or velocity of the object in the horizontal and vertical components. The components of position s are given by the quantities and , and the components of the velocity v are given by = cos and = sin , where is the magnitude of the velocity and is its direction. 2. Analyze the motion of the projectile in the horizontal direction using the following equations: 130 Chapter 3 \| Two-Dimensional Kinematics Horizontal motion( = 0) = 0 + 3. Analyze the motion of the projectile in the vertical direction using the following equations: = 0 = vx = velocity is a constant. Vertical motion(Assuming positive direction is up; = − = −9.80 m/s20 + ) 2 = 0 4. Recombine the horizontal and vertical components of location and/or velocity using the following equations( − 0). = 2 + 2 = tan−1( / ) = 2 2 + v = tan−1( / ). • The maximum height of a projectile launched with initial vertical velocity 0 is given by = 2 0 2 . • The maximum horizontal distance traveled by a projectile is called the range. The range of a projectile on level ground launched at an angle 0 above the horizontal with initial speed 0 is given by = 2 sin 2 0 0 . 3.5 Addition of Velocities • Velocities in two dimensions are added using the same analytical vector techniques, which are rewritten as = cos = sin 2 2 + = = tan−1( / ). • Relative velocity is the velocity of an object as observed from a particular reference frame, and it varies dramatically with reference frame. • Relativity is the study of how different observers measure the same phenomenon, particularly when the observers move relative to one another. Classical relativity is limited to situations where speed is less than about 1% of the speed of light (3000 km/s). Conceptual Questions 3.2 Vector Addition and Subtraction: Graphical Methods 1. Which of the following is a vector: a person's height, the altitude on Mt. Everest, the age of the Earth, the boiling point of water, the cost of this book, the Earth's population, the acceleration of gravity? 2. Give a specific ex
ample of a vector, stating its magnitude, units, and direction. 3. What do vectors and scalars have in common? How do they differ? 4. Two campers in a national park hike from their cabin to the same spot on a lake, each taking a different path, as illustrated below. The total distance traveled along Path 1 is 7.5 km, and that along Path 2 is 8.2 km. What is the final displacement of each camper? This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 3 \| Two-Dimensional Kinematics 131 Figure 3.52 5. If an airplane pilot is told to fly 123 km in a straight line to get from San Francisco to Sacramento, explain why he could end up anywhere on the circle shown in Figure 3.53. What other information would he need to get to Sacramento? Figure 3.53 6. Suppose you take two steps A and B (that is, two nonzero displacements). Under what circumstances can you end up at your starting point? More generally, under what circumstances can two nonzero vectors add to give zero? Is the maximum distance you can end up from the starting point A + B the sum of the lengths of the two steps? 7. Explain why it is not possible to add a scalar to a vector. 8. If you take two steps of different sizes, can you end up at your starting point? More generally, can two vectors with different magnitudes ever add to zero? Can three or more? 3.3 Vector Addition and Subtraction: Analytical Methods 9. Suppose you add two vectors A and B . What relative direction between them produces the resultant with the greatest magnitude? What is the maximum magnitude? What relative direction between them produces the resultant with the smallest magnitude? What is the minimum magnitude? 10. Give an example of a nonzero vector that has a component of zero. 11. Explain why a vector cannot have a component greater than its own magnitude. 12. If the vectors A and B are perpendicular, what is the component of A along the direction of B ? What is the component of B along the direction of A ? 3.4 Projectile Motion 13. Answer the following questions for projectile motion on level ground assuming negligible air resistance (the initial angle being neither 0° nor 90° ): (a) Is the velocity ever zero? (b) When is the velocity a minimum? A maximum? (c) Can the velocity ever be the same as the initial velocity at a time other than at = 0 ? (d) Can the speed ever be the same as the initial speed at a time other than at = 0 ? 132 Chapter 3 \| Two-Dimensional Kinematics 14. Answer the following questions for projectile motion on level ground assuming negligible air resistance (the initial angle being neither 0° nor 90° ): (a) Is the acceleration ever zero? (b) Is the acceleration ever in the same direction as a component of velocity? (c) Is the acceleration ever opposite in direction to a component of velocity? 15. For a fixed initial speed, the range of a projectile is determined by the angle at which it is fired. For all but the maximum, there are two angles that give the same range. Considering factors that might affect the ability of an archer to hit a target, such as wind, explain why the smaller angle (closer to the horizontal) is preferable. When would it be necessary for the archer to use the larger angle? Why does the punter in a football game use the higher trajectory? 16. During a lecture demonstration, a professor places two coins on the edge of a table. She then flicks one of the coins horizontally off the table, simultaneously nudging the other over the edge. Describe the subsequent motion of the two coins, in particular discussing whether they hit the floor at the same time. 3.5 Addition of Velocities 17. What frame or frames of reference do you instinctively use when driving a car? When flying in a commercial jet airplane? 18. A basketball player dribbling down the court usually keeps his eyes fixed on the players around him. He is moving fast. Why doesn't he need to keep his eyes on the ball? 19. If someone is riding in the back of a pickup truck and throws a softball straight backward, is it possible for the ball to fall straight down as viewed by a person standing at the side of the road? Under what condition would this occur? How would the motion of the ball appear to the person who threw it? 20. The hat of a jogger running at constant velocity falls off the back of his head. Draw a sketch showing the path of the hat in the jogger's frame of reference. Draw its path as viewed by a stationary observer. 21. A clod of dirt falls from the bed of a moving truck. It strikes the ground directly below the end of the truck. What is the direction of its velocity relative to the truck just before it hits? Is this the same as the direction of its velocity relative to ground just before it hits? Explain your answers. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 3 \| Two-Dimensional Kinematics 133 Problems & Exercises 3.2 Vector Addition and Subtraction: Graphical Methods Use graphical methods to solve these problems. You may assume data taken from graphs is accurate to three digits. 1. Find the following for path A in Figure 3.54: (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish. Figure 3.54 The various lines represent paths taken by different people walking in a city. All blocks are 120 m on a side. 2. Find the following for path B in Figure 3.54: (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish. 3. Find the north and east components of the displacement for the hikers shown in Figure 3.52. 4. Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements A and B , as in Figure 3.55, then this problem asks you to find their sum R = A + B .) Figure 3.56 6. Repeat the problem above, but reverse the order of the two legs of the walk; show that you get the same final result. That is, you first walk leg B , which is 20.0 m in a direction exactly 40° south of west, and then leg A , which is 12.0 m in a direction exactly 20° west of north. (This problem shows that A + B = B + A .) 7. (a) Repeat the problem two problems prior, but for the second leg you walk 20.0 m in a direction 40.0° north of east (which is equivalent to subtracting B from A —that is, to finding R′ = A − B ). (b) Repeat the problem two problems prior, but now you first walk 20.0 m in a direction 40.0° south of west and then 12.0 m in a direction 20.0° east of south (which is equivalent to subtracting A from B —that is, to finding R′′ = B - A = - R′ ). Show that this is the case. 8. Show that the order of addition of three vectors does not affect their sum. Show this property by choosing any three vectors A , B , and C , all having different lengths and directions. Find the sum A + B + C then find their sum when added in a different order and show the result is the same. (There are five other orders in which A , B , and C can be added; choose only one.) 9. Show that the sum of the vectors discussed in Example 3.2 gives the result shown in Figure 3.24. 10. Find the magnitudes of velocities A and B in Figure 3.57 Figure 3.55 The two displacements A and B add to give a total displacement R having magnitude and direction . 5. Suppose you first walk 12.0 m in a direction 20° west of north and then 20.0 m in a direction 40.0° south of west. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements A and B , as in Figure 3.56, then this problem finds their sum R = A + B .) Figure 3.57 The two velocities vA and vB add to give a total vtot . 11. Find the components of tot along the x- and y-axes in Figure 3.57. 134 Chapter 3 \| Two-Dimensional Kinematics 12. Find the components of tot along a set of perpendicular axes rotated 30° counterclockwise relative to those in Figure 3.57. 3.3 Vector Addition and Subtraction: Analytical Methods 13. Find the following for path C in Figure 3.58: (a) the total distance traveled and (b) the magnitude and direction of the displacement from start to finish. In this part of the problem, explicitly show how you follow the steps of the analytical method of vector addition. Figure 3.58 The various lines represent paths taken by different people walking in a city. All blocks are 120 m on a side. 14. Find the following for path D in Figure 3.58: (a) the total distance traveled and (b) the magnitude and direction of the displacement from start to finish. In this part of the problem, explicitly show how you follow the steps of the analytical method of vector addition. 15. Find the north and east components of the displacement from San Francisco to Sacramento shown in Figure 3.59. Figure 3.59 16. Solve the following problem using analytical techniques: Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements A and B , as in Figure 3.60, then this problem asks you to find their sum R = A + B .) This content is available for free at http://cnx.org/content/col11844/1.13 Figure 3.60 The two displacements A and B add to give a total displacement R having magnitude and direction . Note that you can also solve this graphically. Discuss why the analytical technique for solving this problem is potentially more accurate than the graphical technique. 17. Repeat Exercise 3.16 using analytical techniques, but reverse the order of the two legs of the walk and show that you get the same final result. (This problem shows that adding them in reverse order
gives the same result—that is, B + A = A + B .) Discuss how taking another path to reach the same point might help to overcome an obstacle blocking you other path. 18. You drive 7.50 km in a straight line in a direction 15º east of north. (a) Find the distances you would have to drive straight east and then straight north to arrive at the same point. (This determination is equivalent to find the components of the displacement along the east and north directions.) (b) Show that you still arrive at the same point if the east and north legs are reversed in order. 19. Do Exercise 3.16 again using analytical techniques and change the second leg of the walk to 25.0 m straight south. (This is equivalent to subtracting B from A —that is, finding R′ = A – B ) (b) Repeat again, but now you first walk 25.0 m north and then 18.0 m east. (This is equivalent to subtract A from B —that is, to find A = B + C . Is that consistent with your result?) 20. A new landowner has a triangular piece of flat land she wishes to fence. Starting at the west corner, she measures the first side to be 80.0 m long and the next to be 105 m. These sides are represented as displacement vectors A from B in Figure 3.61. She then correctly calculates the length and orientation of the third side C . What is her result? Chapter 3 \| Two-Dimensional Kinematics 135 Figure 3.61 21. You fly 32.0 km in a straight line in still air in the direction 35.0º south of west. (a) Find the distances you would have to fly straight south and then straight west to arrive at the same point. (This determination is equivalent to finding the components of the displacement along the south and west directions.) (b) Find the distances you would have to fly first in a direction 45.0º south of west and then in a direction 45.0º west of north. These are the components of the displacement along a different set of axes—one rotated 45º . 22. A farmer wants to fence off his four-sided plot of flat land. He measures the first three sides, shown as A B and C in Figure 3.62, and then correctly calculates the length and orientation of the fourth side D . What is his result? Figure 3.62 23. In an attempt to escape his island, Gilligan builds a raft and sets to sea. The wind shifts a great deal during the day, and he is blown along the following straight lines: 2.50 km 45.0º north of west; then 4.70 km 60.0º south of east; then 1.30 km 25.0º south of west; then 5.10 km straight east; then 1.70 km 5.00º east of north; then 7.20 km 55.0º south of west; and finally 2.80 km 10.0º north of east. What is his final position relative to the island? 24. Suppose a pilot flies 40.0 km in a direction 60º north of east and then flies 30.0 km in a direction 15º north of east as shown in Figure 3.63. Find her total distance from the starting point and the direction of the straight-line path to the final position. Discuss qualitatively how this flight would be altered by a wind from the north and how the effect of the wind would depend on both wind speed and the speed of the plane relative to the air mass. Figure 3.63 3.4 Projectile Motion 25. A projectile is launched at ground level with an initial speed of 50.0 m/s at an angle of 30.0° above the horizontal. It strikes a target above the ground 3.00 seconds later. What are the and distances from where the projectile was launched to where it lands? 26. A ball is kicked with an initial velocity of 16 m/s in the horizontal direction and 12 m/s in the vertical direction. (a) At what speed does the ball hit the ground? (b) For how long does the ball remain in the air? (c)What maximum height is attained by the ball? 27. A ball is thrown horizontally from the top of a 60.0-m building and lands 100.0 m from the base of the building. Ignore air resistance. (a) How long is the ball in the air? (b) What must have been the initial horizontal component of the velocity? (c) What is the vertical component of the velocity just before the ball hits the ground? (d) What is the velocity (including both the horizontal and vertical components) of the ball just before it hits the ground? 28. (a) A daredevil is attempting to jump his motorcycle over a line of buses parked end to end by driving up a 32° ramp at a speed of 40.0 m/s (144 km/h) . How many buses can he clear if the top of the takeoff ramp is at the same height as the bus tops and the buses are 20.0 m long? (b) Discuss what your answer implies about the margin of error in this act—that is, consider how much greater the range is than the horizontal distance he must travel to miss the end of the last bus. (Neglect air resistance.) 29. An archer shoots an arrow at a 75.0 m distant target; the bull's-eye of the target is at same height as the release height of the arrow. (a) At what angle must the arrow be released to hit the bull's-eye if its initial speed is 35.0 m/s? In this part of the problem, explicitly show how you follow the steps involved in solving projectile motion problems. (b) There is a large tree halfway between the archer and the target with an overhanging horizontal branch 3.50 m above the release height of the arrow. Will the arrow go over or under the branch? 30. A rugby player passes the ball 7.00 m across the field, where it is caught at the same height as it left his hand. (a) At what angle was the ball thrown if its initial speed was 12.0 m/ s, assuming that the smaller of the two possible angles was used? (b) What other angle gives the same range, and why would it not be used? (c) How long did this pass take? 31. Verify the ranges for the projectiles in Figure 3.41(a) for = 45° and the given initial velocities. 32. Verify the ranges shown for the projectiles in Figure 3.41(b) for an initial velocity of 50 m/s at the given initial angles. 33. The cannon on a battleship can fire a shell a maximum distance of 32.0 km. (a) Calculate the initial velocity of the shell. (b) What maximum height does it reach? (At its highest, the shell is above 60% of the atmosphere—but air resistance is not really negligible as assumed to make this problem easier.) (c) The ocean is not flat, because the Earth is curved. Assume that the radius of the Earth is 6.37×103 km . How many meters lower will its surface be 32.0 km from the ship along a horizontal line parallel to the surface at the ship? Does your answer imply that error introduced by the assumption of a flat Earth in projectile motion is significant here? 136 Chapter 3 \| Two-Dimensional Kinematics 34. An arrow is shot from a height of 1.5 m toward a cliff of height . It is shot with a velocity of 30 m/s at an angle of 60° above the horizontal. It lands on the top edge of the cliff 4.0 s later. (a) What is the height of the cliff? (b) What is the maximum height reached by the arrow along its trajectory? (c) What is the arrow's impact speed just before hitting the cliff? 35. In the standing broad jump, one squats and then pushes off with the legs to see how far one can jump. Suppose the extension of the legs from the crouch position is 0.600 m and the acceleration achieved from this position is 1.25 times the acceleration due to gravity, . How far can they jump? State your assumptions. (Increased range can be achieved by swinging the arms in the direction of the jump.) 36. The world long jump record is 8.95 m (Mike Powell, USA, 1991). Treated as a projectile, what is the maximum range obtainable by a person if he has a take-off speed of 9.5 m/s? State your assumptions. 37. Serving at a speed of 170 km/h, a tennis player hits the ball at a height of 2.5 m and an angle below the horizontal. The service line is 11.9 m from the net, which is 0.91 m high. What is the angle such that the ball just crosses the net? Will the ball land in the service box, whose out line is 6.40 m from the net? 38. A football quarterback is moving straight backward at a speed of 2.00 m/s when he throws a pass to a player 18.0 m straight downfield. (a) If the ball is thrown at an angle of 25° relative to the ground and is caught at the same height as it is released, what is its initial speed relative to the ground? (b) How long does it take to get to the receiver? (c) What is its maximum height above its point of release? 39. Gun sights are adjusted to aim high to compensate for the effect of gravity, effectively making the gun accurate only for a specific range. (a) If a gun is sighted to hit targets that are at the same height as the gun and 100.0 m away, how low will the bullet hit if aimed directly at a target 150.0 m away? The muzzle velocity of the bullet is 275 m/s. (b) Discuss qualitatively how a larger muzzle velocity would affect this problem and what would be the effect of air resistance. 40. An eagle is flying horizontally at a speed of 3.00 m/s when the fish in her talons wiggles loose and falls into the lake 5.00 m below. Calculate the velocity of the fish relative to the water when it hits the water. 41. An owl is carrying a mouse to the chicks in its nest. Its position at that time is 4.00 m west and 12.0 m above the center of the 30.0 cm diameter nest. The owl is flying east at 3.50 m/s at an angle 30.0° below the horizontal when it accidentally drops the mouse. Is the owl lucky enough to have the mouse hit the nest? To answer this question, calculate the horizontal position of the mouse when it has fallen 12.0 m. 42. Suppose a soccer player kicks the ball from a distance 30 m toward the goal. Find the initial speed of the ball if it just passes over the goal, 2.4 m above the ground, given the initial direction to be 40° above the horizontal. 43. Can a goalkeeper at her/ his goal kick a soccer ball into the opponent's goal without the ball touching the ground? The distance will be about 95 m. A goalkeeper can give the ball a speed of 30 m/s. 44. The free throw line in basketball is 4.57 m (15 ft) from the basket, which is 3.05 m (10 ft) above the floor. A player standing on the free throw line throws the ball with an initial This content is available for free at http://cn
x.org/content/col11844/1.13 speed of 7.15 m/s, releasing it at a height of 2.44 m (8 ft) above the floor. At what angle above the horizontal must the ball be thrown to exactly hit the basket? Note that most players will use a large initial angle rather than a flat shot because it allows for a larger margin of error. Explicitly show how you follow the steps involved in solving projectile motion problems. 45. In 2007, Michael Carter (U.S.) set a world record in the shot put with a throw of 24.77 m. What was the initial speed of the shot if he released it at a height of 2.10 m and threw it at an angle of 38.0° above the horizontal? (Although the maximum distance for a projectile on level ground is achieved at 45° when air resistance is neglected, the actual angle to achieve maximum range is smaller; thus, 38° will give a longer range than 45° in the shot put.) 46. A basketball player is running at 5.00 m/s directly toward the basket when he jumps into the air to dunk the ball. He maintains his horizontal velocity. (a) What vertical velocity does he need to rise 0.750 m above the floor? (b) How far from the basket (measured in the horizontal direction) must he start his jump to reach his maximum height at the same time as he reaches the basket? 47. A football player punts the ball at a 45.0° angle. Without an effect from the wind, the ball would travel 60.0 m horizontally. (a) What is the initial speed of the ball? (b) When the ball is near its maximum height it experiences a brief gust of wind that reduces its horizontal velocity by 1.50 m/s. What distance does the ball travel horizontally? 48. Prove that the trajectory of a projectile is parabolic, having the form = + 2 . To obtain this expression, solve the equation = 0 for and substitute it into the expression for = 0 – (1 / 2) 2 (These equations describe the and positions of a projectile that starts at the origin.) You should obtain an equation of the form = + 2 where and are constants. 49. Derive = 2 sin 2θ0 0 for the range of a projectile on level ground by finding the time at which becomes zero and substituting this value of into the expression for − 0 , noting that = − 0 50. Unreasonable Results (a) Find the maximum range of a super cannon that has a muzzle velocity of 4.0 km/s. (b) What is unreasonable about the range you found? (c) Is the premise unreasonable or is the available equation inapplicable? Explain your answer. (d) If such a muzzle velocity could be obtained, discuss the effects of air resistance, thinning air with altitude, and the curvature of the Earth on the range of the super cannon. 51. Construct Your Own Problem Consider a ball tossed over a fence. Construct a problem in which you calculate the ball's needed initial velocity to just clear the fence. Among the things to determine are; the height of the fence, the distance to the fence from the point of release of the ball, and the height at which the ball is released. You should also consider whether it is possible to choose the initial speed for the ball and just calculate the angle at which it is thrown. Also examine the possibility of multiple solutions given the distances and heights you have chosen. Chapter 3 \| Two-Dimensional Kinematics 137 3.5 Addition of Velocities 52. Bryan Allen pedaled a human-powered aircraft across the English Channel from the cliffs of Dover to Cap Gris-Nez on June 12, 1979. (a) He flew for 169 min at an average velocity of 3.53 m/s in a direction 45º south of east. What was his total displacement? (b) Allen encountered a headwind averaging 2.00 m/s almost precisely in the opposite direction of his motion relative to the Earth. What was his average velocity relative to the air? (c) What was his total displacement relative to the air mass? 53. A seagull flies at a velocity of 9.00 m/s straight into the wind. (a) If it takes the bird 20.0 min to travel 6.00 km relative to the Earth, what is the velocity of the wind? (b) If the bird turns around and flies with the wind, how long will he take to return 6.00 km? (c) Discuss how the wind affects the total round-trip time compared to what it would be with no wind. 54. Near the end of a marathon race, the first two runners are separated by a distance of 45.0 m. The front runner has a velocity of 3.50 m/s, and the second a velocity of 4.20 m/s. (a) What is the velocity of the second runner relative to the first? (b) If the front runner is 250 m from the finish line, who will win the race, assuming they run at constant velocity? (c) What distance ahead will the winner be when she crosses the finish line? 55. Verify that the coin dropped by the airline passenger in the Example 3.8 travels 144 m horizontally while falling 1.50 m in the frame of reference of the Earth. 56. A football quarterback is moving straight backward at a speed of 2.00 m/s when he throws a pass to a player 18.0 m straight downfield. The ball is thrown at an angle of 25.0º relative to the ground and is caught at the same height as it is released. What is the initial velocity of the ball relative to the quarterback ? 57. A ship sets sail from Rotterdam, The Netherlands, heading due north at 7.00 m/s relative to the water. The local ocean current is 1.50 m/s in a direction 40.0º north of east. What is the velocity of the ship relative to the Earth? 58. (a) A jet airplane flying from Darwin, Australia, has an air speed of 260 m/s in a direction 5.0º south of west. It is in the jet stream, which is blowing at 35.0 m/s in a direction 15º south of east. What is the velocity of the airplane relative to the Earth? (b) Discuss whether your answers are consistent with your expectations for the effect of the wind on the plane's path. 59. (a) In what direction would the ship in Exercise 3.57 have to travel in order to have a velocity straight north relative to the Earth, assuming its speed relative to the water remains 7.00 m/s ? (b) What would its speed be relative to the Earth? 60. (a) Another airplane is flying in a jet stream that is blowing at 45.0 m/s in a direction 20º south of east (as in Exercise 3.58). Its direction of motion relative to the Earth is 45.0º south of west, while its direction of travel relative to the air is 5.00º south of west. What is the airplane's speed relative to the air mass? (b) What is the airplane's speed relative to the Earth? 61. A sandal is dropped from the top of a 15.0-m-high mast on a ship moving at 1.75 m/s due south. Calculate the velocity of the sandal when it hits the deck of the ship: (a) relative to the ship and (b) relative to a stationary observer on shore. (c) Discuss how the answers give a consistent result for the position at which the sandal hits the deck. 62. The velocity of the wind relative to the water is crucial to sailboats. Suppose a sailboat is in an ocean current that has a velocity of 2.20 m/s in a direction 30.0º east of north relative to the Earth. It encounters a wind that has a velocity of 4.50 m/s in a direction of 50.0º south of west relative to the Earth. What is the velocity of the wind relative to the water? 63. The great astronomer Edwin Hubble discovered that all distant galaxies are receding from our Milky Way Galaxy with velocities proportional to their distances. It appears to an observer on the Earth that we are at the center of an expanding universe. Figure 3.64 illustrates this for five galaxies lying along a straight line, with the Milky Way Galaxy at the center. Using the data from the figure, calculate the velocities: (a) relative to galaxy 2 and (b) relative to galaxy 5. The results mean that observers on all galaxies will see themselves at the center of the expanding universe, and they would likely be aware of relative velocities, concluding that it is not possible to locate the center of expansion with the given information. Figure 3.64 Five galaxies on a straight line, showing their distances and velocities relative to the Milky Way (MW) Galaxy. The distances are in millions of light years (Mly), where a light year is the distance light travels in one year. The velocities are nearly proportional to the distances. The sizes of the galaxies are greatly exaggerated; an average galaxy is about 0.1 Mly across. 64. (a) Use the distance and velocity data in Figure 3.64 to find the rate of expansion as a function of distance. (b) If you extrapolate back in time, how long ago would all of the galaxies have been at approximately the same position? The two parts of this problem give you some idea of how the Hubble constant for universal expansion and the time back to the Big Bang are determined, respectively. 65. An athlete crosses a 25-m-wide river by swimming perpendicular to the water current at a speed of 0.5 m/s relative to the water. He reaches the opposite side at a distance 40 m downstream from his starting point. How fast is the water in the river flowing with respect to the ground? What is the speed of the swimmer with respect to a friend at rest on the ground? 66. A ship sailing in the Gulf Stream is heading 25.0º west of north at a speed of 4.00 m/s relative to the water. Its velocity relative to the Earth is 4.80 m/s 5.00º west of north. What is the velocity of the Gulf Stream? (The velocity obtained is typical for the Gulf Stream a few hundred kilometers off the east coast of the United States.) 67. An ice hockey player is moving at 8.00 m/s when he hits the puck toward the goal. The speed of the puck relative to the player is 29.0 m/s. The line between the center of the goal and the player makes a 90.0º angle relative to his path as shown in Figure 3.65. What angle must the puck's velocity make relative to the player (in his frame of reference) to hit the center of the goal? 138 Chapter 3 \| Two-Dimensional Kinematics Figure 3.65 An ice hockey player moving across the rink must shoot backward to give the puck a velocity toward the goal. 68. Unreasonable Results Suppose you wish to shoot supplies straight up to astronauts in an orbit 36,000 km above the surface of the Ea
rth. (a) At what velocity must the supplies be launched? (b) What is unreasonable about this velocity? (c) Is there a problem with the relative velocity between the supplies and the astronauts when the supplies reach their maximum height? (d) Is the premise unreasonable or is the available equation inapplicable? Explain your answer. 69. Unreasonable Results A commercial airplane has an air speed of 280 m/s due east and flies with a strong tailwind. It travels 3000 km in a direction 5º south of east in 1.50 h. (a) What was the velocity of the plane relative to the ground? (b) Calculate the magnitude and direction of the tailwind's velocity. (c) What is unreasonable about both of these velocities? (d) Which premise is unreasonable? 70. Construct Your Own Problem Consider an airplane headed for a runway in a cross wind. Construct a problem in which you calculate the angle the airplane must fly relative to the air mass in order to have a velocity parallel to the runway. Among the things to consider are the direction of the runway, the wind speed and direction (its velocity) and the speed of the plane relative to the air mass. Also calculate the speed of the airplane relative to the ground. Discuss any last minute maneuvers the pilot might have to perform in order for the plane to land with its wheels pointing straight down the runway. Test Prep for AP® Courses 3.1 Kinematics in Two Dimensions: An Introduction 1. A ball is thrown at an angle of 45 degrees above the horizontal. Which of the following best describes the acceleration of the ball from the instant after it leaves the thrower's hand until the time it hits the ground? a. Always in the same direction as the motion, initially b. positive and gradually dropping to zero by the time it hits the ground Initially positive in the upward direction, then zero at maximum height, then negative from there until it hits the ground c. Always in the opposite direction as the motion, initially positive and gradually dropping to zero by the time it hits the ground d. Always in the downward direction with the same constant value 2. In an experiment, a student launches a ball with an initial horizontal velocity at an elevation 2 meters above ground. The ball follows a parabolic trajectory until it hits the ground. Which of the following accurately describes the graph of the ball's vertical acceleration versus time (taking the downward direction to be negative)? a. A negative value that does not change with time b. A gradually increasing negative value (straight line) This content is available for free at http://cnx.org/content/col11844/1.13 c. An increasing rate of negative values over time (parabolic curve) d. Zero at all times since the initial motion is horizontal 3. A student wishes to design an experiment to show that the acceleration of an object is independent of the object's velocity. To do this, ball A is launched horizontally with some initial speed at an elevation 1.5 meters above the ground, ball B is dropped from rest 1.5 meters above the ground, and ball C is launched vertically with some initial speed at an elevation 1.5 meters above the ground. What information would the student need to collect about each ball in order to test the hypothesis? 3.2 Vector Addition and Subtraction: Graphical Methods 4. A ball is launched vertically upward. The vertical position of the ball is recorded at various points in time in the table shown. Chapter 3 \| Two-Dimensional Kinematics 139 Table 3.1 Height (m) Time (sec) 0.490 0.882 1.176 1.372 1.470 1.470 1.372 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Which of the following correctly describes the graph of the ball's vertical velocity versus time? a. Always positive, steadily decreasing b. Always positive, constant c. Initially positive, steadily decreasing, becoming negative at the end Initially zero, steadily getting more and more negative d. 5. Table 3.2 Height (m) Time (sec) 0.490 0.882 1.176 1.372 1.470 1.470 1.372 0.1 0.2 0.3 0.4 0.5 0.6 0.7 A ball is launched at an angle of 60 degrees above the horizontal, and the vertical position of the ball is recorded at various points in time in the table shown, assuming the ball was at a height of 0 at time t = 0. a. Draw a graph of the ball's vertical velocity versus time. b. Describe the graph of the ball's horizontal velocity. c. Draw a graph of the ball's vertical acceleration versus time. 3.4 Projectile Motion 6. In an experiment, a student launches a ball with an initial horizontal velocity of 5.00 meters/sec at an elevation 2.00 meters above ground. Draw and clearly label with appropriate values and units a graph of the ball's horizontal velocity vs. time and the ball's vertical velocity vs. time. The graph should cover the motion from the instant after the ball is launched until the instant before it hits the ground. Assume the downward direction is negative for this problem. 140 Chapter 3 \| Two-Dimensional Kinematics This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 141 4 DYNAMICS: FORCE AND NEWTON'S LAWS OF MOTION Figure 4.1 Newton’s laws of motion describe the motion of the dolphin’s path. (credit: Jin Jang) Chapter Outline 4.1. Development of Force Concept 4.2. Newton's First Law of Motion: Inertia 4.3. Newton's Second Law of Motion: Concept of a System 4.4. Newton's Third Law of Motion: Symmetry in Forces 4.5. Normal, Tension, and Other Examples of Force 4.6. Problem-Solving Strategies 4.7. Further Applications of Newton's Laws of Motion 4.8. Extended Topic: The Four Basic Forces—An Introduction Connection for AP® Courses Motion draws our attention. Motion itself can be beautiful, causing us to marvel at the forces needed to achieve spectacular motion, such as that of a jumping dolphin, a leaping pole vaulter, a bird in flight, or an orbiting satellite. The study of motion is kinematics, but kinematics only describes the way objects move—their velocity and their acceleration. Dynamics considers the forces that affect the motion of moving objects and systems. Newton’s laws of motion are the foundation of dynamics. These laws provide an example of the breadth and simplicity of principles under which nature functions. They are also universal laws in that they apply to situations on Earth as well as in space. Isaac Newton’s (1642–1727) laws of motion were just one part of the monumental work that has made him legendary. The development of Newton’s laws marks the transition from the Renaissance into the modern era. This transition was characterized by a revolutionary change in the way people thought about the physical universe. For many centuries natural philosophers had debated the nature of the universe based largely on certain rules of logic, with great weight given to the thoughts of earlier 142 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion classical philosophers such as Aristotle (384–322 BC). Among the many great thinkers who contributed to this change were Newton and Galileo Galilei (1564–1647). Figure 4.2 Isaac Newton’s monumental work, Philosophiae Naturalis Principia Mathematica, was published in 1687. It proposed scientific laws that are still used today to describe the motion of objects. (credit: Service commun de la documentation de l'Université de Strasbourg) Galileo was instrumental in establishing observation as the absolute determinant of truth, rather than “logical” argument. Galileo’s use of the telescope was his most notable achievement in demonstrating the importance of observation. He discovered moons orbiting Jupiter and made other observations that were inconsistent with certain ancient ideas and religious dogma. For this reason, and because of the manner in which he dealt with those in authority, Galileo was tried by the Inquisition and punished. He spent the final years of his life under a form of house arrest. Because others before Galileo had also made discoveries by observing the nature of the universe and because repeated observations verified those of Galileo, his work could not be suppressed or denied. After his death, his work was verified by others, and his ideas were eventually accepted by the church and scientific communities. Galileo also contributed to the formulation of what is now called Newton’s first law of motion. Newton made use of the work of his predecessors, which enabled him to develop laws of motion, discover the law of gravity, invent calculus, and make great contributions to the theories of light and color. It is amazing that many of these developments were made by Newton working alone, without the benefit of the usual interactions that take place among scientists today. Newton’s laws are introduced along with Big Idea 3, that interactions can be described by forces. These laws provide a theoretical basis for studying motion depending on interactions between the objects. In particular, Newton's laws are applicable to all forces in inertial frames of references (Enduring Understanding 3.A). We will find that all forces are vectors; that is, forces always have both a magnitude and a direction (Essential Knowledge 3.A.2). Furthermore, we will learn that all forces are a result of interactions between two or more objects (Essential Knowledge 3.A.3). These interactions between any two objects are described by Newton's third law, stating that the forces exerted on these objects are equal in magnitude and opposite in direction to each other (Essential Knowledge 3.A.4). We will discover that there is an empirical cause-effect relationship between the net force exerted on an object of mass m and its acceleration, with this relationship described by Newton's second law (Enduring Understanding 3.B). This supports Big Idea 1, that inertial mass is a property of an object or a system. The mass of an object or a system is one of the factors affecting changes in motion when an object or a system interacts with other objects or systems (Essential Knowledge 1.C.1)
. Another is the net force on an object, which is the vector sum of all the forces exerted on the object (Essential Knowledge 3.B.1). To analyze this, we use free-body diagrams to visualize the forces exerted on a given object in order to find the net force and analyze the object's motion (Essential Knowledge 3.B.2). Thinking of these objects as systems is a concept introduced in this chapter, where a system is a collection of elements that could be considered as a single object without any internal structure (Essential Knowledge 5.A.1). This will support Big Idea 5, that changes that occur to the system due to interactions are governed by conservation laws. These conservation laws will be the focus of later chapters in this book. They explain whether quantities are conserved in the given system or change due to transfer to or from another system due to interactions between the systems (Enduring Understanding 5.A). Furthermore, when a situation involves more than one object, it is important to define the system and analyze the motion of a whole system, not its elements, based on analysis of external forces on the system. This supports Big Idea 4, that interactions between systems cause changes in those systems. All kinematics variables in this case describe the motion of the center of mass of the system (Essential Knowledge 4.A.1, Essential Knowledge 4.A.2). The internal forces between the elements of the system do not affect the velocity of the center of mass (Essential Knowledge 4.A.3). The velocity of the center of mass will change only if there is a net external force exerted on the system (Enduring Understanding 4.A). We will learn that some of these interactions can be explained by the existence of fields extending through space, supporting Big Idea 2. For example, any object that has mass creates a gravitational field in space (Enduring Understanding 2.B). Any material object (one that has mass) placed in the gravitational field will experience gravitational force (Essential Knowledge 2.B.1). This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 143 Forces may be categorized as contact or long-distance (Enduring Understanding 3.C). In this chapter we will work with both. An example of a long-distance force is gravitation (Essential Knowledge 3.C.1). Contact forces, such as tension, friction, normal force, and the force of a spring, result from interatomic electric forces at the microscopic level (Essential Knowledge 3.C.4). It was not until the advent of modern physics early in the twentieth century that it was discovered that Newton’s laws of motion produce a good approximation to motion only when the objects are moving at speeds much, much less than the speed of light and when those objects are larger than the size of most molecules (about 10–9 m in diameter). These constraints define the realm of classical mechanics, as discussed in Introduction to the Nature of Science and Physics. At the beginning of the twentieth century, Albert Einstein (1879–1955) developed the theory of relativity and, along with many other scientists, quantum theory. Quantum theory does not have the constraints present in classical physics. All of the situations we consider in this chapter, and all those preceding the introduction of relativity in Special Relativity, are in the realm of classical physics. The development of special relativity and empirical observations at atomic scales led to the idea that there are four basic forces that account for all known phenomena. These forces are called fundamental (Enduring Understanding 3.G). The properties of gravitational (Essential Knowledge 3.G.1) and electromagnetic (Essential Knowledge 3.G.2) forces are explained in more detail. Big Idea 1 Objects and systems have properties such as mass and charge. Systems may have internal structure. Essential Knowledge 1.C.1 Inertial mass is the property of an object or a system that determines how its motion changes when it interacts with other objects or systems. Big Idea 2 Fields existing in space can be used to explain interactions. Enduring Understanding 2.A A field associates a value of some physical quantity with every point in space. Field models are useful for describing interactions that occur at a distance (long-range forces) as well as a variety of other physical phenomena. Essential Knowledge 2.A.1 A vector field gives, as a function of position (and perhaps time), the value of a physical quantity that is described by a vector. Essential Knowledge 2.A.2 A scalar field gives the value of a physical quantity. Enduring Understanding 2.B A gravitational field is caused by an object with mass. Essential Knowledge 2.B.1 A gravitational field g at the location of an object with mass m causes a gravitational force of magnitude mg to be exerted on the object in the direction of the field. Big Idea 3 The interactions of an object with other objects can be described by forces. Enduring Understanding 3.A All forces share certain common characteristics when considered by observers in inertial reference frames. Essential Knowledge 3.A.2 Forces are described by vectors. Essential Knowledge 3.A.3 A force exerted on an object is always due to the interaction of that object with another object. Essential Knowledge 3.A.4 If one object exerts a force on a second object, the second object always exerts a force of equal magnitude on the first object in the opposite direction. Enduring Understanding 3.B Classically, the acceleration of an object interacting with other objects can be predicted by using = ∑ / . Essential Knowledge 3.B.1 If an object of interest interacts with several other objects, the net force is the vector sum of the individual forces. Essential Knowledge 3.B.2 Free-body diagrams are useful tools for visualizing the forces being exerted on a single object and writing the equations that represent a physical situation. Enduring Understanding 3.C At the macroscopic level, forces can be categorized as either long-range (action-at-a-distance) forces or contact forces. Essential Knowledge 3.C.1 Gravitational force describes the interaction of one object that has mass with another object that has mass. Essential Knowledge 3.C.4 Contact forces result from the interaction of one object touching another object, and they arise from interatomic electric forces. These forces include tension, friction, normal, spring (Physics 1), and buoyant (Physics 2). Enduring Understanding 3.G Certain types of forces are considered fundamental. Essential Knowledge 3.G.1 Gravitational forces are exerted at all scales and dominate at the largest distance and mass scales. Essential Knowledge 3.G.2 Electromagnetic forces are exerted at all scales and can dominate at the human scale. Big Idea 4 Interactions between systems can result in changes in those systems. Enduring Understanding 4.A The acceleration of the center of mass of a system is related to the net force exerted on the system, where = ∑ / . Essential Knowledge 4.A.1 The linear motion of a system can be described by the displacement, velocity, and acceleration of its center of mass. Essential Knowledge 4.A.2 The acceleration is equal to the rate of change of velocity with time, and velocity is equal to the rate of change of position with time. 144 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion Essential Knowledge 4.A.3 Forces that systems exert on each other are due to interactions between objects in the systems. If the interacting objects are parts of the same system, there will be no change in the center-of-mass velocity of that system. Big Idea 5 Changes that occur as a result of interactions are constrained by conservation laws. Enduring Understanding 5.A Certain quantities are conserved, in the sense that the changes of those quantities in a given system are always equal to the transfer of that quantity to or from the system by all possible interactions with other systems. Essential Knowledge 5.A.1 A system is an object or a collection of objects. The objects are treated as having no internal structure. 4.1 Development of Force Concept By the end of this section, you will be able to: • Understand the definition of force. Learning Objectives The information presented in this section supports the following AP® learning objectives and science practices: • 3.A.2.1 The student is able to represent forces in diagrams or mathematically using appropriately labeled vectors with magnitude, direction, and units during the analysis of a situation. (S.P. 1.1) • 3.A.3.2 The student is able to challenge a claim that an object can exert a force on itself. (S.P. 6.1) • 3.A.3.3 The student is able to describe a force as an interaction between two objects and identify both objects for any force. (S.P. 1.4) • 3.B.2.1 The student is able to create and use free-body diagrams to analyze physical situations to solve problems with motion qualitatively and quantitatively. (S.P. 1.1, 1.4, 2.2) Dynamics is the study of the forces that cause objects and systems to move. To understand this, we need a working definition of force. Our intuitive definition of force—that is, a push or a pull—is a good place to start. We know that a push or pull has both magnitude and direction (therefore, it is a vector quantity) and can vary considerably in each regard. For example, a cannon exerts a strong force on a cannonball that is launched into the air. In contrast, Earth exerts only a tiny downward pull on a flea. Our everyday experiences also give us a good idea of how multiple forces add. If two people push in different directions on a third person, as illustrated in Figure 4.3, we might expect the total force to be in the direction shown. Since force is a vector, it adds just like other vectors, as illustrated in Figure 4.3(a) for two ice skaters. Forces, like other vectors, are represented by arrows and can be added using the familiar head-to-tail meth
od or by trigonometric methods. These ideas were developed in Two-Dimensional Kinematics. By definition, force is always the result of an interaction of two or more objects. No object possesses force on its own. For example, a cannon does not possess force, but it can exert force on a cannonball. Earth does not possess force on its own, but exerts force on a football or on any other massive object. The skaters in Figure 4.3 exert force on one another as they interact. No object can exert force on itself. When you clap your hands, one hand exerts force on the other. When a train accelerates, it exerts force on the track and vice versa. A bowling ball is accelerated by the hand throwing it; once the hand is no longer in contact with the bowling ball, it is no longer accelerating the bowling ball or exerting force on it. The ball continues moving forward due to inertia. Figure 4.3 Part (a) shows an overhead view of two ice skaters pushing on a third. Forces are vectors and add like other vectors, so the total force on the third skater is in the direction shown. In part (b), we see a free-body diagram representing the forces acting on the third skater. Figure 4.3(b) is our first example of a free-body diagram, which is a technique used to illustrate all the external forces acting on a body. The body is represented by a single isolated point (or free body), and only those forces acting on the body from the outside (external forces) are shown. (These forces are the only ones shown, because only external forces acting on the body affect its motion. We can ignore any internal forces within the body.) Free-body diagrams are very useful in analyzing forces acting on a system and are employed extensively in the study and application of Newton’s laws of motion. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 145 A more quantitative definition of force can be based on some standard force, just as distance is measured in units relative to a standard distance. One possibility is to stretch a spring a certain fixed distance, as illustrated in Figure 4.4, and use the force it exerts to pull itself back to its relaxed shape—called a restoring force—as a standard. The magnitude of all other forces can be stated as multiples of this standard unit of force. Many other possibilities exist for standard forces. (One that we will encounter in Magnetism is the magnetic force between two wires carrying electric current.) Some alternative definitions of force will be given later in this chapter. Figure 4.4 The force exerted by a stretched spring can be used as a standard unit of force. (a) This spring has a length when undistorted. (b) When stretched a distance Δ , the spring exerts a restoring force, Frestore , which is reproducible. (c) A spring scale is one device that uses a spring to measure force. The force Frestore is exerted on whatever is attached to the hook. Here Frestore has a magnitude of 6 units in the force standard being employed. Take-Home Experiment: Force Standards To investigate force standards and cause and effect, get two identical rubber bands. Hang one rubber band vertically on a hook. Find a small household item that could be attached to the rubber band using a paper clip, and use this item as a weight to investigate the stretch of the rubber band. Measure the amount of stretch produced in the rubber band with one, two, and four of these (identical) items suspended from the rubber band. What is the relationship between the number of items and the amount of stretch? How large a stretch would you expect for the same number of items suspended from two rubber bands? What happens to the amount of stretch of the rubber band (with the weights attached) if the weights are also pushed to the side with a pencil? 4.2 Newton's First Law of Motion: Inertia Learning Objectives By the end of this section, you will be able to: • Define mass and inertia. • Understand Newton's first law of motion. Experience suggests that an object at rest will remain at rest if left alone, and that an object in motion tends to slow down and stop unless some effort is made to keep it moving. What Newton’s first law of motion states, however, is the following: Newton’s First Law of Motion There exists an inertial frame of reference such that a body at rest remains at rest, or, if in motion, remains in motion at a constant velocity unless acted on by a net external force. Note the repeated use of the verb “remains.” We can think of this law as preserving the status quo of motion. The first law of motion postulates the existence of at least one frame of reference which we call an inertial reference frame, relative to which the motion of an object not subject to forces is a straight line at a constant speed. An inertial reference frame is any reference frame that is not itself accelerating. A car traveling at constant velocity is an inertial reference frame. A car slowing down for a stoplight, or speeding up after the light turns green, will be accelerating and is not an inertial reference frame. Finally, when the car goes around a turn, which is due to an acceleration changing the direction of the velocity vector, it is not an inertial reference frame. Note that Newton’s laws of motion are only valid for inertial reference frames. Rather than contradicting our experience, Newton’s first law of motion states that there must be a cause (which is a net external force) for there to be any change in velocity (either a change in magnitude or direction) in an inertial reference frame. We will define net external force in the next section. An object sliding across a table or floor slows down due to the net force of friction acting on the object. If friction disappeared, would the object still slow down? The idea of cause and effect is crucial in accurately describing what happens in various situations. For example, consider what happens to an object sliding along a rough horizontal surface. The object quickly grinds to a halt. If we spray the surface with talcum powder to make the surface smoother, the object slides farther. If we make the surface even smoother by rubbing 146 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion lubricating oil on it, the object slides farther yet. Extrapolating to a frictionless surface, we can imagine the object sliding in a straight line indefinitely. Friction is thus the cause of the slowing (consistent with Newton’s first law). The object would not slow down at all if friction were completely eliminated. Consider an air hockey table. When the air is turned off, the puck slides only a short distance before friction slows it to a stop. However, when the air is turned on, it creates a nearly frictionless surface, and the puck glides long distances without slowing down. Additionally, if we know enough about the friction, we can accurately predict how quickly the object will slow down. Friction is an external force. Newton’s first law is completely general and can be applied to anything from an object sliding on a table to a satellite in orbit to blood pumped from the heart. Experiments have thoroughly verified that any change in velocity (speed or direction) must be caused by an external force. The idea of generally applicable or universal laws is important not only here—it is a basic feature of all laws of physics. Identifying these laws is like recognizing patterns in nature from which further patterns can be discovered. The genius of Galileo, who first developed the idea for the first law, and Newton, who clarified it, was to ask the fundamental question, “What is the cause?” Thinking in terms of cause and effect is a worldview fundamentally different from the typical ancient Greek approach when questions such as “Why does a tiger have stripes?” would have been answered in Aristotelian fashion, “That is the nature of the beast.” True perhaps, but not a useful insight. Mass The property of a body to remain at rest or to remain in motion with constant velocity is called inertia. Newton’s first law is often called the law of inertia. As we know from experience, some objects have more inertia than others. It is obviously more difficult to change the motion of a large boulder than that of a basketball, for example. The inertia of an object is measured by its mass. An object with a small mass will exhibit less inertia and be more affected by other objects. An object with a large mass will exhibit greater inertia and be less affected by other objects. This inertial mass of an object is a measure of how difficult it is to alter the uniform motion of the object by an external force. Roughly speaking, mass is a measure of the amount of “stuff” (or matter) in something. The quantity or amount of matter in an object is determined by the numbers of atoms and molecules of various types it contains. Unlike weight, mass does not vary with location. The mass of an object is the same on Earth, in orbit, or on the surface of the Moon. In practice, it is very difficult to count and identify all of the atoms and molecules in an object, so masses are not often determined in this manner. Operationally, the masses of objects are determined by comparison with the standard kilogram. Check Your Understanding Which has more mass: a kilogram of cotton balls or a kilogram of gold? Solution They are equal. A kilogram of one substance is equal in mass to a kilogram of another substance. The quantities that might differ between them are volume and density. 4.3 Newton's Second Law of Motion: Concept of a System Learning Objectives By the end of this section, you will be able to: • Define net force, external force, and system. • Understand Newton’s second law of motion. • Apply Newton’s second law to determine the weight of an object. Newton’s second law of motion is closely related to Newton’s first law of motion. It mathematically states the cause and effect relationsh
ip between force and changes in motion. Newton’s second law of motion is more quantitative and is used extensively to calculate what happens in situations involving a force. Before we can write down Newton’s second law as a simple equation giving the exact relationship of force, mass, and acceleration, we need to sharpen some ideas that have already been mentioned. First, what do we mean by a change in motion? The answer is that a change in motion is equivalent to a change in velocity. A change in velocity means, by definition, that there is an acceleration. Newton’s first law says that a net external force causes a change in motion; thus, we see that a net external force causes acceleration. Another question immediately arises. What do we mean by an external force? An intuitive notion of external is correct—an external force acts from outside the system of interest. For example, in Figure 4.5(a) the system of interest is the wagon plus the child in it. The two forces exerted by the other children are external forces. An internal force acts between elements of the system. Again looking at Figure 4.5(a), the force the child in the wagon exerts to hang onto the wagon is an internal force between elements of the system of interest. Only external forces affect the motion of a system, according to Newton’s first law. (The internal forces actually cancel, as we shall see in the next section.) You must define the boundaries of the system before you can determine which forces are external. Sometimes the system is obvious, whereas other times identifying the boundaries of a system is more subtle. The concept of a system is fundamental to many areas of physics, as is the correct application of Newton’s laws. This concept will be revisited many times on our journey through physics. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 147 When we describe the acceleration of a system, we are modeling the system as a single point which contains all of the mass of that system. The point we choose for this is the point about which the system’s mass is evenly distributed. For example, in a rigid object, this center of mass is the point where the object will stay balanced even if only supported at this point. For a sphere or disk made of homogenous material, this point is of course at the center. Similarly, for a rod made of homogenous material, the center of mass will be at the midpoint. For the rider in the wagon in Figure 4.5, the center of mass is probably between the rider’s hips. Due to internal forces, the rider’s hand or hair may accelerate slightly differently, but it is the acceleration of the system’s center of mass that interests us. This is true whether the system is a vehicle carrying passengers, a bowl of grapes, or a planet. When we draw a free-body diagram of a system, we represent the system’s center of mass with a single point and use vectors to indicate the forces exerted on that center of mass. (See Figure 4.5.) Figure 4.5 Different forces exerted on the same mass produce different accelerations. (a) Two children push a wagon with a child in it. Arrows representing all external forces are shown. The system of interest is the wagon and its rider. The weight w of the system and the support of the ground N are also shown for completeness and are assumed to cancel. The vector f represents the friction acting on the wagon, and it acts to the left, opposing the motion of the wagon. (b) All of the external forces acting on the system add together to produce a net force, Fnet . The free-body diagram shows all of the forces acting on the system of interest. The dot represents the center of mass of the system. Each force vector extends from this dot. Because there are two forces acting to the right, we draw the vectors collinearly. (c) A larger net external force produces a larger acceleration ( a′ > a ) when an adult pushes the child. Now, it seems reasonable that acceleration should be directly proportional to and in the same direction as the net (total) external force acting on a system. This assumption has been verified experimentally and is illustrated in Figure 4.5. In part (a), a smaller force causes a smaller acceleration than the larger force illustrated in part (c). For completeness, the vertical forces are also shown; they are assumed to cancel since there is no acceleration in the vertical direction. The vertical forces are the weight w and the support of the ground N , and the horizontal force f represents the force of friction. These will be discussed in more detail in later sections. For now, we will define friction as a force that opposes the motion past each other of objects that are touching. Figure 4.5(b) shows how vectors representing the external forces add together to produce a net force, Fnet . To obtain an equation for Newton’s second law, we first write the relationship of acceleration and net external force as the proportionality a ∝ Fnet (4.1) where the symbol ∝ means “proportional to,” and Fnet is the net external force. (The net external force is the vector sum of all external forces and can be determined graphically, using the head-to-tail method, or analytically, using components. The techniques are the same as for the addition of other vectors, and are covered in Two-Dimensional Kinematics.) This proportionality states what we have said in words—acceleration is directly proportional to the net external force. Once the system of interest is chosen, it is important to identify the external forces and ignore the internal ones. It is a tremendous simplification not to have to consider the numerous internal forces acting between objects within the system, such as muscular forces within the child’s body, let alone the myriad of forces between atoms in the objects, but by doing so, we can easily solve some very complex problems with only minimal error due to our simplification 148 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion Now, it also seems reasonable that acceleration should be inversely proportional to the mass of the system. In other words, the larger the mass (the inertia), the smaller the acceleration produced by a given force. And indeed, as illustrated in Figure 4.6, the same net external force applied to a car produces a much smaller acceleration than when applied to a basketball. The proportionality is written as a ∝ 1 (4.2) where is the mass of the system. Experiments have shown that acceleration is exactly inversely proportional to mass, just as it is exactly linearly proportional to the net external force. Figure 4.6 The same force exerted on systems of different masses produces different accelerations. (a) A basketball player pushes on a basketball to make a pass. (The effect of gravity on the ball is ignored.) (b) The same player exerts an identical force on a stalled SUV and produces a far smaller acceleration (even if friction is negligible). (c) The free-body diagrams are identical, permitting direct comparison of the two situations. A series of patterns for the free-body diagram will emerge as you do more problems. Both of these proportionalities have been experimentally verified repeatedly and consistently, for a broad range of systems and scales. Thus, it has been experimentally found that the acceleration of an object depends only on the net external force and the mass of the object. Combining the two proportionalities just given yields Newton's second law of motion. Applying the Science Practices: Testing the Relationship Between Mass, Acceleration, and Force Plan three simple experiments using objects you have at home to test relationships between mass, acceleration, and force. (a) Design an experiment to test the relationship between mass and acceleration. What will be the independent variable in your experiment? What will be the dependent variable? What controls will you put in place to ensure force is constant? (b) Design a similar experiment to test the relationship between mass and force. What will be the independent variable in your experiment? What will be the dependent variable? What controls will you put in place to ensure acceleration is constant? (c) Design a similar experiment to test the relationship between force and acceleration. What will be the independent variable in your experiment? What will be the dependent variable? Will you have any trouble ensuring that the mass is constant? What did you learn? Newton’s Second Law of Motion The acceleration of a system is directly proportional to and in the same direction as the net external force acting on the system, and inversely proportional to its mass. In equation form, Newton’s second law of motion is This is often written in the more familiar form a = Fnet . Fnet = a. When only the magnitude of force and acceleration are considered, this equation is simply net = . (4.3) (4.4) (4.5) Although these last two equations are really the same, the first gives more insight into what Newton’s second law means. The law is a cause and effect relationship among three quantities that is not simply based on their definitions. The validity of the second law is completely based on experimental verification. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 149 Applying the Science Practices: Systems and Free-Body Diagrams First, consider a person on a sled sliding downhill. What is the system in this situation? Try to draw a free-body diagram describing this system, labeling all the forces and their directions. Which of the forces are internal? Which are external? Next, consider a person on a sled being pushed along level ground by a friend. What is the system in this situation? Try to draw a free-body diagram describing this system, labelling all the forces and their directions. Which of the forces are internal? Which are external? Units of Force Fn
et = a is used to define the units of force in terms of the three basic units for mass, length, and time. The SI unit of force is called the newton (abbreviated N) and is the force needed to accelerate a 1-kg system at the rate of 1m/s2 . That is, since Fnet = a , 1 N = 1 kg ⋅ m/s2. (4.6) While almost the entire world uses the newton for the unit of force, in the United States the most familiar unit of force is the pound (lb), where 1 N = 0.225 lb. Weight and the Gravitational Force When an object is dropped, it accelerates toward the center of Earth. Newton’s second law states that a net force on an object is responsible for its acceleration. If air resistance is negligible, the net force on a falling object is the gravitational force, commonly called its weight w . Weight can be denoted as a vector w because it has a direction; down is, by definition, the direction of gravity, and hence weight is a downward force. The magnitude of weight is denoted as . Galileo was instrumental in showing that, in the absence of air resistance, all objects fall with the same acceleration . Using Galileo’s result and Newton’s second law, we can derive an equation for weight. Consider an object with mass falling downward toward Earth. It experiences only the downward force of gravity, which has magnitude . Newton’s second law states that the magnitude of the net external force on an object is net = . Since the object experiences only the downward force of gravity, net = . We know that the acceleration of an object due to gravity is , or = . Substituting these into Newton’s second law gives Weight This is the equation for weight—the gravitational force on a mass : = . Since = 9.80 m/s2 on Earth, the weight of a 1.0 kg object on Earth is 9.8 N, as we see: = = (1.0 kg)(9.80 m/s2 ) = 9.8 N. (4.7) (4.8) Recall that can take a positive or negative value, depending on the positive direction in the coordinate system. Be sure to take this into consideration when solving problems with weight. When the net external force on an object is its weight, we say that it is in free-fall. That is, the only force acting on the object is the force of gravity. In the real world, when objects fall downward toward Earth, they are never truly in free-fall because there is always some upward force from the air acting on the object. The acceleration due to gravity varies slightly over the surface of Earth, so that the weight of an object depends on location and is not an intrinsic property of the object. Weight varies dramatically if one leaves Earth’s surface. On the Moon, for example, the acceleration due to gravity is only 1.67 m/s2 . A 1.0-kg mass thus has a weight of 9.8 N on Earth and only about 1.7 N on the Moon. The broadest definition of weight in this sense is that the weight of an object is the gravitational force on it from the nearest large body, such as Earth, the Moon, the Sun, and so on. This is the most common and useful definition of weight in physics. It differs dramatically, however, from the definition of weight used by NASA and the popular media in relation to space travel and exploration. When they speak of “weightlessness” and “microgravity,” they are really referring to the phenomenon we call “freefall” in physics. We shall use the above definition of weight, and we will make careful distinctions between free-fall and actual weightlessness. It is important to be aware that weight and mass are very different physical quantities, although they are closely related. Mass is the quantity of matter (how much “stuff”) and does not vary in classical physics, whereas weight is the gravitational force and 150 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion does vary depending on gravity. It is tempting to equate the two, since most of our examples take place on Earth, where the weight of an object only varies a little with the location of the object. Furthermore, the terms mass and weight are used interchangeably in everyday language; for example, our medical records often show our “weight” in kilograms, but never in the correct units of newtons. Common Misconceptions: Mass vs. Weight Mass and weight are often used interchangeably in everyday language. However, in science, these terms are distinctly different from one another. Mass is a measure of how much matter is in an object. The typical measure of mass is the kilogram (or the “slug” in English units). Weight, on the other hand, is a measure of the force of gravity acting on an object. Weight is equal to the mass of an object ( ) multiplied by the acceleration due to gravity ( ). Like any other force, weight is measured in terms of newtons (or pounds in English units). Assuming the mass of an object is kept intact, it will remain the same, regardless of its location. However, because weight depends on the acceleration due to gravity, the weight of an object can change when the object enters into a region with stronger or weaker gravity. For example, the acceleration due to gravity on the Moon is 1.67 m/s2 (which is much less than the acceleration due to gravity on Earth, 9.80 m/s2 ). If you measured your weight on Earth and then measured your weight on the Moon, you would find that you “weigh” much less, even though you do not look any skinnier. This is because the force of gravity is weaker on the Moon. In fact, when people say that they are “losing weight,” they really mean that they are losing “mass” (which in turn causes them to weigh less). Take-Home Experiment: Mass and Weight What do bathroom scales measure? When you stand on a bathroom scale, what happens to the scale? It depresses slightly. The scale contains springs that compress in proportion to your weight—similar to rubber bands expanding when pulled. The springs provide a measure of your weight (for an object which is not accelerating). This is a force in newtons (or pounds). In most countries, the measurement is divided by 9.80 to give a reading in mass units of kilograms. The scale measures weight but is calibrated to provide information about mass. While standing on a bathroom scale, push down on a table next to you. What happens to the reading? Why? Would your scale measure the same “mass” on Earth as on the Moon? Example 4.1 What Acceleration Can a Person Produce when Pushing a Lawn Mower? Suppose that the net external force (push minus friction) exerted on a lawn mower is 51 N (about 11 lb) parallel to the ground. The mass of the mower is 24 kg. What is its acceleration? Figure 4.7 The net force on a lawn mower is 51 N to the right. At what rate does the lawn mower accelerate to the right? Strategy Since Fnet and are given, the acceleration can be calculated directly from Newton’s second law as stated in Fnet = a . Solution The magnitude of the acceleration is = net . Entering known values gives = 51 N 24 kg (4.9) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion Substituting the units kg ⋅ m/s2 for N yields = 51 kg ⋅ m/s2 24 kg = 2.1 m/s2. 151 (4.10) Discussion The direction of the acceleration is the same direction as that of the net force, which is parallel to the ground. There is no information given in this example about the individual external forces acting on the system, but we can say something about their relative magnitudes. For example, the force exerted by the person pushing the mower must be greater than the friction opposing the motion (since we know the mower moves forward), and the vertical forces must cancel if there is to be no acceleration in the vertical direction (the mower is moving only horizontally). The acceleration found is small enough to be reasonable for a person pushing a mower. Such an effort would not last too long because the person’s top speed would soon be reached. Example 4.2 What Rocket Thrust Accelerates This Sled? Prior to manned space flights, rocket sleds were used to test aircraft, missile equipment, and physiological effects on human subjects at high speeds. They consisted of a platform that was mounted on one or two rails and propelled by several rockets. Calculate the magnitude of force exerted by each rocket, called its thrust T , for the four-rocket propulsion system shown in Figure 4.8. The sled’s initial acceleration is 49 m/s2, opposing the motion is known to be 650 N. the mass of the system is 2100 kg, and the force of friction Figure 4.8 A sled experiences a rocket thrust that accelerates it to the right. Each rocket creates an identical thrust T . As in other situations where there is only horizontal acceleration, the vertical forces cancel. The ground exerts an upward force N on the system that is equal in magnitude and opposite in direction to its weight, w . The system here is the sled, its rockets, and rider, so none of the forces between these objects are considered. The arrow representing friction ( f ) is drawn larger than scale. Strategy Although there are forces acting vertically and horizontally, we assume the vertical forces cancel since there is no vertical acceleration. This leaves us with only horizontal forces and a simpler one-dimensional problem. Directions are indicated with plus or minus signs, with right taken as the positive direction. See the free-body diagram in the figure. Solution Since acceleration, mass, and the force of friction are given, we start with Newton’s second law and look for ways to find the thrust of the engines. Since we have defined the direction of the force and acceleration as acting “to the right,” we need to consider only the magnitudes of these quantities in the calculations. Hence we begin with net = , (4.11) where net is the net force along the horizontal direction. We can see from Figure 4.8 that the engine thrusts add, while friction opposes the thrust. In equation form, the net external force is net = 4 − . (4.12) 152 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion Substituting this into Newton’s second law gives net
= = 4 − . Using a little algebra, we solve for the total thrust 4T: 4 = + . Substituting known values yields 4 = + = (2100 kg)(49 m/s2 ) + 650 N. So the total thrust is and the individual thrusts are Discussion 4 = 1.0×105 N, = 1.0×105 N 4 = 2.6×104 N. (4.13) (4.14) (4.15) (4.16) (4.17) The numbers are quite large, so the result might surprise you. Experiments such as this were performed in the early 1960s to test the limits of human endurance and the setup designed to protect human subjects in jet fighter emergency ejections. Speeds of 1000 km/h were obtained, with accelerations of 45 's. (Recall that , the acceleration due to gravity, is 9.80 m/s2 . When we say that an acceleration is 45 's, it is 459.80 m/s2 , which is approximately 440 m/s2 .) While living subjects are not used any more, land speeds of 10,000 km/h have been obtained with rocket sleds. In this example, as in the preceding one, the system of interest is obvious. We will see in later examples that choosing the system of interest is crucial—and the choice is not always obvious. Newton’s second law of motion is more than a definition; it is a relationship among acceleration, force, and mass. It can help us make predictions. Each of those physical quantities can be defined independently, so the second law tells us something basic and universal about nature. The next section introduces the third and final law of motion. Applying the Science Practices: Sums of Forces Recall that forces are vector quantities, and therefore the net force acting on a system should be the vector sum of the forces. (a) Design an experiment to test this hypothesis. What sort of a system would be appropriate and convenient to have multiple forces applied to it? What features of the system should be held constant? What could be varied? Can forces be arranged in multiple directions so that, while the hypothesis is still tested, the resulting calculations are not too inconvenient? (b) Another group of students has done such an experiment, using a motion capture system looking down at an air hockey table to measure the motion of the 0.10-kg puck. The table was aligned with the cardinal directions, and a compressed air hose was placed in the center of each side, capable of varying levels of force output and fixed so that it was aimed at the center of the table. Table 4.1 Forces Measured acceleration (magnitudes) 3 N north, 4 N west 5 N south, 12 N east 48 ± 4 m/s2 132 ± 6 m/s2 6 N north, 12 N east, 4 N west 99 ± 3 m/s2 Given the data in the table, is the hypothesis confirmed? What were the directions of the accelerations? 4.4 Newton's Third Law of Motion: Symmetry in Forces Learning Objectives By the end of this section, you will be able to: • Understand Newton's third law of motion. • Apply Newton's third law to define systems and solve problems of motion. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 153 The information presented in this section supports the following AP® learning objectives and science practices: • 3.A.2.1 The student is able to represent forces in diagrams or mathematically using appropriately labeled vectors with magnitude, direction, and units during the analysis of a situation. (S.P. 1.1) • 3.A.3.1 The student is able to analyze a scenario and make claims (develop arguments, justify assertions) about the forces exerted on an object by other objects for different types of forces or components of forces. (S.P. 6.4, 7.2) • 3.A.3.3 The student is able to describe a force as an interaction between two objects and identify both objects for any force. (S.P. 1.4) • 3.A.4.1 The student is able to construct explanations of physical situations involving the interaction of bodies using Newton's third law and the representation of action-reaction pairs of forces. (S.P. 1.4, 6.2) • 3.A.4.2 The student is able to use Newton's third law to make claims and predictions about the action-reaction pairs of forces when two objects interact. (S.P. 6.4, 7.2) • 3.A.4.3 The student is able to analyze situations involving interactions among several objects by using free-body diagrams that include the application of Newton's third law to identify forces. (S.P. 1.4) • 3.B.2.1 The student is able to create and use free-body diagrams to analyze physical situations to solve problems with motion qualitatively and quantitatively. (S.P. 1.1, 1.4, 2.2) • 4.A.2.1 The student is able to make predictions about the motion of a system based on the fact that acceleration is equal to the change in velocity per unit time, and velocity is equal to the change in position per unit time. (S.P. 6.4) • 4.A.2.2 The student is able to evaluate using given data whether all the forces on a system or whether all the parts of a system have been identified. (S.P. 5.3) • 4.A.3.1 The student is able to apply Newton's second law to systems to calculate the change in the center-of-mass velocity when an external force is exerted on the system. (S.P. 2.2) There is a passage in the musical Man of la Mancha that relates to Newton’s third law of motion. Sancho, in describing a fight with his wife to Don Quixote, says, “Of course I hit her back, Your Grace, but she’s a lot harder than me and you know what they say, ‘Whether the stone hits the pitcher or the pitcher hits the stone, it’s going to be bad for the pitcher.’” This is exactly what happens whenever one body exerts a force on another—the first also experiences a force (equal in magnitude and opposite in direction). Numerous common experiences, such as stubbing a toe or throwing a ball, confirm this. It is precisely stated in Newton’s third law of motion. Newton’s Third Law of Motion Whenever one body exerts a force on a second body, the first body experiences a force that is equal in magnitude and opposite in direction to the force that it exerts. This law represents a certain symmetry in nature: Forces always occur in pairs, and one body cannot exert a force on another without experiencing a force itself. We sometimes refer to this law loosely as “action-reaction,” where the force exerted is the action and the force experienced as a consequence is the reaction. Newton’s third law has practical uses in analyzing the origin of forces and understanding which forces are external to a system. We can readily see Newton’s third law at work by taking a look at how people move about. Consider a swimmer pushing off from the side of a pool, as illustrated in Figure 4.9. She pushes against the pool wall with her feet and accelerates in the direction opposite to that of her push. The wall has exerted an equal and opposite force back on the swimmer. You might think that two equal and opposite forces would cancel, but they do not because they act on different systems. In this case, there are two systems that we could investigate: the swimmer or the wall. If we select the swimmer to be the system of interest, as in the figure, then Fwall on feet is an external force on this system and affects its motion. The swimmer moves in the direction of Fwall on feet . In contrast, the force Ffeet on wall acts on the wall and not on our system of interest. Thus Ffeet on wall does not directly affect the motion of the system and does not cancel Fwall on feet . Note that the swimmer pushes in the direction opposite to that in which she wishes to move. The reaction to her push is thus in the desired direction. 154 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion Figure 4.9 When the swimmer exerts a force Ffeet on wall on the wall, she accelerates in the direction opposite to that of her push. This means the net external force on her is in the direction opposite to Ffeet on wall . This opposition occurs because, in accordance with Newton’s third law of motion, the wall exerts a force Fwall on feet on her, equal in magnitude but in the direction opposite to the one she exerts on it. The line around the swimmer indicates the system of interest. Note that Ffeet on wall does not act on this system (the swimmer) and, thus, does not cancel Fwall on feet . Thus the free-body diagram shows only Fwall on feet , w , the gravitational force, and BF , the buoyant force of the water supporting the swimmer’s weight. The vertical forces w and BF cancel since there is no vertical motion. Similarly, when a person stands on Earth, the Earth exerts a force on the person, pulling the person toward the Earth. As stated by Newton’s third law of motion, the person also exerts a force that is equal in magnitude, but opposite in direction, pulling the Earth up toward the person. Since the mass of the Earth is so great, however, and = , the acceleration of the Earth toward the person is not noticeable. Other examples of Newton’s third law are easy to find. As a professor paces in front of a whiteboard, she exerts a force backward on the floor. The floor exerts a reaction force forward on the professor that causes her to accelerate forward. Similarly, a car accelerates because the ground pushes forward on the drive wheels in reaction to the drive wheels pushing backward on the ground. You can see evidence of the wheels pushing backward when tires spin on a gravel road and throw rocks backward. In another example, rockets move forward by expelling gas backward at high velocity. This means the rocket exerts a large backward force on the gas in the rocket combustion chamber, and the gas therefore exerts a large reaction force forward on the rocket. This reaction force is called thrust. It is a common misconception that rockets propel themselves by pushing on the ground or on the air behind them. They actually work better in a vacuum, where they can more readily expel the exhaust gases. Helicopters similarly create lift by pushing air down, thereby experiencing an upward reaction force. Birds and airplanes also fly by exerting force on air in a direction opposite to that of whatever force they need. For example, the wings of a b
ird force air downward and backward in order to get lift and move forward. An octopus propels itself in the water by ejecting water through a funnel from its body, similar to a jet ski. In a situation similar to Sancho’s, professional cage fighters experience reaction forces when they punch, sometimes breaking their hand by hitting an opponent’s body. Example 4.3 Getting Up To Speed: Choosing the Correct System A physics professor pushes a cart of demonstration equipment to a lecture hall, as seen in Figure 4.10. Her mass is 65.0 kg, the cart’s is 12.0 kg, and the equipment’s is 7.0 kg. Calculate the acceleration produced when the professor exerts a backward force of 150 N on the floor. All forces opposing the motion, such as friction on the cart’s wheels and air resistance, total 24.0 N. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 155 Figure 4.10 A professor pushes a cart of demonstration equipment. The lengths of the arrows are proportional to the magnitudes of the forces (except for f , since it is too small to draw to scale). Different questions are asked in each example; thus, the system of interest must be defined differently for each. System 1 is appropriate for Example 4.4, since it asks for the acceleration of the entire group of objects. Only Ffloor and f are external forces acting on System 1 along the line of motion. All other forces either cancel or act on the outside world. System 2 is chosen for this example so that Fprof will be an external force and enter into Newton’s second law. Note that the free-body diagrams, which allow us to apply Newton’s second law, vary with the system chosen. Strategy Since they accelerate as a unit, we define the system to be the professor, cart, and equipment. This is System 1 in Figure 4.10. The professor pushes backward with a force Ffoot of 150 N. According to Newton’s third law, the floor exerts a forward reaction force Ffloor of 150 N on System 1. Because all motion is horizontal, we can assume there is no net force in the vertical direction. The problem is therefore one-dimensional along the horizontal direction. As noted, f opposes the motion and is thus in the opposite direction of Ffloor . Note that we do not include the forces Fprof or Fcart because these are internal forces, and we do not include Ffoot because it acts on the floor, not on the system. There are no other significant forces acting on System 1. If the net external force can be found from all this information, we can use Newton’s second law to find the acceleration as requested. See the free-body diagram in the figure. Solution Newton’s second law is given by = net . The net external force on System 1 is deduced from Figure 4.10 and the discussion above to be net = floor − = 150 N − 24.0 N = 126 N. The mass of System 1 is = (65.0 + 12.0 + 7.0) kg = 84 kg. These values of net and produce an acceleration of net = = 126 N 84 kg = 1.5 m/s2 . Discussion (4.18) (4.19) (4.20) (4.21) 156 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion None of the forces between components of System 1, such as between the professor’s hands and the cart, contribute to the net external force because they are internal to System 1. Another way to look at this is to note that forces between components of a system cancel because they are equal in magnitude and opposite in direction. For example, the force exerted by the professor on the cart results in an equal and opposite force back on her. In this case both forces act on the same system and, therefore, cancel. Thus internal forces (between components of a system) cancel. Choosing System 1 was crucial to solving this problem. Example 4.4 Force on the Cart—Choosing a New System Calculate the force the professor exerts on the cart in Figure 4.10 using data from the previous example if needed. Strategy If we now define the system of interest to be the cart plus equipment (System 2 in Figure 4.10), then the net external force on System 2 is the force the professor exerts on the cart minus friction. The force she exerts on the cart, Fprof , is an external force acting on System 2. Fprof was internal to System 1, but it is external to System 2 and will enter Newton’s second law for System 2. Solution Newton’s second law can be used to find Fprof . Starting with = net and noting that the magnitude of the net external force on System 2 is net = prof − , we solve for prof , the desired quantity: prof = net + . (4.22) (4.23) (4.24) The value of is given, so we must calculate net net . That can be done since both the acceleration and mass of System 2 are known. Using Newton’s second law we see that net = , (4.25) where the mass of System 2 is 19.0 kg ( = 12.0 kg + 7.0 kg) and its acceleration was found to be = 1.5 m/s2 in the previous example. Thus, Now we can find the desired force: net = , net = (19.0 kg)(1.5 m/s2 ) = 29 N. prof = net + , prof = 29 N+24.0 N = 53 N. (4.26) (4.27) (4.28) (4.29) Discussion It is interesting that this force is significantly less than the 150-N force the professor exerted backward on the floor. Not all of that 150-N force is transmitted to the cart; some of it accelerates the professor. The choice of a system is an important analytical step both in solving problems and in thoroughly understanding the physics of the situation (which is not necessarily the same thing). PhET Explorations: Gravity Force Lab Visualize the gravitational force that two objects exert on each other. Change properties of the objects in order to see how it changes the gravity force. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 157 Figure 4.11 Gravity Force Lab (http://cnx.org/content/m54849/1.2/gravity-force-lab_en.jar) 4.5 Normal, Tension, and Other Examples of Force Learning Objectives By the end of this section, you will be able to: • Define normal and tension forces. • Apply Newton's laws of motion to solve problems involving a variety of forces. • Use trigonometric identities to resolve weight into components. The information presented in this section supports the following AP® learning objectives and science practices: • 2.B.1.1 The student is able to apply = to calculate the gravitational force on an object with mass m in a gravitational field of strength g in the context of the effects of a net force on objects and systems. (S.P. 2.2, 7.2) • 3.A.2.1 The student is able to represent forces in diagrams or mathematically using appropriately labeled vectors with magnitude, direction, and units during the analysis of a situation. (S.P. 1.1) • 3.A.3.1 The student is able to analyze a scenario and make claims (develop arguments, justify assertions) about the forces exerted on an object by other objects for different types of forces or components of forces. (S.P. 6.4, 7.2) • 3.A.3.3 The student is able to describe a force as an interaction between two objects and identify both objects for any force. (S.P. 1.4) • 3.A.4.1 The student is able to construct explanations of physical situations involving the interaction of bodies using Newton's third law and the representation of action-reaction pairs of forces. (S.P. 1.4, 6.2) • 3.A.4.2 The student is able to use Newton's third law to make claims and predictions about the action-reaction pairs of forces when two objects interact. (S.P. 6.4, 7.2) • 3.A.4.3 The student is able to analyze situations involving interactions among several objects by using free-body diagrams that include the application of Newton's third law to identify forces. (S.P. 1.4) • 3.B.1.3 The student is able to re-express a free-body diagram representation into a mathematical representation and solve the mathematical representation for the acceleration of the object. (S.P. 1.5, 2.2) • 3.B.2.1 The student is able to create and use free-body diagrams to analyze physical situations to solve problems with motion qualitatively and quantitatively. (S.P. 1.1, 1.4, 2.2) Forces are given many names, such as push, pull, thrust, lift, weight, friction, and tension. Traditionally, forces have been grouped into several categories and given names relating to their source, how they are transmitted, or their effects. The most important of these categories are discussed in this section, together with some interesting applications. Further examples of forces are discussed later in this text. Normal Force Weight (also called force of gravity) is a pervasive force that acts at all times and must be counteracted to keep an object from falling. You definitely notice that you must support the weight of a heavy object by pushing up on it when you hold it stationary, as illustrated in Figure 4.12(a). But how do inanimate objects like a table support the weight of a mass placed on them, such as shown in Figure 4.12(b)? When the bag of dog food is placed on the table, the table actually sags slightly under the load. This would be noticeable if the load were placed on a card table, but even rigid objects deform when a force is applied to them. Unless the object is deformed beyond its limit, it will exert a restoring force much like a deformed spring (or trampoline or diving board). The greater the deformation, the greater the restoring force. So when the load is placed on the table, the table sags until the restoring force becomes as large as the weight of the load. At this point the net external force on the load is zero. That is the situation when the load is stationary on the table. The table sags quickly, and the sag is slight so we do not notice it. But it is similar to the sagging of a trampoline when you climb onto it. 158 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion Figure 4.12 (a) The person holding the bag of dog food must supply an upward force Fhand equal in magnitude and opposite in direction to the weight of the food w . (b) The card table sags when the dog food is placed on it, much like
a stiff trampoline. Elastic restoring forces in the table grow as it sags until they supply a force N equal in magnitude and opposite in direction to the weight of the load. We must conclude that whatever supports a load, be it animate or not, must supply an upward force equal to the weight of the load, as we assumed in a few of the previous examples. If the force supporting a load is perpendicular to the surface of contact between the load and its support, this force is defined to be a normal force and here is given the symbol N . (This is not the unit for force N.) The word normal means perpendicular to a surface. The normal force can be less than the object’s weight if the object is on an incline, as you will see in the next example. Common Misconception: Normal Force (N) vs. Newton (N) In this section we have introduced the quantity normal force, which is represented by the variable N . This should not be confused with the symbol for the newton, which is also represented by the letter N. These symbols are particularly important to distinguish because the units of a normal force ( N ) happen to be newtons (N). For example, the normal force N that the floor exerts on a chair might be N = 100 N . One important difference is that normal force is a vector, while the newton is simply a unit. Be careful not to confuse these letters in your calculations! You will encounter more similarities among variables and units as you proceed in physics. Another example of this is the quantity work ( ) and the unit watts (W). Example 4.5 Weight on an Incline, a Two-Dimensional Problem Consider the skier on a slope shown in Figure 4.13. Her mass including equipment is 60.0 kg. (a) What is her acceleration if friction is negligible? (b) What is her acceleration if friction is known to be 45.0 N? This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 159 Figure 4.13 Since motion and friction are parallel to the slope, it is most convenient to project all forces onto a coordinate system where one axis is parallel to the slope and the other is perpendicular (axes shown to left of skier). N is perpendicular to the slope and f is parallel to the slope, but w has components along both axes, namely w⊥ and w ∥ . N is equal in magnitude to w⊥ , so that there is no motion perpendicular to the slope, but is less than ∥ , so that there is a downslope acceleration (along the parallel axis). Strategy This is a two-dimensional problem, since the forces on the skier (the system of interest) are not parallel. The approach we have used in two-dimensional kinematics also works very well here. Choose a convenient coordinate system and project the vectors onto its axes, creating two connected one-dimensional problems to solve. The most convenient coordinate system for motion on an incline is one that has one coordinate parallel to the slope and one perpendicular to the slope. (Remember that motions along mutually perpendicular axes are independent.) We use the symbols ⊥ and ∥ to represent perpendicular and parallel, respectively. This choice of axes simplifies this type of problem, because there is no motion perpendicular to the slope and because friction is always parallel to the surface between two objects. The only external forces acting on the system are the skier’s weight, friction, and the support of the slope, respectively labeled w , f , and N in Figure 4.13. N is always perpendicular to the slope, and f is parallel to it. But w is not in the direction of either axis, and so the first step we take is to project it into components along the chosen axes, defining ∥ to be the component of weight parallel to the slope and ⊥ the component of weight perpendicular to the slope. Once this is done, we can consider the two separate problems of forces parallel to the slope and forces perpendicular to the slope. Solution The magnitude of the component of the weight parallel to the slope is ∥ = sin (25º) = sin (25º) , and the magnitude of the component of the weight perpendicular to the slope is ⊥ = cos (25º) = cos (25º) . (a) Neglecting friction. Since the acceleration is parallel to the slope, we need only consider forces parallel to the slope. (Forces perpendicular to the slope add to zero, since there is no acceleration in that direction.) The forces parallel to the slope are the amount of the skier’s weight parallel to the slope ∥ and friction . Using Newton’s second law, with subscripts to denote quantities parallel to the slope, where net ∥ = ∥ = sin (25º) , assuming no friction for this part, so that ∥ = net ∥ ∥ = net ∥ = sin (25º) (9.80 m/s2)(0.4226) = 4.14 m/s2 = sin (25º) (4.30) (4.31) (4.32) is the acceleration. (b) Including friction. We now have a given value for friction, and we know its direction is parallel to the slope and it opposes motion between surfaces in contact. So the net external force is now net ∥ = ∥ − , (4.33) and substituting this into Newton’s second law, ∥ = net ∥ , gives 160 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion ∥ = net ∣ ∣ = ∥ − = sin (25º) − . We substitute known values to obtain ∥ = (60.0 kg)(9.80 m/s2)(0.4226) − 45.0 N 60.0 kg , which yields ∥ = 3.39 m/s2, (4.34) (4.35) (4.36) which is the acceleration parallel to the incline when there is 45.0 N of opposing friction. Discussion Since friction always opposes motion between surfaces, the acceleration is smaller when there is friction than when there is none. In fact, it is a general result that if friction on an incline is negligible, then the acceleration down the incline is = sin , regardless of mass. This is related to the previously discussed fact that all objects fall with the same acceleration in the absence of air resistance. Similarly, all objects, regardless of mass, slide down a frictionless incline with the same acceleration (if the angle is the same). Resolving Weight into Components Figure 4.14 An object rests on an incline that makes an angle θ with the horizontal. When an object rests on an incline that makes an angle with the horizontal, the force of gravity acting on the object is divided into two components: a force acting perpendicular to the plane, w⊥ , and a force acting parallel to the plane, w ∥ . The perpendicular force of weight, w⊥ , is typically equal in magnitude and opposite in direction to the normal force, N . The force acting parallel to the plane, w ∥ , causes the object to accelerate down the incline. The force of friction, f , opposes the motion of the object, so it acts upward along the plane. It is important to be careful when resolving the weight of the object into components. If the angle of the incline is at an angle to the horizontal, then the magnitudes of the weight components are and ∥ = sin () = sin () ⊥ = cos () = cos (). (4.37) (4.38) Instead of memorizing these equations, it is helpful to be able to determine them from reason. To do this, draw the right triangle formed by the three weight vectors. Notice that the angle of the incline is the same as the angle formed between w and w⊥ . Knowing this property, you can use trigonometry to determine the magnitude of the weight components: cos () = ⊥ ⊥ = cos () = cos () ∥ sin () = ∥ = sin () = sin () (4.39) (4.40) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 161 Take-Home Experiment: Force Parallel To investigate how a force parallel to an inclined plane changes, find a rubber band, some objects to hang from the end of the rubber band, and a board you can position at different angles. How much does the rubber band stretch when you hang the object from the end of the board? Now place the board at an angle so that the object slides off when placed on the board. How much does the rubber band extend if it is lined up parallel to the board and used to hold the object stationary on the board? Try two more angles. What does this show? Tension A tension is a force along the length of a medium, especially a force carried by a flexible medium, such as a rope or cable. The word “tension” comes from a Latin word meaning “to stretch.” Not coincidentally, the flexible cords that carry muscle forces to other parts of the body are called tendons. Any flexible connector, such as a string, rope, chain, wire, or cable, can exert pulls only parallel to its length; thus, a force carried by a flexible connector is a tension with direction parallel to the connector. It is important to understand that tension is a pull in a connector. In contrast, consider the phrase: “You can’t push a rope.” The tension force pulls outward along the two ends of a rope. Consider a person holding a mass on a rope as shown in Figure 4.15. Figure 4.15 When a perfectly flexible connector (one requiring no force to bend it) such as this rope transmits a force T , that force must be parallel to the length of the rope, as shown. The pull such a flexible connector exerts is a tension. Note that the rope pulls with equal force but in opposite directions on the hand and the supported mass (neglecting the weight of the rope). This is an example of Newton’s third law. The rope is the medium that carries the equal and opposite forces between the two objects. The tension anywhere in the rope between the hand and the mass is equal. Once you have determined the tension in one location, you have determined the tension at all locations along the rope. Tension in the rope must equal the weight of the supported mass, as we can prove using Newton’s second law. If the 5.00-kg mass in the figure is stationary, then its acceleration is zero, and thus Fnet = 0 . The only external forces acting on the mass are its weight w and the tension T supplied by the rope. Thus, net = − = 0, (4.41) where and are the magnitudes of the tension and weight and their signs indicate direction, with up being positive here. Thus, just as you would expect, the tension equals the weigh
t of the supported mass: For a 5.00-kg mass, then (neglecting the mass of the rope) we see that = = (5.00 kg)(9.80 m/s2 ) = 49.0 N. = = . (4.42) (4.43) If we cut the rope and insert a spring, the spring would extend a length corresponding to a force of 49.0 N, providing a direct observation and measure of the tension force in the rope. Flexible connectors are often used to transmit forces around corners, such as in a hospital traction system, a finger joint, or a bicycle brake cable. If there is no friction, the tension is transmitted undiminished. Only its direction changes, and it is always parallel to the flexible connector. This is illustrated in Figure 4.16 (a) and (b). 162 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion Figure 4.16 (a) Tendons in the finger carry force T from the muscles to other parts of the finger, usually changing the force’s direction, but not its magnitude (the tendons are relatively friction free). (b) The brake cable on a bicycle carries the tension T from the handlebars to the brake mechanism. Again, the direction but not the magnitude of T is changed. Example 4.6 What Is the Tension in a Tightrope? Calculate the tension in the wire supporting the 70.0-kg tightrope walker shown in Figure 4.17. Figure 4.17 The weight of a tightrope walker causes a wire to sag by 5.0 degrees. The system of interest here is the point in the wire at which the tightrope walker is standing. Strategy As you can see in the figure, the wire is not perfectly horizontal (it cannot be!), but is bent under the person’s weight. Thus, the tension on either side of the person has an upward component that can support his weight. As usual, forces are vectors represented pictorially by arrows having the same directions as the forces and lengths proportional to their magnitudes. The system is the tightrope walker, and the only external forces acting on him are his weight w and the two tensions TL (left tension) and TR (right tension), as illustrated. It is reasonable to neglect the weight of the wire itself. The net external force is zero since the system is stationary. A little trigonometry can now be used to find the tensions. One conclusion is possible at the outset—we can see from part (b) of the figure that the magnitudes of the tensions L and R must be equal. This is because there is no horizontal acceleration in the rope, and the only forces acting to the left and right are L and . Thus, the magnitude of those forces must be equal so that they cancel each other out. Whenever we have two-dimensional vector problems in which no two vectors are parallel, the easiest method of solution is to pick a convenient coordinate system and project the vectors onto its axes. In this case the best coordinate system has one axis horizontal and the other vertical. We call the horizontal the -axis and the vertical the -axis. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 163 Solution First, we need to resolve the tension vectors into their horizontal and vertical components. It helps to draw a new free-body diagram showing all of the horizontal and vertical components of each force acting on the system. Figure 4.18 When the vectors are projected onto vertical and horizontal axes, their components along those axes must add to zero, since the tightrope walker is stationary. The small angle results in being much greater than . Consider the horizontal components of the forces (denoted with a subscript ): net = L − R. The net external horizontal force net = 0 , since the person is stationary. Thus, net = 0 = L − R L = R . Now, observe Figure 4.18. You can use trigonometry to determine the magnitude of L and R . Notice that: cos (5.0º) = L L L = L cos (5.0º) cos (5.0º) = R R R = R cos (5.0º). L cos (5.0º) = R cos (5.0º). L = R = , Equating L and R : Thus, (4.44) (4.45) (4.46) (4.47) (4.48) as predicted. Now, considering the vertical components (denoted by a subscript ), we can solve for . Again, since the person is stationary, Newton’s second law implies that net = 0 . Thus, as illustrated in the free-body diagram in Figure 4.18, net = L + R − = 0. Observing Figure 4.18, we can use trigonometry to determine the relationship between L , R , and . As we determined from the analysis in the horizontal direction, L = R = : sin (5.0º) = L L L = L sin (5.0º) = sin (5.0º) sin (5.0º) = R R R = R sin (5.0º) = sin (5.0º). (4.49) (4.50) 164 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion Now, we can substitute the values for L and R , into the net force equation in the vertical direction: = L + R − = 0 = sin (5.0º) + sin (5.0º) − = 0 net net 2 sin (5.0º) − = 0 = 2 sin (5.0º) and so that and the tension is Discussion = 2 sin (5.0º) = 2 sin (5.0º) , = (70.0 kg)(9.80 m/s2) 2(0.0872) , = 3900 N. (4.51) (4.52) (4.53) (4.54) Note that the vertical tension in the wire acts as a normal force that supports the weight of the tightrope walker. The tension is almost six times the 686-N weight of the tightrope walker. Since the wire is nearly horizontal, the vertical component of its tension is only a small fraction of the tension in the wire. The large horizontal components are in opposite directions and cancel, and so most of the tension in the wire is not used to support the weight of the tightrope walker. If we wish to create a very large tension, all we have to do is exert a force perpendicular to a flexible connector, as illustrated in Figure 4.19. As we saw in the last example, the weight of the tightrope walker acted as a force perpendicular to the rope. We saw that the tension in the roped related to the weight of the tightrope walker in the following way: = 2 sin () . (4.55) We can extend this expression to describe the tension created when a perpendicular force ( F⊥ ) is exerted at the middle of a flexible connector: = ⊥ 2 sin () . (4.56) Note that is the angle between the horizontal and the bent connector. In this case, becomes very large as approaches zero. Even the relatively small weight of any flexible connector will cause it to sag, since an infinite tension would result if it were horizontal (i.e., = 0 and sin = 0 ). (See Figure 4.19.) Figure 4.19 We can create a very large tension in the chain by pushing on it perpendicular to its length, as shown. Suppose we wish to pull a car out of the mud when no tow truck is available. Each time the car moves forward, the chain is tightened to keep it as nearly straight as possible. The tension in the chain is given by = ⊥ 2 sin () ; since is small, is very large. This situation is analogous to the tightrope walker shown in Figure 4.17, except that the tensions shown here are those transmitted to the car and the tree rather than those acting at the point where F⊥ is applied. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 165 Figure 4.20 Unless an infinite tension is exerted, any flexible connector—such as the chain at the bottom of the picture—will sag under its own weight, giving a characteristic curve when the weight is evenly distributed along the length. Suspension bridges—such as the Golden Gate Bridge shown in this image—are essentially very heavy flexible connectors. The weight of the bridge is evenly distributed along the length of flexible connectors, usually cables, which take on the characteristic shape. (credit: Leaflet, Wikimedia Commons) Extended Topic: Real Forces and Inertial Frames There is another distinction among forces in addition to the types already mentioned. Some forces are real, whereas others are not. Real forces are those that have some physical origin, such as the gravitational pull. Contrastingly, fictitious forces are those that arise simply because an observer is in an accelerating frame of reference, such as one that rotates (like a merry-go-round) or undergoes linear acceleration (like a car slowing down). For example, if a satellite is heading due north above Earth’s northern hemisphere, then to an observer on Earth it will appear to experience a force to the west that has no physical origin. Of course, what is happening here is that Earth is rotating toward the east and moves east under the satellite. In Earth’s frame this looks like a westward force on the satellite, or it can be interpreted as a violation of Newton’s first law (the law of inertia). An inertial frame of reference is one in which all forces are real and, equivalently, one in which Newton’s laws have the simple forms given in this chapter. Earth’s rotation is slow enough that Earth is nearly an inertial frame. You ordinarily must perform precise experiments to observe fictitious forces and the slight departures from Newton’s laws, such as the effect just described. On the large scale, such as for the rotation of weather systems and ocean currents, the effects can be easily observed. The crucial factor in determining whether a frame of reference is inertial is whether it accelerates or rotates relative to a known inertial frame. Unless stated otherwise, all phenomena discussed in this text are considered in inertial frames. All the forces discussed in this section are real forces, but there are a number of other real forces, such as lift and thrust, that are not discussed in this section. They are more specialized, and it is not necessary to discuss every type of force. It is natural, however, to ask where the basic simplicity we seek to find in physics is in the long list of forces. Are some more basic than others? Are some different manifestations of the same underlying force? The answer to both questions is yes, as will be seen in the next (extended) section and in the treatment of modern physics later in the text. PhET Explorations: Forces in 1 Dimension Explore the forces at work when you try to push a filing cabinet. Create an applied force and see the resulting friction force and total force acting on the cabine
t. Charts show the forces, position, velocity, and acceleration vs. time. View a free-body diagram of all the forces (including gravitational and normal forces). Figure 4.21 Forces in 1 Dimension (http://cnx.org/content/m54857/1.4/forces-1d_en.jar) 4.6 Problem-Solving Strategies By the end of this section, you will be able to: Learning Objectives 166 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion • Apply a problem-solving procedure to solve problems using Newton's laws of motion The information presented in this section supports the following AP® learning objectives and science practices: • 3.A.2.1 The student is able to represent forces in diagrams or mathematically using appropriately labeled vectors with magnitude, direction, and units during the analysis of a situation. (S.P. 1.1) • 3.A.3.3 The student is able to describe a force as an interaction between two objects and identify both objects for any force. (S.P. 1.4) • 3.B.1.1 The student is able to predict the motion of an object subject to forces exerted by several objects using an application of Newton's second law in a variety of physical situations with acceleration in one dimension. (S.P. 6.4, 7.2) • 3.B.1.3 The student is able to re-express a free-body diagram representation into a mathematical representation and solve the mathematical representation for the acceleration of the object. (S.P. 1.5, 2.2) • 3.B.2.1 The student is able to create and use free-body diagrams to analyze physical situations to solve problems with motion qualitatively and quantitatively. (S.P. 1.1, 1.4, 2.2) Success in problem solving is obviously necessary to understand and apply physical principles, not to mention the more immediate need of passing exams. The basics of problem solving, presented earlier in this text, are followed here, but specific strategies useful in applying Newton’s laws of motion are emphasized. These techniques also reinforce concepts that are useful in many other areas of physics. Many problem-solving strategies are stated outright in the worked examples, and so the following techniques should reinforce skills you have already begun to develop. Problem-Solving Strategy for Newton’s Laws of Motion Step 1. As usual, it is first necessary to identify the physical principles involved. Once it is determined that Newton’s laws of motion are involved (if the problem involves forces), it is particularly important to draw a careful sketch of the situation. Such a sketch is shown in Figure 4.22(a). Then, as in Figure 4.22(b), use arrows to represent all forces, label them carefully, and make their lengths and directions correspond to the forces they represent (whenever sufficient information exists). Figure 4.22 (a) A sketch of Tarzan hanging from a vine. (b) Arrows are used to represent all forces. T is the tension in the vine above Tarzan, FT is the force he exerts on the vine, and w is his weight. All other forces, such as the nudge of a breeze, are assumed negligible. (c) Suppose we are given the ape man’s mass and asked to find the tension in the vine. We then define the system of interest as shown and draw a free-body diagram. FT is no longer shown, because it is not a force acting on the system of interest; rather, FT acts on the outside world. (d) Showing only the arrows, the head-to-tail method of addition is used. It is apparent that T = - w , if Tarzan is stationary. Step 2. Identify what needs to be determined and what is known or can be inferred from the problem as stated. That is, make a list of knowns and unknowns. Then carefully determine the system of interest. This decision is a crucial step, since Newton’s second law involves only external forces. Once the system of interest has been identified, it becomes possible to determine which forces are external and which are internal, a necessary step to employ Newton’s second law. (See Figure 4.22(c).) Newton’s third law may be used to identify whether forces are exerted between components of a system (internal) or between the system and something outside (external). As illustrated earlier in this chapter, the system of interest depends on what This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 167 question we need to answer. This choice becomes easier with practice, eventually developing into an almost unconscious process. Skill in clearly defining systems will be beneficial in later chapters as well. A diagram showing the system of interest and all of the external forces is called a free-body diagram. Only forces are shown on free-body diagrams, not acceleration or velocity. We have drawn several of these in worked examples. Figure 4.22(c) shows a free-body diagram for the system of interest. Note that no internal forces are shown in a free-body diagram. Step 3. Once a free-body diagram is drawn, Newton’s second law can be applied to solve the problem. This is done in Figure 4.22(d) for a particular situation. In general, once external forces are clearly identified in free-body diagrams, it should be a straightforward task to put them into equation form and solve for the unknown, as done in all previous examples. If the problem is one-dimensional—that is, if all forces are parallel—then they add like scalars. If the problem is two-dimensional, then it must be broken down into a pair of one-dimensional problems. This is done by projecting the force vectors onto a set of axes chosen for convenience. As seen in previous examples, the choice of axes can simplify the problem. For example, when an incline is involved, a set of axes with one axis parallel to the incline and one perpendicular to it is most convenient. It is almost always convenient to make one axis parallel to the direction of motion, if this is known. Applying Newton’s Second Law Before you write net force equations, it is critical to determine whether the system is accelerating in a particular direction. If the acceleration is zero in a particular direction, then the net force is zero in that direction. Similarly, if the acceleration is nonzero in a particular direction, then the net force is described by the equation: net = . For example, if the system is accelerating in the horizontal direction, but it is not accelerating in the vertical direction, then you will have the following conclusions: net = , net = 0. (4.57) (4.58) You will need this information in order to determine unknown forces acting in a system. Step 4. As always, check the solution to see whether it is reasonable. In some cases, this is obvious. For example, it is reasonable to find that friction causes an object to slide down an incline more slowly than when no friction exists. In practice, intuition develops gradually through problem solving, and with experience it becomes progressively easier to judge whether an answer is reasonable. Another way to check your solution is to check the units. If you are solving for force and end up with units of m/s, then you have made a mistake. 4.7 Further Applications of Newton's Laws of Motion Learning Objectives By the end of this section, you will be able to: • Apply problem-solving techniques to solve for quantities in more complex systems of forces. • Integrate concepts from kinematics to solve problems using Newton's laws of motion. The information presented in this section supports the following AP® learning objectives and science practices: • 3.A.2.1 The student is able to represent forces in diagrams or mathematically using appropriately labeled vectors with magnitude, direction, and units during the analysis of a situation. (S.P. 1.1) • 3.A.3.1 The student is able to analyze a scenario and make claims (develop arguments, justify assertions) about the forces exerted on an object by other objects for different types of forces or components of forces. (S.P. 6.4, 7.2) • 3.A.3.3 The student is able to describe a force as an interaction between two objects and identify both objects for any force. (S.P. 1.4) • 3.B.1.1 The student is able to predict the motion of an object subject to forces exerted by several objects using an application of Newton's second law in a variety of physical situations with acceleration in one dimension. (S.P. 6.4, 7.2) • 3.B.1.3 The student is able to re-express a free-body diagram representation into a mathematical representation and solve the mathematical representation for the acceleration of the object. (S.P. 1.5, 2.2) • 3.B.2.1 The student is able to create and use free-body diagrams to analyze physical situations to solve problems with motion qualitatively and quantitatively. (S.P. 1.1, 1.4, 2.2) There are many interesting applications of Newton’s laws of motion, a few more of which are presented in this section. These serve also to illustrate some further subtleties of physics and to help build problem-solving skills. 168 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion Example 4.7 Drag Force on a Barge Suppose two tugboats push on a barge at different angles, as shown in Figure 4.23. The first tugboat exerts a force of 2.7×105 N in the x-direction, and the second tugboat exerts a force of 3.6×105 N in the y-direction. Figure 4.23 (a) A view from above of two tugboats pushing on a barge. (b) The free-body diagram for the ship contains only forces acting in the plane of the water. It omits the two vertical forces—the weight of the barge and the buoyant force of the water supporting it cancel and are not shown. Since the applied forces are perpendicular, the x- and y-axes are in the same direction as F and F . The problem quickly becomes a one-dimensional problem along the direction of Fapp , since friction is in the direction opposite to Fapp . If the mass of the barge is 5.0×106 kg and its acceleration is observed to be 7.5×10−2 m/s2 in the direction shown, what is the drag force of the water on the barge resisting the motion? (Note: drag force is a frictional force exerted by fluids, such as air or
water. The drag force opposes the motion of the object.) Strategy The directions and magnitudes of acceleration and the applied forces are given in Figure 4.23(a). We will define the total force of the tugboats on the barge as Fapp so that: Fapp =F + F (4.59) Since the barge is flat bottomed, the drag of the water FD will be in the direction opposite to Fapp , as shown in the freebody diagram in Figure 4.23(b). The system of interest here is the barge, since the forces on it are given as well as its acceleration. Our strategy is to find the magnitude and direction of the net applied force Fapp , and then apply Newton’s second law to solve for the drag force FD . Solution Since F and F are perpendicular, the magnitude and direction of Fapp are easily found. First, the resultant magnitude is given by the Pythagorean theorem: 2 2 + F app = F app = (2.7×105 N)2 + (3.6×105 N)2 = 4.5×105 N. The angle is given by = tan−1 3.6×105 N = tan−1 2.7×105 N = 53º (4.60) (4.61) which we know, because of Newton’s first law, is the same direction as the acceleration. FD is in the opposite direction of Fapp , since it acts to slow down the acceleration. Therefore, the net external force is in the same direction as Fapp , but its magnitude is slightly less than Fapp . The problem is now one-dimensional. From Figure 4.23(b), we can see that net = app − D. (4.62) But Newton’s second law states that This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion Thus, net = . app − D = . This can be solved for the magnitude of the drag force of the water D in terms of known quantities: D = app − . 169 (4.63) (4.64) (4.65) Substituting known values gives FD = (4.5×105 N) − (5.0×106 kg)(7.5×10–2 m/s2 ) = 7.5×104 N. The direction of FD has already been determined to be in the direction opposite to Fapp , or at an angle of 53º south of west. (4.66) Discussion The numbers used in this example are reasonable for a moderately large barge. It is certainly difficult to obtain larger accelerations with tugboats, and small speeds are desirable to avoid running the barge into the docks. Drag is relatively small for a well-designed hull at low speeds, consistent with the answer to this example, where D is less than 1/600th of the weight of the ship. In the earlier example of a tightrope walker we noted that the tensions in wires supporting a mass were equal only because the angles on either side were equal. Consider the following example, where the angles are not equal; slightly more trigonometry is involved. Example 4.8 Different Tensions at Different Angles Consider the traffic light (mass 15.0 kg) suspended from two wires as shown in Figure 4.24. Find the tension in each wire, neglecting the masses of the wires. 170 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion Figure 4.24 A traffic light is suspended from two wires. (b) Some of the forces involved. (c) Only forces acting on the system are shown here. The free-body diagram for the traffic light is also shown. (d) The forces projected onto vertical (y) and horizontal (x) axes. The horizontal components of the tensions must cancel, and the sum of the vertical components of the tensions must equal the weight of the traffic light. (e) The free-body diagram shows the vertical and horizontal forces acting on the traffic light. Strategy The system of interest is the traffic light, and its free-body diagram is shown in Figure 4.24(c). The three forces involved are not parallel, and so they must be projected onto a coordinate system. The most convenient coordinate system has one axis vertical and one horizontal, and the vector projections on it are shown in part (d) of the figure. There are two unknowns in this problem ( 1 and 2 ), so two equations are needed to find them. These two equations come from applying Newton’s second law along the vertical and horizontal axes, noting that the net external force is zero along each axis because acceleration is zero. Solution First consider the horizontal or x-axis: Thus, as you might expect, net = 2 − 1 = 0. (4.67) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion This gives us the following relationship between 1 and 2 : 1 = 2. Thus, 1 cos (30º) = 2 cos (45º). 2 = (1.225)1. 171 (4.68) (4.69) (4.70) Note that 1 and 2 are not equal in this case, because the angles on either side are not equal. It is reasonable that 2 ends up being greater than 1 , because it is exerted more vertically than 1 . Now consider the force components along the vertical or y-axis: This implies net = 1 + 2 − = 0. 1 + 2 = . Substituting the expressions for the vertical components gives 1 sin (30º) + 2 sin (45º) = . There are two unknowns in this equation, but substituting the expression for 2 in terms of 1 reduces this to one equation with one unknown: which yields 1(0.500) + (1.2251)(0.707) = = , (1.366)1 = (15.0 kg)(9.80 m/s2). Solving this last equation gives the magnitude of 1 to be 1 = 108 N. (4.71) (4.72) (4.73) (4.74) (4.75) (4.76) Finally, the magnitude of 2 is determined using the relationship between them, 2 = 1.225 1 , found above. Thus we obtain 2 = 132 N. (4.77) Discussion Both tensions would be larger if both wires were more horizontal, and they will be equal if and only if the angles on either side are the same (as they were in the earlier example of a tightrope walker). The bathroom scale is an excellent example of a normal force acting on a body. It provides a quantitative reading of how much it must push upward to support the weight of an object. But can you predict what you would see on the dial of a bathroom scale if you stood on it during an elevator ride? Will you see a value greater than your weight when the elevator starts up? What about when the elevator moves upward at a constant speed: will the scale still read more than your weight at rest? Consider the following example. Example 4.9 What Does the Bathroom Scale Read in an Elevator? Figure 4.25 shows a 75.0-kg man (weight of about 165 lb) standing on a bathroom scale in an elevator. Calculate the scale reading: (a) if the elevator accelerates upward at a rate of 1.20 m/s2 , and (b) if the elevator moves upward at a constant speed of 1 m/s. 172 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion Figure 4.25 (a) The various forces acting when a person stands on a bathroom scale in an elevator. The arrows are approximately correct for when the elevator is accelerating upward—broken arrows represent forces too large to be drawn to scale. T is the tension in the supporting cable, w is the weight of the person, ws is the weight of the scale, we is the weight of the elevator, Fs is the force of the scale on the person, Fp is the force of the person on the scale, Ft is the force of the scale on the floor of the elevator, and N is the force of the floor upward on the scale. (b) The free-body diagram shows only the external forces acting on the designated system of interest—the person. Strategy If the scale is accurate, its reading will equal p , the magnitude of the force the person exerts downward on it. Figure 4.25(a) shows the numerous forces acting on the elevator, scale, and person. It makes this one-dimensional problem look much more formidable than if the person is chosen to be the system of interest and a free-body diagram is drawn as in Figure 4.25(b). Analysis of the free-body diagram using Newton’s laws can produce answers to both parts (a) and (b) of this example, as well as some other questions that might arise. The only forces acting on the person are his weight w and the upward force of the scale Fs . According to Newton’s third law Fp and Fs are equal in magnitude and opposite in direction, so that we need to find s in order to find what the scale reads. We can do this, as usual, by applying Newton’s second law, net = . From the free-body diagram we see that net = s − , so that s − = . Solving for s gives an equation with only one unknown: or, because = , simply s = + , s = + . (4.78) (4.79) (4.80) (4.81) No assumptions were made about the acceleration, and so this solution should be valid for a variety of accelerations in addition to the ones in this exercise. Solution for (a) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 173 In this part of the problem, = 1.20 m/s2 , so that s = (75.0 kg)(1.20 m/s2 ) + (75.0 kg)(9.80 m/s2), yielding Discussion for (a) s = 825 N. (4.82) (4.83) This is about 185 lb. What would the scale have read if he were stationary? Since his acceleration would be zero, the force of the scale would be equal to his weight: net = = 0 = s − s = = = (75.0 kg)(9.80 m/s2) = 735 N. s s (4.84) So, the scale reading in the elevator is greater than his 735-N (165 lb) weight. This means that the scale is pushing up on the person with a force greater than his weight, as it must in order to accelerate him upward. Clearly, the greater the acceleration of the elevator, the greater the scale reading, consistent with what you feel in rapidly accelerating versus slowly accelerating elevators. Solution for (b) Now, what happens when the elevator reaches a constant upward velocity? Will the scale still read more than his weight? For any constant velocity—up, down, or stationary—acceleration is zero because = Δ Δ , and Δ = 0 . Thus, Now which gives Discussion for (b75.0 kg)(9.80 m/s2), s = 735 N. (4.85) (4.86) (4.87) The scale reading is 735 N, which equals the person’s weight. This will be the case whenever the elevator has a constant velocity—moving up, moving down, or stationary. The solution to the previous example also applies to an elevator accelerating downward, as mentioned. When an elevator accelerates downward, is negative, and the scale reading is less than the weight of the person, until a constant downward velocity is reached, at which time the sca
le reading again becomes equal to the person’s weight. If the elevator is in free-fall and accelerating downward at , then the scale reading will be zero and the person will appear to be weightless. Integrating Concepts: Newton’s Laws of Motion and Kinematics Physics is most interesting and most powerful when applied to general situations that involve more than a narrow set of physical principles. Newton’s laws of motion can also be integrated with other concepts that have been discussed previously in this text to solve problems of motion. For example, forces produce accelerations, a topic of kinematics, and hence the relevance of earlier chapters. When approaching problems that involve various types of forces, acceleration, velocity, and/or position, use the following steps to approach the problem: Problem-Solving Strategy Step 1. Identify which physical principles are involved. Listing the givens and the quantities to be calculated will allow you to identify the principles involved. Step 2. Solve the problem using strategies outlined in the text. If these are available for the specific topic, you should refer to them. You should also refer to the sections of the text that deal with a particular topic. The following worked example illustrates how these strategies are applied to an integrated concept problem. 174 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion Example 4.10 What Force Must a Soccer Player Exert to Reach Top Speed? A soccer player starts from rest and accelerates forward, reaching a velocity of 8.00 m/s in 2.50 s. (a) What was his average acceleration? (b) What average force did he exert backward on the ground to achieve this acceleration? The player’s mass is 70.0 kg, and air resistance is negligible. Strategy 1. To solve an integrated concept problem, we must first identify the physical principles involved and identify the chapters in which they are found. Part (a) of this example considers acceleration along a straight line. This is a topic of kinematics. Part (b) deals with force, a topic of dynamics found in this chapter. 2. The following solutions to each part of the example illustrate how the specific problem-solving strategies are applied. These involve identifying knowns and unknowns, checking to see if the answer is reasonable, and so forth. Solution for (a) We are given the initial and final velocities (zero and 8.00 m/s forward); thus, the change in velocity is Δ = 8.00 m/s . We are given the elapsed time, and so Δ = 2.50 s . The unknown is acceleration, which can be found from its definition: = Δ Δ . (4.88) Substituting the known values yields = 8.00 m/s 2.50 s = 3.20 m/s2. (4.89) Discussion for (a) This is an attainable acceleration for an athlete in good condition. Solution for (b) Here we are asked to find the average force the player exerts backward to achieve this forward acceleration. Neglecting air resistance, this would be equal in magnitude to the net external force on the player, since this force causes his acceleration. Since we now know the player’s acceleration and are given his mass, we can use Newton’s second law to find the force exerted. That is, Substituting the known values of and gives net = . net = (70.0 kg)(3.20 m/s2) = 224 N. Discussion for (b) This is about 50 pounds, a reasonable average force. (4.90) (4.91) This worked example illustrates how to apply problem-solving strategies to situations that include topics from different chapters. The first step is to identify the physical principles involved in the problem. The second step is to solve for the unknown using familiar problem-solving strategies. These strategies are found throughout the text, and many worked examples show how to use them for single topics. You will find these techniques for integrated concept problems useful in applications of physics outside of a physics course, such as in your profession, in other science disciplines, and in everyday life. The following problems will build your skills in the broad application of physical principles. 4.8 Extended Topic: The Four Basic Forces—An Introduction By the end of this section, you will be able to: • Understand the four basic forces that underlie the processes in nature. Learning Objectives The information presented in this section supports the following AP® learning objectives and science practices: • 3.C.4.1 The student is able to make claims about various contact forces between objects based on the microscopic cause of those forces. (S.P. 6.1) • 3.C.4.2 The student is able to explain contact forces (tension, friction, normal, buoyant, spring) as arising from interatomic electric forces and that they therefore have certain directions. (S.P. 6.2) • 3.G.1.1 The student is able to articulate situations when the gravitational force is the dominant force and when the electromagnetic, weak, and strong forces can be ignored. (S.P. 7.1) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 175 One of the most remarkable simplifications in physics is that only four distinct forces account for all known phenomena. In fact, nearly all of the forces we experience directly are due to only one basic force, called the electromagnetic force. (The gravitational force is the only force we experience directly that is not electromagnetic.) This is a tremendous simplification of the myriad of apparently different forces we can list, only a few of which were discussed in the previous section. As we will see, the basic forces are all thought to act through the exchange of microscopic carrier particles, and the characteristics of the basic forces are determined by the types of particles exchanged. Action at a distance, such as the gravitational force of Earth on the Moon, is explained by the existence of a force field rather than by “physical contact.” The four basic forces are the gravitational force, the electromagnetic force, the weak nuclear force, and the strong nuclear force. Their properties are summarized in Table 4.2. Since the weak and strong nuclear forces act over an extremely short range, the size of a nucleus or less, we do not experience them directly, although they are crucial to the very structure of matter. These forces determine which nuclei are stable and which decay, and they are the basis of the release of energy in certain nuclear reactions. Nuclear forces determine not only the stability of nuclei, but also the relative abundance of elements in nature. The properties of the nucleus of an atom determine the number of electrons it has and, thus, indirectly determine the chemistry of the atom. More will be said of all of these topics in later chapters. Concept Connections: The Four Basic Forces The four basic forces will be encountered in more detail as you progress through the text. The gravitational force is defined in Uniform Circular Motion and Gravitation, electric force in Electric Charge and Electric Field, magnetic force in Magnetism, and nuclear forces in Radioactivity and Nuclear Physics. On a macroscopic scale, electromagnetism and gravity are the basis for all forces. The nuclear forces are vital to the substructure of matter, but they are not directly experienced on the macroscopic scale. Table 4.2 Properties of the Four Basic Forces[1] Force Approximate Relative Strengths Range Attraction/Repulsion Carrier Particle Gravitational 10−38 Electromagnetic 10 – 2 Weak nuclear 10 – 13 Strong nuclear 1 ∞ ∞ attractive only Graviton attractive and repulsive Photon < 10–18 m attractive and repulsive W+ , W – , Z0 < 10–15 m attractive and repulsive gluons The gravitational force is surprisingly weak—it is only because gravity is always attractive that we notice it at all. Our weight is the gravitational force due to the entire Earth acting on us. On the very large scale, as in astronomical systems, the gravitational force is the dominant force determining the motions of moons, planets, stars, and galaxies. The gravitational force also affects the nature of space and time. As we shall see later in the study of general relativity, space is curved in the vicinity of very massive bodies, such as the Sun, and time actually slows down near massive bodies. Take a good look at the ranges for the four fundamental forces listed in Table 4.2. The range of the strong nuclear force, 10−15 m, is approximately the size of the nucleus of an atom; the weak nuclear force has an even shorter range. At scales on the order of 10−10 m, approximately the size of an atom, both nuclear forces are completely dominated by the electromagnetic force. Notice that this scale is still utterly tiny compared to our everyday experience. At scales that we do experience daily, electromagnetism tends to be negligible, due to its attractive and repulsive properties canceling each other out. That leaves gravity, which is usually the strongest of the forces at scales above ~10−4 m, and hence includes our everyday activities, such as throwing, climbing stairs, and walking. Electromagnetic forces can be either attractive or repulsive. They are long-range forces, which act over extremely large distances, and they nearly cancel for macroscopic objects. (Remember that it is the net external force that is important.) If they did not cancel, electromagnetic forces would completely overwhelm the gravitational force. The electromagnetic force is a combination of electrical forces (such as those that cause static electricity) and magnetic forces (such as those that affect a compass needle). These two forces were thought to be quite distinct until early in the 19th century, when scientists began to discover that they are different manifestations of the same force. This discovery is a classical case of the unification of forces. Similarly, friction, tension, and all of the other classes of forces we experience directly (except gravity, of course) are due to electromagnetic interactions of atoms and molecules. It
is still convenient to consider these forces separately in specific applications, however, because of the ways they manifest themselves. 1. The graviton is a proposed particle, though it has not yet been observed by scientists. See the discussion of gravitational waves later in this section. The particles W+ are called vector bosons; these were predicted by theory and first observed in 1983. There are eight types of gluons proposed by scientists, and their existence is indicated by meson exchange in the nuclei of atoms. , W− , and Z0 176 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion Concept Connections: Unifying Forces Attempts to unify the four basic forces are discussed in relation to elementary particles later in this text. By “unify” we mean finding connections between the forces that show that they are different manifestations of a single force. Even if such unification is achieved, the forces will retain their separate characteristics on the macroscopic scale and may be identical only under extreme conditions such as those existing in the early universe. Physicists are now exploring whether the four basic forces are in some way related. Attempts to unify all forces into one come under the rubric of Grand Unified Theories (GUTs), with which there has been some success in recent years. It is now known that under conditions of extremely high density and temperature, such as existed in the early universe, the electromagnetic and weak nuclear forces are indistinguishable. They can now be considered to be different manifestations of one force, called the electroweak force. So the list of four has been reduced in a sense to only three. Further progress in unifying all forces is proving difficult—especially the inclusion of the gravitational force, which has the special characteristics of affecting the space and time in which the other forces exist. While the unification of forces will not affect how we discuss forces in this text, it is fascinating that such underlying simplicity exists in the face of the overt complexity of the universe. There is no reason that nature must be simple—it simply is. Action at a Distance: Concept of a Field All forces act at a distance. This is obvious for the gravitational force. Earth and the Moon, for example, interact without coming into contact. It is also true for all other forces. Friction, for example, is an electromagnetic force between atoms that may not actually touch. What is it that carries forces between objects? One way to answer this question is to imagine that a force field surrounds whatever object creates the force. A second object (often called a test object) placed in this field will experience a force that is a function of location and other variables. The field itself is the “thing” that carries the force from one object to another. The field is defined so as to be a characteristic of the object creating it; the field does not depend on the test object placed in it. Earth’s gravitational field, for example, is a function of the mass of Earth and the distance from its center, independent of the presence of other masses. The concept of a field is useful because equations can be written for force fields surrounding objects (for gravity, this yields = at Earth’s surface), and motions can be calculated from these equations. (See Figure 4.26.) Figure 4.26 The electric force field between a positively charged particle and a negatively charged particle. When a positive test charge is placed in the field, the charge will experience a force in the direction of the force field lines. Concept Connections: Force Fields The concept of a force field is also used in connection with electric charge and is presented in Electric Charge and Electric Field. It is also a useful idea for all the basic forces, as will be seen in Particle Physics. Fields help us to visualize forces and how they are transmitted, as well as to describe them with precision and to link forces with subatomic carrier particles. Making Connections: Vector and Scalar Fields These fields may be either scalar or vector fields. Gravity and electromagnetism are examples of vector fields. A test object placed in such a field will have both the magnitude and direction of the resulting force on the test object completely defined by the object’s location in the field. We will later cover examples of scalar fields, which have a magnitude but no direction. The field concept has been applied very successfully; we can calculate motions and describe nature to high precision using field equations. As useful as the field concept is, however, it leaves unanswered the question of what carries the force. It has been proposed in recent decades, starting in 1935 with Hideki Yukawa’s (1907–1981) work on the strong nuclear force, that all forces are transmitted by the exchange of elementary particles. We can visualize particle exchange as analogous to macroscopic phenomena such as two people passing a basketball back and forth, thereby exerting a repulsive force without touching one another. (See Figure 4.27.) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 177 Figure 4.27 The exchange of masses resulting in repulsive forces. (a) The person throwing the basketball exerts a force Fp1 on it toward the other person and feels a reaction force FB away from the second person. (b) The person catching the basketball exerts a force Fp2 on it to stop the ball and feels a reaction force F′B away from the first person. (c) The analogous exchange of a meson between a proton and a neutron carries the strong nuclear forces Fexch and F′exch between them. An attractive force can also be exerted by the exchange of a mass—if person 2 pulled the basketball away from the first person as he tried to retain it, then the force between them would be attractive. This idea of particle exchange deepens rather than contradicts field concepts. It is more satisfying philosophically to think of something physical actually moving between objects acting at a distance. Table 4.2 lists the exchange or carrier particles, both observed and proposed, that carry the four forces. But the real fruit of the particle-exchange proposal is that searches for Yukawa’s proposed particle found it and a number of others that were completely unexpected, stimulating yet more research. All of this research eventually led to the proposal of quarks as the underlying substructure of matter, which is a basic tenet of GUTs. If successful, these theories will explain not only forces, but also the structure of matter itself. Yet physics is an experimental science, so the test of these theories must lie in the domain of the real world. As of this writing, scientists at the CERN laboratory in Switzerland are starting to test these theories using the world’s largest particle accelerator: the Large Hadron Collider. This accelerator (27 km in circumference) allows two high-energy proton beams, traveling in opposite directions, to collide. An energy of 14 trillion electron volts will be available. It is anticipated that some new particles, possibly force carrier particles, will be found. (See Figure 4.28.) One of the force carriers of high interest that researchers hope to detect is the Higgs boson. The observation of its properties might tell us why different particles have different masses. Figure 4.28 The world’s largest particle accelerator spans the border between Switzerland and France. Two beams, traveling in opposite directions close to the speed of light, collide in a tube similar to the central tube shown here. External magnets determine the beam’s path. Special detectors will analyze particles created in these collisions. Questions as broad as what is the origin of mass and what was matter like the first few seconds of our universe will be explored. This accelerator began preliminary operation in 2008. (credit: Frank Hommes) 178 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion Tiny particles also have wave-like behavior, something we will explore more in a later chapter. To better understand force-carrier particles from another perspective, let us consider gravity. The search for gravitational waves has been going on for a number of years. Almost 100 years ago, Einstein predicted the existence of these waves as part of his general theory of relativity. Gravitational waves are created during the collision of massive stars, in black holes, or in supernova explosions—like shock waves. These gravitational waves will travel through space from such sites much like a pebble dropped into a pond sends out ripples—except these waves move at the speed of light. A detector apparatus has been built in the U.S., consisting of two large installations nearly 3000 km apart—one in Washington state and one in Louisiana! The facility is called the Laser Interferometer Gravitational-Wave Observatory (LIGO). Each installation is designed to use optical lasers to examine any slight shift in the relative positions of two masses due to the effect of gravity waves. The two sites allow simultaneous measurements of these small effects to be separated from other natural phenomena, such as earthquakes. Initial operation of the detectors began in 2002, and work is proceeding on increasing their sensitivity. Similar installations have been built in Italy (VIRGO), Germany (GEO600), and Japan (TAMA300) to provide a worldwide network of gravitational wave detectors. International collaboration in this area is moving into space with the joint EU/US project LISA (Laser Interferometer Space Antenna). Earthquakes and other Earthly noises will be no problem for these monitoring spacecraft. LISA will complement LIGO by looking at much more massive black holes through the observation of gravitational-wave sources emitting much larger wavelengths. Three satellites will be placed in space above Earth in an equilateral triangle
(with 5,000,000-km sides) (Figure 4.29). The system will measure the relative positions of each satellite to detect passing gravitational waves. Accuracy to within 10% of the size of an atom will be needed to detect any waves. The launch of this project might be as early as 2018. “I’m sure LIGO will tell us something about the universe that we didn’t know before. The history of science tells us that any time you go where you haven’t been before, you usually find something that really shakes the scientific paradigms of the day. Whether gravitational wave astrophysics will do that, only time will tell.” —David Reitze, LIGO Input Optics Manager, University of Florida Figure 4.29 Space-based future experiments for the measurement of gravitational waves. Shown here is a drawing of LISA’s orbit. Each satellite of LISA will consist of a laser source and a mass. The lasers will transmit a signal to measure the distance between each satellite’s test mass. The relative motion of these masses will provide information about passing gravitational waves. (credit: NASA) The ideas presented in this section are but a glimpse into topics of modern physics that will be covered in much greater depth in later chapters. Glossary acceleration: the rate at which an object’s velocity changes over a period of time carrier particle: a fundamental particle of nature that is surrounded by a characteristic force field; photons are carrier particles of the electromagnetic force dynamics: the study of how forces affect the motion of objects and systems external force: a force acting on an object or system that originates outside of the object or system force: a push or pull on an object with a specific magnitude and direction; can be represented by vectors; can be expressed as a multiple of a standard force force field: a region in which a test particle will experience a force free-body diagram: a sketch showing all of the external forces acting on an object or system; the system is represented by a dot, and the forces are represented by vectors extending outward from the dot free-fall: a situation in which the only force acting on an object is the force due to gravity friction: a force past each other of objects that are touching; examples include rough surfaces and air resistance This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 179 inertia: the tendency of an object to remain at rest or remain in motion inertial frame of reference: a coordinate system that is not accelerating; all forces acting in an inertial frame of reference are real forces, as opposed to fictitious forces that are observed due to an accelerating frame of reference law of inertia: see Newton’s first law of motion mass: the quantity of matter in a substance; measured in kilograms net external force: the vector sum of all external forces acting on an object or system; causes a mass to accelerate Newton’s first law of motion: in an inertial frame of reference, a body at rest remains at rest, or, if in motion, remains in motion at a constant velocity unless acted on by a net external force; also known as the law of inertia Newton’s second law of motion: the net external force Fnet on an object with mass is proportional to and in the same direction as the acceleration of the object, a , and inversely proportional to the mass; defined mathematically as a = Fnet Newton’s third law of motion: whenever one body exerts a force on a second body, the first body experiences a force that is equal in magnitude and opposite in direction to the force that the first body exerts normal force: the force that a surface applies to an object to support the weight of the object; acts perpendicular to the surface on which the object rests system: defined by the boundaries of an object or collection of objects being observed; all forces originating from outside of the system are considered external forces tension: the pulling force that acts along a medium, especially a stretched flexible connector, such as a rope or cable; when a rope supports the weight of an object, the force on the object due to the rope is called a tension force thrust: a reaction force that pushes a body forward in response to a backward force; rockets, airplanes, and cars are pushed forward by a thrust reaction force weight: the force w due to gravity acting on an object of mass ; defined mathematically as: w g , where g is the magnitude and direction of the acceleration due to gravity Section Summary 4.1 Development of Force Concept • Dynamics is the study of how forces affect the motion of objects. • Force is a push or pull that can be defined in terms of various standards, and it is a vector having both magnitude and direction. • External forces are any outside forces that act on a body. A free-body diagram is a drawing of all external forces acting on a body. 4.2 Newton's First Law of Motion: Inertia • Newton’s first law of motion states that in an inertial frame of reference a body at rest remains at rest, or, if in motion, remains in motion at a constant velocity unless acted on by a net external force. This is also known as the law of inertia. Inertia is the tendency of an object to remain at rest or remain in motion. Inertia is related to an object’s mass. • • Mass is the quantity of matter in a substance. 4.3 Newton's Second Law of Motion: Concept of a System • Acceleration, a , is defined as a change in velocity, meaning a change in its magnitude or direction, or both. • An external force is one acting on a system from outside the system, as opposed to internal forces, which act between components within the system. • Newton’s second law of motion states that the acceleration of a system is directly proportional to and in the same direction as the net external force acting on the system, and inversely proportional to its mass. • In equation form, Newton’s second law of motion is a = Fnet . • This is often written in the more familiar form: Fnet = a . • The weight w of an object is defined as the force of gravity acting on an object of mass . The object experiences an acceleration due to gravity g : • If the only force acting on an object is due to gravity, the object is in free fall. w = g. 180 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion • Friction is a force that opposes the motion past each other of objects that are touching. 4.4 Newton's Third Law of Motion: Symmetry in Forces • Newton’s third law of motion represents a basic symmetry in nature. It states: Whenever one body exerts a force on a second body, the first body experiences a force that is equal in magnitude and opposite in direction to the force that the first body exerts. • A thrust is a reaction force that pushes a body forward in response to a backward force. Rockets, airplanes, and cars are pushed forward by a thrust reaction force. 4.5 Normal, Tension, and Other Examples of Force • When objects rest on a surface, the surface applies a force to the object that supports the weight of the object. This supporting force acts perpendicular to and away from the surface. It is called a normal force, N . • When objects rest on a non-accelerating horizontal surface, the magnitude of the normal force is equal to the weight of the object: • When objects rest on an inclined plane that makes an angle with the horizontal surface, the weight of the object can be resolved into components that act perpendicular ( w⊥ ) and parallel ( w ∥ ) to the surface of the plane. These components can be calculated using: = . • The pulling force that acts along a stretched flexible connector, such as a rope or cable, is called tension, T . When a rope ∥ = sin () = sin () ⊥ = cos () = cos (). supports the weight of an object that is at rest, the tension in the rope is equal to the weight of the object: • = . In any inertial frame of reference (one that is not accelerated or rotated), Newton’s laws have the simple forms given in this chapter and all forces are real forces having a physical origin. 4.6 Problem-Solving Strategies • To solve problems involving Newton’s laws of motion, follow the procedure described: 1. Draw a sketch of the problem. 2. Identify known and unknown quantities, and identify the system of interest. Draw a free-body diagram, which is a sketch showing all of the forces acting on an object. The object is represented by a dot, and the forces are represented by vectors extending in different directions from the dot. If vectors act in directions that are not horizontal or vertical, resolve the vectors into horizontal and vertical components and draw them on the free-body diagram. 3. Write Newton’s second law in the horizontal and vertical directions and add the forces acting on the object. If the object does not accelerate in a particular direction (for example, the -direction) then net = 0 . If the object does accelerate in that direction, net = . 4. Check your answer. Is the answer reasonable? Are the units correct? 4.7 Further Applications of Newton's Laws of Motion • Newton’s laws of motion can be applied in numerous situations to solve problems of motion. • Some problems will contain multiple force vectors acting in different directions on an object. Be sure to draw diagrams, resolve all force vectors into horizontal and vertical components, and draw a free-body diagram. Always analyze the direction in which an object accelerates so that you can determine whether net = or net = 0 . • The normal force on an object is not always equal in magnitude to the weight of the object. If an object is accelerating, the normal force will be less than or greater than the weight of the object. Also, if the object is on an inclined plane, the normal force will always be less than the full weight of the object. • Some problems will contain various physical quantities, such as forces, acceleration, velocity, or position. You can apply concepts from kinematics and dynamics in ord
er to solve these problems of motion. 4.8 Extended Topic: The Four Basic Forces—An Introduction • The various types of forces that are categorized for use in many applications are all manifestations of the four basic forces in nature. • The properties of these forces are summarized in Table 4.2. • Everything we experience directly without sensitive instruments is due to either electromagnetic forces or gravitational forces. The nuclear forces are responsible for the submicroscopic structure of matter, but they are not directly sensed because of their short ranges. Attempts are being made to show all four forces are different manifestations of a single unified force. • A force field surrounds an object creating a force and is the carrier of that force. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 181 Conceptual Questions 4.1 Development of Force Concept 1. Propose a force standard different from the example of a stretched spring discussed in the text. Your standard must be capable of producing the same force repeatedly. 2. What properties do forces have that allow us to classify them as vectors? 4.2 Newton's First Law of Motion: Inertia 3. How are inertia and mass related? 4. What is the relationship between weight and mass? Which is an intrinsic, unchanging property of a body? 4.3 Newton's Second Law of Motion: Concept of a System 5. Which statement is correct? (a) Net force causes motion. (b) Net force causes change in motion. Explain your answer and give an example. 6. Why can we neglect forces such as those holding a body together when we apply Newton’s second law of motion? 7. Explain how the choice of the “system of interest” affects which forces must be considered when applying Newton’s second law of motion. 8. Describe a situation in which the net external force on a system is not zero, yet its speed remains constant. 9. A system can have a nonzero velocity while the net external force on it is zero. Describe such a situation. 10. A rock is thrown straight up. What is the net external force acting on the rock when it is at the top of its trajectory? 11. (a) Give an example of different net external forces acting on the same system to produce different accelerations. (b) Give an example of the same net external force acting on systems of different masses, producing different accelerations. (c) What law accurately describes both effects? State it in words and as an equation. 12. If the acceleration of a system is zero, are no external forces acting on it? What about internal forces? Explain your answers. 13. If a constant, nonzero force is applied to an object, what can you say about the velocity and acceleration of the object? 14. The gravitational force on the basketball in Figure 4.6 is ignored. When gravity is taken into account, what is the direction of the net external force on the basketball—above horizontal, below horizontal, or still horizontal? 4.4 Newton's Third Law of Motion: Symmetry in Forces 15. When you take off in a jet aircraft, there is a sensation of being pushed back into the seat. Explain why you move backward in the seat—is there really a force backward on you? (The same reasoning explains whiplash injuries, in which the head is apparently thrown backward.) 16. A device used since the 1940s to measure the kick or recoil of the body due to heart beats is the “ballistocardiograph.” What physics principle(s) are involved here to measure the force of cardiac contraction? How might we construct such a device? 17. Describe a situation in which one system exerts a force on another and, as a consequence, experiences a force that is equal in magnitude and opposite in direction. Which of Newton’s laws of motion apply? 18. Why does an ordinary rifle recoil (kick backward) when fired? The barrel of a recoilless rifle is open at both ends. Describe how Newton’s third law applies when one is fired. Can you safely stand close behind one when it is fired? 19. An American football lineman reasons that it is senseless to try to out-push the opposing player, since no matter how hard he pushes he will experience an equal and opposite force from the other player. Use Newton’s laws and draw a free-body diagram of an appropriate system to explain how he can still out-push the opposition if he is strong enough. 20. Newton’s third law of motion tells us that forces always occur in pairs of equal and opposite magnitude. Explain how the choice of the “system of interest” affects whether one such pair of forces cancels. 4.5 Normal, Tension, and Other Examples of Force 21. If a leg is suspended by a traction setup as shown in Figure 4.30, what is the tension in the rope? 182 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion Figure 4.30 A leg is suspended by a traction system in which wires are used to transmit forces. Frictionless pulleys change the direction of the force T without changing its magnitude. 22. In a traction setup for a broken bone, with pulleys and rope available, how might we be able to increase the force along the tibia using the same weight? (See Figure 4.30.) (Note that the tibia is the shin bone shown in this image.) 4.7 Further Applications of Newton's Laws of Motion 23. To simulate the apparent weightlessness of space orbit, astronauts are trained in the hold of a cargo aircraft that is accelerating downward at . Why will they appear to be weightless, as measured by standing on a bathroom scale, in this accelerated frame of reference? Is there any difference between their apparent weightlessness in orbit and in the aircraft? 24. A cartoon shows the toupee coming off the head of an elevator passenger when the elevator rapidly stops during an upward ride. Can this really happen without the person being tied to the floor of the elevator? Explain your answer. 4.8 Extended Topic: The Four Basic Forces—An Introduction 25. Explain, in terms of the properties of the four basic forces, why people notice the gravitational force acting on their bodies if it is such a comparatively weak force. 26. What is the dominant force between astronomical objects? Why are the other three basic forces less significant over these very large distances? 27. Give a detailed example of how the exchange of a particle can result in an attractive force. (For example, consider one child pulling a toy out of the hands of another.) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 183 Problems & Exercises 4.3 Newton's Second Law of Motion: Concept of a System You may assume data taken from illustrations is accurate to three digits. 1. A 63.0-kg sprinter starts a race with an acceleration of 4.20 m/s2 . What is the net external force on him? Figure 4.32 2. If the sprinter from the previous problem accelerates at that rate for 20 m, and then maintains that velocity for the remainder of the 100-m dash, what will be his time for the race? 8. What is the deceleration of the rocket sled if it comes to rest in 1.1 s from a speed of 1000 km/h? (Such deceleration caused one test subject to black out and have temporary blindness.) 3. A cleaner pushes a 4.50-kg laundry cart in such a way that the net external force on it is 60.0 N. Calculate the magnitude of its acceleration. 4. Since astronauts in orbit are apparently weightless, a clever method of measuring their masses is needed to monitor their mass gains or losses to adjust diets. One way to do this is to exert a known force on an astronaut and measure the acceleration produced. Suppose a net external force of 50.0 N is exerted and the astronaut’s acceleration is measured to be 0.893 m/s2 . (a) Calculate her mass. (b) By exerting a force on the astronaut, the vehicle in which they orbit experiences an equal and opposite force. Discuss how this would affect the measurement of the astronaut’s acceleration. Propose a method in which recoil of the vehicle is avoided. 5. In Figure 4.7, the net external force on the 24-kg mower is stated to be 51 N. If the force of friction opposing the motion is 24 N, what force (in newtons) is the person exerting on the mower? Suppose the mower is moving at 1.5 m/s when the force is removed. How far will the mower go before stopping? 6. The same rocket sled drawn in Figure 4.31 is decelerated at a rate of 196 m/s2 . What force is necessary to produce this deceleration? Assume that the rockets are off. The mass of the system is 2100 kg. 9. Suppose two children push horizontally, but in exactly opposite directions, on a third child in a wagon. The first child exerts a force of 75.0 N, the second a force of 90.0 N, friction is 12.0 N, and the mass of the third child plus wagon is 23.0 kg. (a) What is the system of interest if the acceleration of the child in the wagon is to be calculated? (b) Draw a free-body diagram, including all forces acting on the system. (c) Calculate the acceleration. (d) What would the acceleration be if friction were 15.0 N? 10. A powerful motorcycle can produce an acceleration of 3.50 m/s2 while traveling at 90.0 km/h. At that speed the forces resisting motion, including friction and air resistance, total 400 N. (Air resistance is analogous to air friction. It always opposes the motion of an object.) What is the magnitude of the force the motorcycle exerts backward on the ground to produce its acceleration if the mass of the motorcycle with rider is 245 kg? 11. The rocket sled shown in Figure 4.33 accelerates at a rate of 49.0 m/s2 . Its passenger has a mass of 75.0 kg. (a) Calculate the horizontal component of the force the seat exerts against his body. Compare this with his weight by using a ratio. (b) Calculate the direction and magnitude of the total force the seat exerts against his body. Figure 4.31 7. (a) If the rocket sled shown in Figure 4.32 starts with only one rocket burning, what is the magnitude of its acceleration? Assume that the mass of t
he system is 2100 kg, the thrust T is 2.4×104 N, and the force of friction opposing the motion is known to be 650 N. (b) Why is the acceleration not onefourth of what it is with all rockets burning? Figure 4.33 12. Repeat the previous problem for the situation in which the rocket sled decelerates at a rate of 201 m/s2 . In this problem, the forces are exerted by the seat and restraining belts. 13. The weight of an astronaut plus his space suit on the Moon is only 250 N. How much do they weigh on Earth? What is the mass on the Moon? On Earth? 14. Suppose the mass of a fully loaded module in which astronauts take off from the Moon is 10,000 kg. The thrust of its engines is 30,000 N. (a) Calculate its the magnitude of acceleration in a vertical takeoff from the Moon. (b) Could it lift off from Earth? If not, why not? If it could, calculate the magnitude of its acceleration. 184 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 4.4 Newton's Third Law of Motion: Symmetry in Forces 15. What net external force is exerted on a 1100-kg artillery shell fired from a battleship if the shell is accelerated at 2.40104 m/s2 ? What is the magnitude of the force exerted on the ship by the artillery shell? 16. A brave but inadequate rugby player is being pushed backward by an opposing player who is exerting a force of 800 N on him. The mass of the losing player plus equipment is 90.0 kg, and he is accelerating at 1.20 m/s2 backward. (a) What is the force of friction between the losing player’s feet and the grass? (b) What force does the winning player exert on the ground to move forward if his mass plus equipment is 110 kg? (c) Draw a sketch of the situation showing the system of interest used to solve each part. For this situation, draw a free-body diagram and write the net force equation. 4.5 Normal, Tension, and Other Examples of Force 17. Two teams of nine members each engage in a tug of war. Each of the first team’s members has an average mass of 68 kg and exerts an average force of 1350 N horizontally. Each of the second team’s members has an average mass of 73 kg and exerts an average force of 1365 N horizontally. (a) What is magnitude of the acceleration of the two teams? (b) What is the tension in the section of rope between the teams? 18. What force does a trampoline have to apply to a 45.0-kg gymnast to accelerate her straight up at 7.50 m/s2 ? Note that the answer is independent of the velocity of the gymnast—she can be moving either up or down, or be stationary. 19. (a) Calculate the tension in a vertical strand of spider web if a spider of mass 8.00×10−5 kg hangs motionless on it. (b) Calculate the tension in a horizontal strand of spider web if the same spider sits motionless in the middle of it much like the tightrope walker in Figure 4.17. The strand sags at an angle of 12º below the horizontal. Compare this with the tension in the vertical strand (find their ratio). 20. Suppose a 60.0-kg gymnast climbs a rope. (a) What is the tension in the rope if he climbs at a constant speed? (b) What is the tension in the rope if he accelerates upward at a rate of 1.50 m/s2 ? 21. Show that, as stated in the text, a force F⊥ exerted on a flexible medium at its center and perpendicular to its length (such as on the tightrope wire in Figure 4.17) gives rise to a ⊥ 2 sin () tension of magnitude = . 22. Consider the baby being weighed in Figure 4.34. (a) What is the mass of the child and basket if a scale reading of 55 N is observed? (b) What is the tension 1 in the cord attaching the baby to the scale? (c) What is the tension 2 in the cord attaching the scale to the ceiling, if the scale has a mass of 0.500 kg? (d) Draw a sketch of the situation indicating the system of interest used to solve each part. The masses of the cords are negligible. This content is available for free at http://cnx.org/content/col11844/1.13 Figure 4.34 A baby is weighed using a spring scale. 4.6 Problem-Solving Strategies 23. A 5.00×105-kg rocket is accelerating straight up. Its engines produce 1.250×107 N of thrust, and air resistance is 4.50×106 N . What is the rocket’s acceleration? Explicitly show how you follow the steps in the Problem-Solving Strategy for Newton’s laws of motion. 24. The wheels of a midsize car exert a force of 2100 N backward on the road to accelerate the car in the forward direction. If the force of friction including air resistance is 250 N and the acceleration of the car is 1.80 m/s2 , what is the mass of the car plus its occupants? Explicitly show how you follow the steps in the Problem-Solving Strategy for Newton’s laws of motion. For this situation, draw a free-body diagram and write the net force equation. 25. Calculate the force a 70.0-kg high jumper must exert on the ground to produce an upward acceleration 4.00 times the acceleration due to gravity. Explicitly show how you follow the steps in the Problem-Solving Strategy for Newton’s laws of motion. 26. When landing after a spectacular somersault, a 40.0-kg gymnast decelerates by pushing straight down on the mat. Calculate the force she must exert if her deceleration is 7.00 times the acceleration due to gravity. Explicitly show how you follow the steps in the Problem-Solving Strategy for Newton’s laws of motion. Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 185 27. A freight train consists of two 8.00×104 -kg engines and 45 cars with average masses of 5.50×104 kg . (a) What force must each engine exert backward on the track to accelerate the train at a rate of 5.00×10–2 m/s2 if the force of friction is 7.50×105 N , assuming the engines exert identical forces? This is not a large frictional force for such a massive system. Rolling friction for trains is small, and consequently trains are very energy-efficient transportation systems. (b) What is the force in the coupling between the 37th and 38th cars (this is the force each exerts on the other), assuming all cars have the same mass and that friction is evenly distributed among all of the cars and engines? 28. Commercial airplanes are sometimes pushed out of the passenger loading area by a tractor. (a) An 1800-kg tractor exerts a force of 1.75×104 N backward on the pavement, and the system experiences forces resisting motion that total 2400 N. If the acceleration is 0.150 m/s2 , what is the mass of the airplane? (b) Calculate the force exerted by the tractor on the airplane, assuming 2200 N of the friction is experienced by the airplane. (c) Draw two sketches showing the systems of interest used to solve each part, including the free-body diagrams for each. 29. A 1100-kg car pulls a boat on a trailer. (a) What total force resists the motion of the car, boat, and trailer, if the car exerts a 1900-N force on the road and produces an acceleration of 0.550 m/s2 ? The mass of the boat plus trailer is 700 kg. (b) What is the force in the hitch between the car and the trailer if 80% of the resisting forces are experienced by the boat and trailer? 30. (a) Find the magnitudes of the forces F1 and F2 that add to give the total force Ftot shown in Figure 4.35. This may be done either graphically or by using trigonometry. (b) Show graphically that the same total force is obtained independent of the order of addition of F1 and F2 . (c) Find the direction and magnitude of some other pair of vectors that add to give Ftot . Draw these to scale on the same drawing used in part (b) or a similar picture. Figure 4.35 31. Two children pull a third child on a snow saucer sled exerting forces F1 and F2 as shown from above in Figure 4.36. Find the acceleration of the 49.00-kg sled and child system. Note that the direction of the frictional force is unspecified; it will be in the opposite direction of the sum of F1 and F2 . Figure 4.36 An overhead view of the horizontal forces acting on a child’s snow saucer sled. 32. Suppose your car was mired deeply in the mud and you wanted to use the method illustrated in Figure 4.37 to pull it out. (a) What force would you have to exert perpendicular to the center of the rope to produce a force of 12,000 N on the car if the angle is 2.00°? In this part, explicitly show how you follow the steps in the Problem-Solving Strategy for Newton’s laws of motion. (b) Real ropes stretch under such forces. What force would be exerted on the car if the angle increases to 7.00° and you still apply the force found in part (a) to its center? Figure 4.37 33. What force is exerted on the tooth in Figure 4.38 if the tension in the wire is 25.0 N? Note that the force applied to the tooth is smaller than the tension in the wire, but this is necessitated by practical considerations of how force can be applied in the mouth. Explicitly show how you follow steps in the Problem-Solving Strategy for Newton’s laws of motion. Figure 4.38 Braces are used to apply forces to teeth to realign them. Shown in this figure are the tensions applied by the wire to the protruding tooth. The total force applied to the tooth by the wire, Fapp , points straight toward the back of the mouth. 34. Figure 4.39 shows Superhero and Trusty Sidekick hanging motionless from a rope. Superhero’s mass is 90.0 kg, while Trusty Sidekick’s is 55.0 kg, and the mass of the rope is negligible. (a) Draw a free-body diagram of the situation showing all forces acting on Superhero, Trusty Sidekick, and the rope. (b) Find the tension in the rope above Superhero. (c) Find the tension in the rope between 186 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion Superhero and Trusty Sidekick. Indicate on your free-body diagram the system of interest used to solve each part. is unreasonable about the result? (c) Which premise is unreasonable, and why is it unreasonable? 39. Unreasonable Results (a) What is the initial acceleration of a rocket that has a mass of 1.50×106 kg at takeoff, the engines of which produce a thrust of 2.00×106 N ? Do not neglect gravity. (b) What is unreasonable about the result? (This result has been unintentionally achieved by se
veral real rockets.) (c) Which premise is unreasonable, or which premises are inconsistent? (You may find it useful to compare this problem to the rocket problem earlier in this section.) 4.7 Further Applications of Newton's Laws of Motion 40. A flea jumps by exerting a force of 1.20×10−5 N straight down on the ground. A breeze blowing on the flea parallel to the ground exerts a force of 0.500×10−6 N on the flea. Find the direction and magnitude of the acceleration of the flea if its mass is 6.00×10−7 kg . Do not neglect the gravitational force. 41. Two muscles in the back of the leg pull upward on the Achilles tendon, as shown in Figure 4.40. (These muscles are called the medial and lateral heads of the gastrocnemius muscle.) Find the magnitude and direction of the total force on the Achilles tendon. What type of movement could be caused by this force? Figure 4.40 Achilles tendon 42. A 76.0-kg person is being pulled away from a burning building as shown in Figure 4.41. Calculate the tension in the two ropes if the person is momentarily motionless. Include a free-body diagram in your solution. Figure 4.39 Superhero and Trusty Sidekick hang motionless on a rope as they try to figure out what to do next. Will the tension be the same everywhere in the rope? 35. A nurse pushes a cart by exerting a force on the handle at a downward angle 35.0º below the horizontal. The loaded cart has a mass of 28.0 kg, and the force of friction is 60.0 N. (a) Draw a free-body diagram for the system of interest. (b) What force must the nurse exert to move at a constant velocity? 36. Construct Your Own Problem Consider the tension in an elevator cable during the time the elevator starts from rest and accelerates its load upward to some cruising velocity. Taking the elevator and its load to be the system of interest, draw a free-body diagram. Then calculate the tension in the cable. Among the things to consider are the mass of the elevator and its load, the final velocity, and the time taken to reach that velocity. 37. Construct Your Own Problem Consider two people pushing a toboggan with four children on it up a snowcovered slope. Construct a problem in which you calculate the acceleration of the toboggan and its load. Include a freebody diagram of the appropriate system of interest as the basis for your analysis. Show vector forces and their components and explain the choice of coordinates. Among the things to be considered are the forces exerted by those pushing, the angle of the slope, and the masses of the toboggan and children. 38. Unreasonable Results (a) Repeat Exercise 4.29, but assume an acceleration of 1.20 m/s2 is produced. (b) What This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 187 average force on the shell in the mortar? Express your answer in newtons and as a ratio to the weight of the shell. 48. Integrated Concepts Repeat Exercise 4.47 for a shell fired at an angle 10.0º from the vertical. 49. Integrated Concepts An elevator filled with passengers has a mass of 1700 kg. (a) The elevator accelerates upward from rest at a rate of 1.20 m/s2 for 1.50 s. Calculate the tension in the cable supporting the elevator. (b) The elevator continues upward at constant velocity for 8.50 s. What is the tension in the cable during this time? (c) The elevator decelerates at a rate of 0.600 m/s2 for 3.00 s. What is the tension in the cable during deceleration? (d) How high has the elevator moved above its original starting point, and what is its final velocity? 50. Unreasonable Results (a) What is the final velocity of a car originally traveling at 50.0 km/h that decelerates at a rate of 0.400 m/s2 for 50.0 s? (b) What is unreasonable about the result? (c) Which premise is unreasonable, or which premises are inconsistent? 51. Unreasonable Results A 75.0-kg man stands on a bathroom scale in an elevator that accelerates from rest to 30.0 m/s in 2.00 s. (a) Calculate the scale reading in newtons and compare it with his weight. (The scale exerts an upward force on him equal to its reading.) (b) What is unreasonable about the result? (c) Which premise is unreasonable, or which premises are inconsistent? 4.8 Extended Topic: The Four Basic Forces—An Introduction 52. (a) What is the strength of the weak nuclear force relative to the strong nuclear force? (b) What is the strength of the weak nuclear force relative to the electromagnetic force? Since the weak nuclear force acts at only very short distances, such as inside nuclei, where the strong and electromagnetic forces also act, it might seem surprising that we have any knowledge of it at all. We have such knowledge because the weak nuclear force is responsible for beta decay, a type of nuclear decay not explained by other forces. 53. (a) What is the ratio of the strength of the gravitational force to that of the strong nuclear force? (b) What is the ratio of the strength of the gravitational force to that of the weak nuclear force? (c) What is the ratio of the strength of the gravitational force to that of the electromagnetic force? What do your answers imply about the influence of the gravitational force on atomic nuclei? 54. What is the ratio of the strength of the strong nuclear force to that of the electromagnetic force? Based on this ratio, you might expect that the strong force dominates the nucleus, which is true for small nuclei. Large nuclei, however, have sizes greater than the range of the strong nuclear force. At these sizes, the electromagnetic force begins to affect nuclear stability. These facts will be used to explain nuclear fusion and fission later in this text. Figure 4.41 The force T2 needed to hold steady the person being rescued from the fire is less than her weight and less than the force T1 in the other rope, since the more vertical rope supports a greater part of her weight (a vertical force). 43. Integrated Concepts A 35.0-kg dolphin decelerates from 12.0 to 7.50 m/s in 2.30 s to join another dolphin in play. What average force was exerted to slow him if he was moving horizontally? (The gravitational force is balanced by the buoyant force of the water.) 44. Integrated Concepts When starting a foot race, a 70.0-kg sprinter exerts an average force of 650 N backward on the ground for 0.800 s. (a) What is his final speed? (b) How far does he travel? 45. Integrated Concepts A large rocket has a mass of 2.00×106 kg at takeoff, and its engines produce a thrust of 3.50×107 N . (a) Find its initial acceleration if it takes off vertically. (b) How long does it take to reach a velocity of 120 km/h straight up, assuming constant mass and thrust? (c) In reality, the mass of a rocket decreases significantly as its fuel is consumed. Describe qualitatively how this affects the acceleration and time for this motion. 46. Integrated Concepts A basketball player jumps straight up for a ball. To do this, he lowers his body 0.300 m and then accelerates through this distance by forcefully straightening his legs. This player leaves the floor with a vertical velocity sufficient to carry him 0.900 m above the floor. (a) Calculate his velocity when he leaves the floor. (b) Calculate his acceleration while he is straightening his legs. He goes from zero to the velocity found in part (a) in a distance of 0.300 m. (c) Calculate the force he exerts on the floor to do this, given that his mass is 110 kg. 47. Integrated Concepts A 2.50-kg fireworks shell is fired straight up from a mortar and reaches a height of 110 m. (a) Neglecting air resistance (a poor assumption, but we will make it for this example), calculate the shell’s velocity when it leaves the mortar. (b) The mortar itself is a tube 0.450 m long. Calculate the average acceleration of the shell in the tube as it goes from zero to the velocity found in (a). (c) What is the 188 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion Test Prep for AP® Courses 4.1 Development of Force Concept 1. Figure 4.42 The figure above represents a racetrack with semicircular sections connected by straight sections. Each section has length d, and markers along the track are spaced d/4 apart. Two people drive cars counterclockwise around the track, as shown. Car X goes around the curves at constant speed vc, increases speed at constant acceleration for half of each straight section to reach a maximum speed of 2vc, then brakes at constant acceleration for the other half of each straight section to return to speed vc. Car Y also goes around the curves at constant speed vc, increases its speed at constant acceleration for one-fourth of each straight section to reach the same maximum speed 2vc, stays at that speed for half of each straight section, then brakes at constant acceleration for the remaining fourth of each straight section to return to speed vc. (a) On the figures below, draw an arrow showing the direction of the net force on each of the cars at the positions noted by the dots. If the net force is zero at any position, label the dot with 0. Figure 4.43 The position of the six dots on the Car Y track on the right are as follows: The first dot on the left center of the track is at the same position as it is on the Car X track. The second dot is just slight to the right of the Car X dot (less than a dash) past three perpendicular hash marks moving to the right. The third dot is about one and two-thirds perpendicular hash marks to the right of the center top perpendicular has mark. The fourth dot is in the same position as the Car X figure (one perpendicular hash mark above the center right perpendicular hash mark). The fifth dot is about one and two-third perpendicular hash marks to the right of the center bottom perpendicular hash mark. The sixth dot is in the same position as the Car Y dot (one and two third perpendicular hash marks to the left of the center bottom hash mark). (b) i. Indicate which car, if either, completes one trip around the track in less time, a
nd justify your answer qualitatively without using equations. This content is available for free at http://cnx.org/content/col11844/1.13 ii. Justify your answer about which car, if either, completes one trip around the track in less time quantitatively with appropriate equations. 2. Which of the following is an example of a body exerting a force on itself? a. a person standing up from a seated position b. a car accelerating while driving c. both of the above d. none of the above 3. A hawk accelerates as it glides in the air. Does the force causing the acceleration come from the hawk itself? Explain. 4. What causes the force that moves a boat forward when someone rows it? a. The force is caused by the rower’s arms. b. The force is caused by an interaction between the oars and gravity. c. The force is caused by an interaction between the oars and the water the boat is traveling in. d. The force is caused by friction. 4.4 Newton's Third Law of Motion: Symmetry in Forces 5. What object or objects commonly exert forces on the following objects in motion? (a) a soccer ball being kicked, (b) a dolphin jumping, (c) a parachutist drifting to Earth. 6. A ball with a mass of 0.25 kg hits a gym ceiling with a force of 78.0 N. What happens next? a. The ball accelerates downward with a force of 80.5 N. b. The ball accelerates downward with a force of 78.0 N. c. The ball accelerates downward with a force of 2.45 N. d. It depends on the height of the ceiling. 7. Which of the following is true? a. Earth exerts a force due to gravity on your body, and your body exerts a smaller force on the Earth, because your mass is smaller than the mass of the Earth. b. The Moon orbits the Earth because the Earth exerts a force on the Moon and the Moon exerts a force equal in magnitude and direction on the Earth. c. A rocket taking off exerts a force on the Earth equal to the force the Earth exerts on the rocket. d. An airplane cruising at a constant speed is not affected by gravity. 8. Stationary skater A pushes stationary skater B, who then accelerates at 5.0 m/s2. Skater A does not move. Since forces act in action-reaction pairs, explain why Skater A did not move? 9. The current in a river exerts a force of 9.0 N on a balloon floating in the river. A wind exerts a force of 13.0 N on the balloon in the opposite direction. Draw a free-body diagram to show the forces acting on the balloon. Use your free-body diagram to predict the effect on the balloon. 10. A force is applied to accelerate an object on a smooth icy surface. When the force stops, which of the following will be true? (Assume zero friction.) a. The object’s acceleration becomes zero. b. The object’s speed becomes zero. c. The object’s acceleration continues to increase at a constant rate. d. The object accelerates, but in the opposite direction. 11. A parachutist’s fall to Earth is determined by two opposing forces. A gravitational force of 539 N acts on the parachutist. After 2 s, she opens her parachute and experiences an air resistance of 615 N. At what speed is the parachutist falling after 10 s? Chapter 4 \| Dynamics: Force and Newton's Laws of Motion 189 12. A flight attendant pushes a cart down the aisle of a plane in flight. In determining the acceleration of the cart relative to the plane, which factor do you not need to consider? a. The friction of the cart’s wheels. b. The force with which the flight attendant’s feet push on the floor. c. The velocity of the plane. d. The mass of the items in the cart. 13. A landscaper is easing a wheelbarrow full of soil down a hill. Define the system you would analyze and list all the forces that you would need to include to calculate the acceleration of the wheelbarrow. 14. Two water-skiers, with masses of 48 kg and 61 kg, are preparing to be towed behind the same boat. When the boat accelerates, the rope the skiers hold onto accelerates with it and exerts a net force of 290 N on the skiers. At what rate will the skiers accelerate? a. 10.8 m/s2 b. 2.7 m/s2 c. 6.0 m/s2 and 4.8 m/s2 d. 5.3 m/s2 15. A figure skater has a mass of 40 kg and her partner's mass is 50 kg. She pushes against the ice with a force of 120 N, causing her and her partner to move forward. Calculate the pair’s acceleration. Assume that all forces opposing the motion, such as friction and air resistance, total 5.0 N. 4.5 Normal, Tension, and Other Examples of Force 16. An archer shoots an arrow straight up with a force of 24.5 N. The arrow has a mass of 0.4 kg. What is the force of gravity on the arrow? a. 9.8 m/s2 b. 9.8 N c. 61.25 N d. 3.9 N 17. A cable raises a mass of 120.0 kg with an acceleration of 1.3 m/s2. What force of tension is in the cable? 18. A child pulls a wagon along a grassy field. Define the system, the pairs of forces at work, and the results. 19. Two teams are engaging in a tug–of-war. The rope suddenly snaps. Which statement is true about the forces involved? a. The forces exerted by the two teams are no longer equal; the teams will accelerate in opposite directions as a result. b. The forces exerted by the players are no longer balanced by the force of tension in the rope; the teams will accelerate in opposite directions as a result. c. The force of gravity balances the forces exerted by the players; the teams will fall as a result d. The force of tension in the rope is transferred to the players; the teams will accelerate in opposite directions as a result. 20. The following free-body diagram represents a toboggan on a hill. What acceleration would you expect, and why? Figure 4.44 a. Acceleration down the hill; the force due to being pushed, together with the downhill component of gravity, overcomes the opposing force of friction. b. Acceleration down the hill; friction is less than the opposing component of force due to gravity. c. No movement; friction is greater than the force due to being pushed. It depends on how strong the force due to friction is. p d. 21. Draw a free-body diagram to represent the forces acting on a kite on a string that is floating stationary in the air. Label the forces in your diagram. 22. A car is sliding down a hill with a slope of 20°. The mass of the car is 965 kg. When a cable is used to pull the car up the slope, a force of 4215 N is applied. What is the car’s acceleration, ignoring friction? 4.6 Problem-Solving Strategies 23. A toboggan with two riders has a total mass of 85.0 kg. A third person is pushing the toboggan with a force of 42.5 N at the top of a hill with an angle of 15°. The force of friction on the toboggan is 31.0 N. Which statement describes an accurate free-body diagram to represent the situation? a. An arrow of magnitude 10.5 N points down the slope of the hill. b. An arrow of magnitude 833 N points straight down. c. An arrow of magnitude 833 N points perpendicular to the slope of the hill. d. An arrow of magnitude 73.5 N points down the slope of the hill. 24. A mass of 2.0 kg is suspended from the ceiling of an elevator by a rope. What is the tension in the rope when the elevator (i) accelerates upward at 1.5 m/s2? (ii) accelerates downward at 1.5 m/s2? a. b. Because the mass is hanging from the elevator itself, the tension in the rope will not change in either case. (i) 22.6 N; (ii) 19.6 N (i) 16.6 N; (ii) 19.6 N (i) 22.6 N; (ii) 16.6 N c. d. 25. Which statement is true about drawing free-body diagrams? 190 Chapter 4 \| Dynamics: Force and Newton's Laws of Motion a. Drawing a free-body diagram should be the last step in solving a problem about forces. b. Drawing a free-body diagram helps you compare forces 30. Explain which of the four fundamental forces is responsible for a ball bouncing off the ground after it hits, and why this force has this effect. quantitatively. c. The forces in a free-body diagram should always balance. d. Drawing a free-body diagram can help you determine the net force. 4.7 Further Applications of Newton's Laws of Motion 26. A basketball player jumps as he shoots the ball. Describe the forces that are acting on the ball and on the basketball player. What are the results? 27. Two people push on a boulder to try to move it. The mass of the boulder is 825 kg. One person pushes north with a force of 64 N. The other pushes west with a force of 38 N. Predict the magnitude of the acceleration of the boulder. Assume that friction is negligible. 28. 31. Which of the basic forces best explains tension in a rope being pulled between two people? Is the acting force causing attraction or repulsion in this instance? a. gravity; attraction b. electromagnetic; attraction c. weak and strong nuclear; attraction d. weak and strong nuclear; repulsion 32. Explain how interatomic electric forces produce the normal force, and why it has the direction it does. 33. The gravitational force is the weakest of the four basic forces. In which case can the electromagnetic, strong, and weak forces be ignored because the gravitational force is so strongly dominant? a. a person jumping on a trampoline b. a rocket blasting off from Earth c. a log rolling down a hill d. all of the above 34. Describe a situation in which gravitational force is the dominant force. Why can the other three basic forces be ignored in the situation you described? Figure 4.45 The figure shows the forces exerted on a block that is sliding on a horizontal surface: the gravitational force of 40 N, the 40 N normal force exerted by the surface, and a frictional force exerted to the left. The coefficient of friction between the block and the surface is 0.20. The acceleration of the block is most nearly a. 1.0 m/s2 to the right b. 1.0 m/s2 to the left c. 2.0 m/s2 to the right d. 2.0 m/s2 to the left 4.8 Extended Topic: The Four Basic Forces—An Introduction 29. Which phenomenon correctly describes the direction and magnitude of normal forces? a. electromagnetic attraction b. electromagnetic repulsion c. gravitational attraction d. gravitational repulsion This content is available for free at http://cnx.org/content/col11844/1.13 C
hapter 5 \| Further Applications of Newton's Laws: Friction, Drag, and Elasticity 191 FURTHER APPLICATIONS OF NEWTON'S 5 LAWS: FRICTION, DRAG, AND ELASTICITY Figure 5.1 Total hip replacement surgery has become a common procedure. The head (or ball) of the patient's femur fits into a cup that has a hard plastic-like inner lining. (credit: National Institutes of Health, via Wikimedia Commons) Chapter Outline 5.1. Friction 5.2. Drag Forces 5.3. Elasticity: Stress and Strain Connection for AP® Courses Have you ever wondered why it is difficult to walk on a smooth surface like ice? The interaction between you and the surface is a result of forces that affect your motion. In the previous chapter, you learned Newton's laws of motion and examined how net force affects the motion, position and shape of an object. Now we will look at some interesting and common forces that will provide further applications of Newton's laws of motion. The information presented in this chapter supports learning objectives covered under Big Idea 3 of the AP Physics Curriculum Framework, which refer to the nature of forces and their roles in interactions among objects. The chapter discusses examples of specific contact forces, such as friction, air or liquid drag, and elasticity that may affect the motion or shape of an object. It also discusses the nature of forces on both macroscopic and microscopic levels (Enduring Understanding 3.C and Essential Knowledge 3.C.4). In addition, Newton's laws are applied to describe the motion of an object (Enduring Understanding 3.B) and to examine relationships between contact forces and other forces exerted on an object (Enduring Understanding 3.A, 3.A.3 and 192 Chapter 5 \| Further Applications of Newton's Laws: Friction, Drag, and Elasticity Essential Knowledge 3.A.4). The examples in this chapter give you practice in using vector properties of forces (Essential Knowledge 3.A.2) and free-body diagrams (Essential Knowledge 3.B.2) to determine net force (Essential Knowledge 3.B.1). Big Idea 3 The interactions of an object with other objects can be described by forces. Enduring Understanding 3.A All forces share certain common characteristics when considered by observers in inertial reference frames. Essential Knowledge 3.A.2 Forces are described by vectors. Essential Knowledge 3.A.3 A force exerted on an object is always due to the interaction of that object with another object. Essential Knowledge 3.A.4 If one object exerts a force on a second object, the second object always exerts a force of equal magnitude on the ﬁrst object in the opposite direction. Enduring Understanding 3.B Classically, the acceleration of an object interacting with other objects can be predicted by using → = → ∑ . Essential Knowledge 3.B.1 If an object of interest interacts with several other objects, the net force is the vector sum of the individual forces. Essential Knowledge 3.B.2 Free-body diagrams are useful tools for visualizing forces being exerted on a single object and writing the equations that represent a physical situation. Enduring Understanding 3.C At the macroscopic level, forces can be categorized as either long-range (action-at-a-distance) forces or contact forces. Essential Knowledge 3.C.4 Contact forces result from the interaction of one object touching another object, and they arise from interatomic electric forces. These forces include tension, friction, normal, spring (Physics 1), and buoyant (Physics 2). 5.1 Friction Learning Objectives By the end of this section, you will be able to: • Discuss the general characteristics of friction. • Describe the various types of friction. • Calculate the magnitudes of static and kinetic frictional forces. The information presented in this section supports the following AP® learning objectives and science practices: • 3.C.4.1 The student is able to make claims about various contact forces between objects based on the microscopic cause of those forces. (S.P. 6.1) • 3.C.4.2 The student is able to explain contact forces (tension, friction, normal, buoyant, spring) as arising from interatomic electric forces and that they therefore have certain directions. (S.P. 6.2) Friction is a force that is around us all the time that opposes relative motion between systems in contact but also allows us to move (which you have discovered if you have ever tried to walk on ice). While a common force, the behavior of friction is actually very complicated and is still not completely understood. We have to rely heavily on observations for whatever understandings we can gain. However, we can still deal with its more elementary general characteristics and understand the circumstances in which it behaves. Friction Friction is a force that opposes relative motion between systems in contact. One of the simpler characteristics of friction is that it is parallel to the contact surface between systems and always in a direction that opposes motion or attempted motion of the systems relative to each other. If two systems are in contact and moving relative to one another, then the friction between them is called kinetic friction. For example, friction slows a hockey puck sliding on ice. But when objects are stationary, static friction can act between them; the static friction is usually greater than the kinetic friction between the objects. Kinetic Friction If two systems are in contact and moving relative to one another, then the friction between them is called kinetic friction. Imagine, for example, trying to slide a heavy crate across a concrete floor—you may push harder and harder on the crate and not move it at all. This means that the static friction responds to what you do—it increases to be equal to and in the opposite direction of your push. But if you finally push hard enough, the crate seems to slip suddenly and starts to move. Once in motion it is easier to keep it in motion than it was to get it started, indicating that the kinetic friction force is less than the static friction This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 5 \| Further Applications of Newton's Laws: Friction, Drag, and Elasticity 193 force. If you add mass to the crate, say by placing a box on top of it, you need to push even harder to get it started and also to keep it moving. Furthermore, if you oiled the concrete you would find it to be easier to get the crate started and keep it going (as you might expect). Figure 5.2 is a crude pictorial representation of how friction occurs at the interface between two objects. Close-up inspection of these surfaces shows them to be rough. So when you push to get an object moving (in this case, a crate), you must raise the object until it can skip along with just the tips of the surface hitting, break off the points, or do both. A considerable force can be resisted by friction with no apparent motion. The harder the surfaces are pushed together (such as if another box is placed on the crate), the more force is needed to move them. Part of the friction is due to adhesive forces between the surface molecules of the two objects, which explain the dependence of friction on the nature of the substances. Adhesion varies with substances in contact and is a complicated aspect of surface physics. Once an object is moving, there are fewer points of contact (fewer molecules adhering), so less force is required to keep the object moving. At small but nonzero speeds, friction is nearly independent of speed. Figure 5.2 Frictional forces, such as , always oppose motion or attempted motion between objects in contact. Friction arises in part because of the roughness of the surfaces in contact, as seen in the expanded view. In order for the object to move, it must rise to where the peaks can skip along the bottom surface. Thus a force is required just to set the object in motion. Some of the peaks will be broken off, also requiring a force to maintain motion. Much of the friction is actually due to attractive forces between molecules making up the two objects, so that even perfectly smooth surfaces are not friction-free. Such adhesive forces also depend on the substances the surfaces are made of, explaining, for example, why rubber-soled shoes slip less than those with leather soles. The magnitude of the frictional force has two forms: one for static situations (static friction), the other for when there is motion (kinetic friction). When there is no motion between the objects, the magnitude of static friction fs is s ≤ s, (5.1) where s is the coefficient of static friction and is the magnitude of the normal force (the force perpendicular to the surface). Magnitude of Static Friction Magnitude of static friction s is where s is the coefficient of static friction and is the magnitude of the normal force. s ≤ s, (5.2) The symbol ≤ means less than or equal to, implying that static friction can have a minimum and a maximum value of s . Static friction is a responsive force that increases to be equal and opposite to whatever force is exerted, up to its maximum limit. Once the applied force exceeds s(max) , the object will move. Thus Once an object is moving, the magnitude of kinetic friction fk is given by k = k, s(max) = s. (5.3) (5.4) where k is the coefficient of kinetic friction. A system in which k = k is described as a system in which friction behaves simply. Magnitude of Kinetic Friction The magnitude of kinetic friction k is given by 194 Chapter 5 \| Further Applications of Newton's Laws: Friction, Drag, and Elasticity where k is the coefficient of kinetic friction. k = k, (5.5) As seen in Table 5.1, the coefficients of kinetic friction are less than their static counterparts. That values of in Table 5.1 are stated to only one or, at most, two digits is an indication of the approximate description of friction given by the above two equations. Table 5.1 Coefficients of Static and Kinetic Friction System Static friction μs Kinetic friction μk Rubber on dry concrete
Rubber on wet concrete Wood on wood Waxed wood on wet snow Metal on wood Steel on steel (dry) Steel on steel (oiled) Teflon on steel 1.0 0.7 0.5 0.14 0.5 0.6 0.05 0.04 Bone lubricated by synovial fluid 0.016 Shoes on wood Shoes on ice Ice on ice Steel on ice 0.9 0.1 0.1 0.4 0.7 0.5 0.3 0.1 0.3 0.3 0.03 0.04 0.015 0.7 0.05 0.03 0.02 The equations given earlier include the dependence of friction on materials and the normal force. The direction of friction is always opposite that of motion, parallel to the surface between objects, and perpendicular to the normal force. For example, if the crate you try to push (with a force parallel to the floor) has a mass of 100 kg, then the normal force would be equal to its weight, = = (100 kg)(9.80 m/s2) = 980 N , perpendicular to the floor. If the coefficient of static friction is 0.45, you would have to exert a force parallel to the floor greater than s(max) = s = (0.45)(980 N) = 440 N to move the crate. Once there is motion, friction is less and the coefficient of kinetic friction might be 0.30, so that a force of only 290 N ( k = k = (0.30)(980 N) = 290 N ) would keep it moving at a constant speed. If the floor is lubricated, both coefficients are considerably less than they would be without lubrication. Coefficient of friction is a unit less quantity with a magnitude usually between 0 and 1.0. The coefficient of the friction depends on the two surfaces that are in contact. Take-Home Experiment Find a small plastic object (such as a food container) and slide it on a kitchen table by giving it a gentle tap. Now spray water on the table, simulating a light shower of rain. What happens now when you give the object the same-sized tap? Now add a few drops of (vegetable or olive) oil on the surface of the water and give the same tap. What happens now? This latter situation is particularly important for drivers to note, especially after a light rain shower. Why? Many people have experienced the slipperiness of walking on ice. However, many parts of the body, especially the joints, have much smaller coefficients of friction—often three or four times less than ice. A joint is formed by the ends of two bones, which are connected by thick tissues. The knee joint is formed by the lower leg bone (the tibia) and the thighbone (the femur). The hip is a ball (at the end of the femur) and socket (part of the pelvis) joint. The ends of the bones in the joint are covered by cartilage, which provides a smooth, almost glassy surface. The joints also produce a fluid (synovial fluid) that reduces friction and wear. A damaged or arthritic joint can be replaced by an artificial joint (Figure 5.3). These replacements can be made of metals (stainless steel or titanium) or plastic (polyethylene), also with very small coefficients of friction. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 5 \| Further Applications of Newton's Laws: Friction, Drag, and Elasticity 195 Figure 5.3 Artificial knee replacement is a procedure that has been performed for more than 20 years. In this figure, we see the post-op x rays of the right knee joint replacement. (credit: Mike Baird, Flickr) Other natural lubricants include saliva produced in our mouths to aid in the swallowing process, and the slippery mucus found between organs in the body, allowing them to move freely past each other during heartbeats, during breathing, and when a person moves. Artificial lubricants are also common in hospitals and doctor's clinics. For example, when ultrasonic imaging is carried out, the gel that couples the transducer to the skin also serves to to lubricate the surface between the transducer and the skin—thereby reducing the coefficient of friction between the two surfaces. This allows the transducer to mover freely over the skin. Example 5.1 Skiing Exercise A skier with a mass of 62 kg is sliding down a snowy slope. Find the coefficient of kinetic friction for the skier if friction is known to be 45.0 N. Strategy The magnitude of kinetic friction was given in to be 45.0 N. Kinetic friction is related to the normal force N as k = k ; thus, the coefficient of kinetic friction can be found if we can find the normal force of the skier on a slope. The normal force is always perpendicular to the surface, and since there is no motion perpendicular to the surface, the normal force should equal the component of the skier's weight perpendicular to the slope. (See the skier and free-body diagram in Figure 5.4.) 196 Chapter 5 \| Further Applications of Newton's Laws: Friction, Drag, and Elasticity Figure 5.4 The motion of the skier and friction are parallel to the slope and so it is most convenient to project all forces onto a coordinate system where one axis is parallel to the slope and the other is perpendicular (axes shown to left of skier). N (the normal force) is perpendicular to the slope, and f (the friction) is parallel to the slope, but w (the skier's weight) has components along both axes, namely w⊥ and W// . N is equal in magnitude to w⊥ , so there is no motion perpendicular to the slope. However, f is less than W// in magnitude, so there is acceleration down the slope (along the x-axis). That is, = ⊥ = cos 25º = cos 25º. Substituting this into our expression for kinetic friction, we get k = k cos 25º, which can now be solved for the coefficient of kinetic friction k . Solution Solving for k gives k = k = k = k cos 25º cos 25º. Substituting known values on the right-hand side of the equation, k = 45.0 N (62 kg)(9.80 m/s2)(0.906) = 0.082. Discussion (5.6) (5.7) (5.8) (5.9) This result is a little smaller than the coefficient listed in Table 5.1 for waxed wood on snow, but it is still reasonable since values of the coefficients of friction can vary greatly. In situations like this, where an object of mass slides down a slope that makes an angle with the horizontal, friction is given by k = k cos . All objects will slide down a slope with constant acceleration under these circumstances. Proof of this is left for this chapter's Problems and Exercises. Take-Home Experiment An object will slide down an inclined plane at a constant velocity if the net force on the object is zero. We can use this fact to measure the coefficient of kinetic friction between two objects. As shown in Example 5.1, the kinetic friction on a slope k = k cos . The component of the weight down the slope is equal to sin (see the free-body diagram in Figure 5.4). These forces act in opposite directions, so when they have equal magnitude, the acceleration is zero. Writing these out: Solving for k , we find that k = k cos = sin . k = sin cos = tan . (5.10) (5.11) (5.12) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 5 \| Further Applications of Newton's Laws: Friction, Drag, and Elasticity 197 Put a coin on a book and tilt it until the coin slides at a constant velocity down the book. You might need to tap the book lightly to get the coin to move. Measure the angle of tilt relative to the horizontal and find k . Note that the coin will not start to slide at all until an angle greater than is attained, since the coefficient of static friction is larger than the coefficient of kinetic friction. Discuss how this may affect the value for k and its uncertainty. We have discussed that when an object rests on a horizontal surface, there is a normal force supporting it equal in magnitude to its weight. Furthermore, simple friction is always proportional to the normal force. Making Connections: Submicroscopic Explanations of Friction The simpler aspects of friction dealt with so far are its macroscopic (large-scale) characteristics. Great strides have been made in the atomic-scale explanation of friction during the past several decades. Researchers are finding that the atomic nature of friction seems to have several fundamental characteristics. These characteristics not only explain some of the simpler aspects of friction—they also hold the potential for the development of nearly friction-free environments that could save hundreds of billions of dollars in energy which is currently being converted (unnecessarily) to heat. Figure 5.5 illustrates one macroscopic characteristic of friction that is explained by microscopic (small-scale) research. We have noted that friction is proportional to the normal force, but not to the area in contact, a somewhat counterintuitive notion. When two rough surfaces are in contact, the actual contact area is a tiny fraction of the total area since only high spots touch. When a greater normal force is exerted, the actual contact area increases, and it is found that the friction is proportional to this area. Figure 5.5 Two rough surfaces in contact have a much smaller area of actual contact than their total area. When there is a greater normal force as a result of a greater applied force, the area of actual contact increases as does friction. But the atomic-scale view promises to explain far more than the simpler features of friction. The mechanism for how heat is generated is now being determined. In other words, why do surfaces get warmer when rubbed? Essentially, atoms are linked with one another to form lattices. When surfaces rub, the surface atoms adhere and cause atomic lattices to vibrate—essentially creating sound waves that penetrate the material. The sound waves diminish with distance and their energy is converted into heat. Chemical reactions that are related to frictional wear can also occur between atoms and molecules on the surfaces. Figure 5.6 shows how the tip of a probe drawn across another material is deformed by atomic-scale friction. The force needed to drag the tip can be measured and is found to be related to shear stress, which will be discussed later in this chapter. The variation in shear stress is remarkable (more than a factor of 1012 ) and difficult to predict theoretically, but shear stress is yielding a fundamental understanding of a large-scale ph
enomenon known since ancient times—friction. 198 Chapter 5 \| Further Applications of Newton's Laws: Friction, Drag, and Elasticity Figure 5.6 The tip of a probe is deformed sideways by frictional force as the probe is dragged across a surface. Measurements of how the force varies for different materials are yielding fundamental insights into the atomic nature of friction. PhET Explorations: Forces and Motion Explore the forces at work when you try to push a filing cabinet. Create an applied force and see the resulting friction force and total force acting on the cabinet. Charts show the forces, position, velocity, and acceleration vs. time. Draw a free-body diagram of all the forces (including gravitational and normal forces). Figure 5.7 Forces and Motion (http://cnx.org/content/m54899/1.2/forces-and-motion_en.jar) 5.2 Drag Forces Learning Objectives By the end of this section, you will be able to: • Define drag force and model it mathematically. • Discuss the applications of drag force. • Define terminal velocity. • Perform calculations to find terminal velocity. Another interesting force in everyday life is the force of drag on an object when it is moving in a fluid (either a gas or a liquid). You feel the drag force when you move your hand through water. You might also feel it if you move your hand during a strong wind. The faster you move your hand, the harder it is to move. You feel a smaller drag force when you tilt your hand so only the side goes through the air—you have decreased the area of your hand that faces the direction of motion. Like friction, the drag force always opposes the motion of an object. Unlike simple friction, the drag force is proportional to some function of the velocity of the object in that fluid. This functionality is complicated and depends upon the shape of the object, its size, its velocity, and the fluid it is in. For most large objects such as bicyclists, cars, and baseballs not moving too slowly, the magnitude of the drag force D is found to be proportional to the square of the speed of the object. We can write this relationship mathematically as . When taking into account other factors, this relationship becomes D = 1 2 Cρ 2, (5.13) where is the drag coefficient, is the area of the object facing the fluid, and is the density of the fluid. (Recall that density is mass per unit volume.) This equation can also be written in a more generalized fashion as D = 2 , where is a constant equivalent to 0.5 . We have set the exponent for these equations as 2 because, when an object is moving at high velocity This content is available for free at http://cnx.org/content/col11844/1.13 212 Chapter 5 \| Further Applications of Newton's Laws: Friction, Drag, and Elasticity Although measurable, this is not a significant decrease in volume considering that the force per unit area is about 500 atmospheres (1 million pounds per square foot). Liquids and solids are extraordinarily difficult to compress. Conversely, very large forces are created by liquids and solids when they try to expand but are constrained from doing so—which is equivalent to compressing them to less than their normal volume. This often occurs when a contained material warms up, since most materials expand when their temperature increases. If the materials are tightly constrained, they deform or break their container. Another very common example occurs when water freezes. Water, unlike most materials, expands when it freezes, and it can easily fracture a boulder, rupture a biological cell, or crack an engine block that gets in its way. Other types of deformations, such as torsion or twisting, behave analogously to the tension, shear, and bulk deformations considered here. Glossary deformation: change in shape due to the application of force drag force: D , found to be proportional to the square of the speed of the object; mathematically D ∝ 2 D = 1 where is the drag coefficient, is the area of the object facing the fluid, and is the density of the fluid 2 2, friction: a force that opposes relative motion or attempts at motion between systems in contact Hooke's law: proportional relationship between the force on a material and the deformation Δ it causes, = Δ kinetic friction: a force that opposes the motion of two systems that are in contact and moving relative to one another magnitude of kinetic friction: k = k , where k is the coefficient of kinetic friction magnitude of static friction: s ≤ s , where s is the coefficient of static friction and is the magnitude of the normal force shear deformation: deformation perpendicular to the original length of an object static friction: a force that opposes the motion of two systems that are in contact and are not moving relative to one another Stokes' law: s = 6 , where is the radius of the object, is the viscosity of the fluid, and is the object's velocity strain: ratio of change in length to original length stress: ratio of force to area tensile strength: measure of deformation for a given tension or compression Section Summary 5.1 Friction • Friction is a contact force between systems that opposes the motion or attempted motion between them. Simple friction is proportional to the normal force pushing the systems together. (A normal force is always perpendicular to the contact surface between systems.) Friction depends on both of the materials involved. The magnitude of static friction s between systems stationary relative to one another is given by s ≤ s, where s is the coefficient of static friction, which depends on both of the materials. • The kinetic friction force k between systems moving relative to one another is given by where k is the coefficient of kinetic friction, which also depends on both materials. k = k, 5.2 Drag Forces This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 5 \| Further Applications of Newton's Laws: Friction, Drag, and Elasticity 213 • Drag forces acting on an object moving in a fluid oppose the motion. For larger objects (such as a baseball) moving at a velocity in air, the drag force is given by D = 1 2 2, where is the drag coefficient (typical values are given in Table 5.2), is the area of the object facing the fluid, and is the fluid density. • For small objects (such as a bacterium) moving in a denser medium (such as water), the drag force is given by Stokes' law, s = 6, where is the radius of the object, is the fluid viscosity, and is the object's velocity. 5.3 Elasticity: Stress and Strain • Hooke's law is given by = Δ, where Δ is the amount of deformation (the change in length), is the applied force, and is a proportionality constant that depends on the shape and composition of the object and the direction of the force. The relationship between the deformation and the applied force can also be written as Δ = 1 0, where is Young's modulus, which depends on the substance, is the cross-sectional area, and 0 is the original length. • The ratio of force to area, , is defined as stress, measured in N/m2. • The ratio of the change in length to length, Δ 0 , is defined as strain (a unitless quantity). In other words, • The expression for shear deformation is stress = ×strain. Δ = 1 where is the shear modulus and is the force applied perpendicular to 0 and parallel to the cross-sectional area . 0, • The relationship of the change in volume to other physical quantities is given by Δ = 1 where is the bulk modulus, 0 is the original volume, and surfaces. 0, is the force per unit area applied uniformly inward on all Conceptual Questions 5.1 Friction 1. Define normal force. What is its relationship to friction when friction behaves simply? 2. The glue on a piece of tape can exert forces. Can these forces be a type of simple friction? Explain, considering especially that tape can stick to vertical walls and even to ceilings. 3. When you learn to drive, you discover that you need to let up slightly on the brake pedal as you come to a stop or the car will stop with a jerk. Explain this in terms of the relationship between static and kinetic friction. 4. When you push a piece of chalk across a chalkboard, it sometimes screeches because it rapidly alternates between slipping and sticking to the board. Describe this process in more detail, in particular explaining how it is related to the fact that kinetic friction is less than static friction. (The same slip-grab process occurs when tires screech on pavement.) 5.2 Drag Forces 5. Athletes such as swimmers and bicyclists wear body suits in competition. Formulate a list of pros and cons of such suits. 6. Two expressions were used for the drag force experienced by a moving object in a liquid. One depended upon the speed, while the other was proportional to the square of the speed. In which types of motion would each of these expressions be more applicable than the other one? 7. As cars travel, oil and gasoline leaks onto the road surface. If a light rain falls, what does this do to the control of the car? Does a heavy rain make any difference? 8. Why can a squirrel jump from a tree branch to the ground and run away undamaged, while a human could break a bone in such a fall? 214 Chapter 5 \| Further Applications of Newton's Laws: Friction, Drag, and Elasticity 5.3 Elasticity: Stress and Strain 9. The elastic properties of the arteries are essential for blood flow. Explain the importance of this in terms of the characteristics of the flow of blood (pulsating or continuous). 10. What are you feeling when you feel your pulse? Measure your pulse rate for 10 s and for 1 min. Is there a factor of 6 difference? 11. Examine different types of shoes, including sports shoes and thongs. In terms of physics, why are the bottom surfaces designed as they are? What differences will dry and wet conditions make for these surfaces? 12. Would you expect your height to be different depending upon the time of day? Why or why not? 13. Why can a squirrel jump from a tree branch to the ground and run away und
amaged, while a human could break a bone in such a fall? 14. Explain why pregnant women often suffer from back strain late in their pregnancy. 15. An old carpenter's trick to keep nails from bending when they are pounded into hard materials is to grip the center of the nail firmly with pliers. Why does this help? 16. When a glass bottle full of vinegar warms up, both the vinegar and the glass expand, but vinegar expands significantly more with temperature than glass. The bottle will break if it was filled to its tightly capped lid. Explain why, and also explain how a pocket of air above the vinegar would prevent the break. (This is the function of the air above liquids in glass containers.) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 5 \| Further Applications of Newton's Laws: Friction, Drag, and Elasticity 215 Problems & Exercises 5.1 Friction 1. A physics major is cooking breakfast when he notices that the frictional force between his steel spatula and his Teflon frying pan is only 0.200 N. Knowing the coefficient of kinetic friction between the two materials, he quickly calculates the normal force. What is it? 2. (a) When rebuilding her car's engine, a physics major must exert 300 N of force to insert a dry steel piston into a steel cylinder. What is the magnitude of the normal force between the piston and cylinder? (b) What is the magnitude of the force would she have to exert if the steel parts were oiled? 3. (a) What is the maximum frictional force in the knee joint of a person who supports 66.0 kg of her mass on that knee? (b) During strenuous exercise it is possible to exert forces to the joints that are easily ten times greater than the weight being supported. What is the maximum force of friction under such conditions? The frictional forces in joints are relatively small in all circumstances except when the joints deteriorate, such as from injury or arthritis. Increased frictional forces can cause further damage and pain. 4. Suppose you have a 120-kg wooden crate resting on a wood floor. (a) What maximum force can you exert horizontally on the crate without moving it? (b) If you continue to exert this force once the crate starts to slip, what will the magnitude of its acceleration then be? 5. (a) If half of the weight of a small 1.00×103 kg utility truck is supported by its two drive wheels, what is the magnitude of the maximum acceleration it can achieve on dry concrete? (b) Will a metal cabinet lying on the wooden bed of the truck slip if it accelerates at this rate? (c) Solve both problems assuming the truck has four-wheel drive. 6. A team of eight dogs pulls a sled with waxed wood runners on wet snow (mush!). The dogs have average masses of 19.0 kg, and the loaded sled with its rider has a mass of 210 kg. (a) Calculate the magnitude of the acceleration starting from rest if each dog exerts an average force of 185 N backward on the snow. (b) What is the magnitude of the acceleration once the sled starts to move? (c) For both situations, calculate the magnitude of the force in the coupling between the dogs and the sled. 7. Consider the 65.0-kg ice skater being pushed by two others shown in Figure 5.21. (a) Find the direction and magnitude of Ftot , the total force exerted on her by the others, given that the magnitudes 1 and 2 are 26.4 N and 18.6 N, respectively. (b) What is her initial acceleration if she is initially stationary and wearing steel-bladed skates that point in the direction of Ftot ? (c) What is her acceleration assuming she is already moving in the direction of Ftot ? (Remember that friction always acts in the direction opposite that of motion or attempted motion between surfaces in contact.) Figure 5.21 8. Show that the acceleration of any object down a frictionless incline that makes an angle with the horizontal is = sin . (Note that this acceleration is independent of mass.) 9. Show that the acceleration of any object down an incline where friction behaves simply (that is, where k = k ) is = ( sin − kcos ). Note that the acceleration is independent of mass and reduces to the expression found in the previous problem when friction becomes negligibly small ( k = 0). 10. Calculate the deceleration of a snow boarder going up a 5.0º , slope assuming the coefficient of friction for waxed wood on wet snow. The result of Exercise 5.9 may be useful, but be careful to consider the fact that the snow boarder is going uphill. Explicitly show how you follow the steps in Problem-Solving Strategies. 11. (a) Calculate the acceleration of a skier heading down a 10.0º slope, assuming the coefficient of friction for waxed wood on wet snow. (b) Find the angle of the slope down which this skier could coast at a constant velocity. You can neglect air resistance in both parts, and you will find the result of Exercise 5.9 to be useful. Explicitly show how you follow the steps in the Problem-Solving Strategies. 12. If an object is to rest on an incline without slipping, then friction must equal the component of the weight of the object parallel to the incline. This requires greater and greater friction for steeper slopes. Show that the maximum angle of an incline above the horizontal for which an object will not slide down is = tan–1 μs . You may use the result of the previous problem. Assume that = 0 and that static friction has reached its maximum value. 13. Calculate the maximum deceleration of a car that is heading down a 6º slope (one that makes an angle of 6º with the horizontal) under the following road conditions. You may assume that the weight of the car is evenly distributed on all four tires and that the coefficient of static friction is involved—that is, the tires are not allowed to slip during the deceleration. (Ignore rolling.) Calculate for a car: (a) On dry concrete. (b) On wet concrete. (c) On ice, assuming that s = 0.100 , the same as for shoes on ice. 14. Calculate the maximum acceleration of a car that is heading up a 4º slope (one that makes an angle of 4º with the horizontal) under the following road conditions. Assume that only half the weight of the car is supported by the two drive wheels and that the coefficient of static friction is involved—that is, the tires are not allowed to slip during the acceleration. (Ignore rolling.) (a) On dry concrete. (b) On wet 216 Chapter 5 \| Further Applications of Newton's Laws: Friction, Drag, and Elasticity concrete. (c) On ice, assuming that μs = 0.100 , the same as for shoes on ice. 15. Repeat Exercise 5.14 for a car with four-wheel drive. 16. A freight train consists of two 8.00×105-kg engines and 45 cars with average masses of 5.50×105 kg . (a) What force must each engine exert backward on the track to accelerate the train at a rate of 5.00×10−2 m / s2 if the force of friction is 7.50×105 N , assuming the engines exert identical forces? This is not a large frictional force for such a massive system. Rolling friction for trains is small, and consequently trains are very energy-efficient transportation systems. (b) What is the magnitude of the force in the coupling between the 37th and 38th cars (this is the force each exerts on the other), assuming all cars have the same mass and that friction is evenly distributed among all of the cars and engines? 17. Consider the 52.0-kg mountain climber in Figure 5.22. (a) Find the tension in the rope and the force that the mountain climber must exert with her feet on the vertical rock face to remain stationary. Assume that the force is exerted parallel to her legs. Also, assume negligible force exerted by her arms. (b) What is the minimum coefficient of friction between her shoes and the cliff? Figure 5.22 Part of the climber's weight is supported by her rope and part by friction between her feet and the rock face. 18. A contestant in a winter sporting event pushes a 45.0-kg block of ice across a frozen lake as shown in Figure 5.23(a). (a) Calculate the minimum force he must exert to get the block moving. (b) What is the magnitude of its acceleration once it starts to move, if that force is maintained? 19. Repeat Exercise 5.18 with the contestant pulling the block of ice with a rope over his shoulder at the same angle above the horizontal as shown in Figure 5.23(b). This content is available for free at http://cnx.org/content/col11844/1.13 Figure 5.23 Which method of sliding a block of ice requires less force—(a) pushing or (b) pulling at the same angle above the horizontal? 5.2 Drag Forces 20. The terminal velocity of a person falling in air depends upon the weight and the area of the person facing the fluid. Find the terminal velocity (in meters per second and kilometers per hour) of an 80.0-kg skydiver falling in a pike (headfirst) position with a surface area of 0.140 m2 . 21. A 60-kg and a 90-kg skydiver jump from an airplane at an altitude of 6000 m, both falling in the pike position. Make some assumption on their frontal areas and calculate their terminal velocities. How long will it take for each skydiver to reach the ground (assuming the time to reach terminal velocity is small)? Assume all values are accurate to three significant digits. 22. A 560-g squirrel with a surface area of 930 cm2 falls from a 5.0-m tree to the ground. Estimate its terminal velocity. (Use a drag coefficient for a horizontal skydiver.) What will be the velocity of a 56-kg person hitting the ground, assuming no drag contribution in such a short distance? 23. To maintain a constant speed, the force provided by a car's engine must equal the drag force plus the force of friction of the road (the rolling resistance). (a) What are the magnitudes of drag forces at 70 km/h and 100 km/h for a Toyota Camry? (Drag area is 0.70 m2 ) (b) What is the magnitude of drag force at 70 km/h and 100 km/h for a Hummer H2? (Drag area is 2.44 m2 ) Assume all values are accurate to three significant digits. 24. By what factor does the drag force on a car increase as it goes from 65 to 110 km/h? 25. Calcula
te the speed a spherical rain drop would achieve falling from 5.00 km (a) in the absence of air drag (b) with air drag. Take the size across of the drop to be 4 mm, the density to be 1.00×103 kg/m3 , and the surface area to be 2 . Chapter 5 \| Further Applications of Newton's Laws: Friction, Drag, and Elasticity 217 26. Using Stokes' law, verify that the units for viscosity are kilograms per meter per second. 27. Find the terminal velocity of a spherical bacterium (diameter 2.00 μm ) falling in water. You will first need to note that the drag force is equal to the weight at terminal velocity. Take the density of the bacterium to be 1.10×103 kg/m3 . 28. Stokes' law describes sedimentation of particles in liquids and can be used to measure viscosity. Particles in liquids achieve terminal velocity quickly. One can measure the time it takes for a particle to fall a certain distance and then use Stokes' law to calculate the viscosity of the liquid. Suppose a steel ball bearing (density 7.8×103 kg/m3 3.0 mm ) is dropped in a container of motor oil. It takes 12 s to fall a distance of 0.60 m. Calculate the viscosity of the oil. , diameter 5.3 Elasticity: Stress and Strain 29. During a circus act, one performer swings upside down hanging from a trapeze holding another, also upside-down, performer by the legs. If the upward force on the lower performer is three times her weight, how much do the bones (the femurs) in her upper legs stretch? You may assume each is equivalent to a uniform rod 35.0 cm long and 1.80 cm in radius. Her mass is 60.0 kg. 30. During a wrestling match, a 150 kg wrestler briefly stands on one hand during a maneuver designed to perplex his already moribund adversary. By how much does the upper arm bone shorten in length? The bone can be represented by a uniform rod 38.0 cm in length and 2.10 cm in radius. 31. (a) The “lead” in pencils is a graphite composition with a Young's modulus of about 1×109 N / m2 . Calculate the change in length of the lead in an automatic pencil if you tap it straight into the pencil with a force of 4.0 N. The lead is 0.50 mm in diameter and 60 mm long. (b) Is the answer reasonable? That is, does it seem to be consistent with what you have observed when using pencils? 32. TV broadcast antennas are the tallest artificial structures on Earth. In 1987, a 72.0-kg physicist placed himself and 400 kg of equipment at the top of one 610-m high antenna to perform gravity experiments. By how much was the antenna compressed, if we consider it to be equivalent to a steel cylinder 0.150 m in radius? 33. (a) By how much does a 65.0-kg mountain climber stretch her 0.800-cm diameter nylon rope when she hangs 35.0 m below a rock outcropping? (b) Does the answer seem to be consistent with what you have observed for nylon ropes? Would it make sense if the rope were actually a bungee cord? 34. A 20.0-m tall hollow aluminum flagpole is equivalent in strength to a solid cylinder 4.00 cm in diameter. A strong wind bends the pole much as a horizontal force of 900 N exerted at the top would. How far to the side does the top of the pole flex? 35. As an oil well is drilled, each new section of drill pipe supports its own weight and that of the pipe and drill bit beneath it. Calculate the stretch in a new 6.00 m length of steel pipe that supports 3.00 km of pipe having a mass of 20.0 kg/m and a 100-kg drill bit. The pipe is equivalent in strength to a solid cylinder 5.00 cm in diameter. 36. Calculate the force a piano tuner applies to stretch a steel piano wire 8.00 mm, if the wire is originally 0.850 mm in diameter and 1.35 m long. 37. A vertebra is subjected to a shearing force of 500 N. Find the shear deformation, taking the vertebra to be a cylinder 3.00 cm high and 4.00 cm in diameter. 38. A disk between vertebrae in the spine is subjected to a shearing force of 600 N. Find its shear deformation, taking it to have the shear modulus of 1×109 N / m2 . The disk is equivalent to a solid cylinder 0.700 cm high and 4.00 cm in diameter. 39. When using a pencil eraser, you exert a vertical force of 6.00 N at a distance of 2.00 cm from the hardwood-eraser joint. The pencil is 6.00 mm in diameter and is held at an angle of 20.0º to the horizontal. (a) By how much does the wood flex perpendicular to its length? (b) How much is it compressed lengthwise? 40. To consider the effect of wires hung on poles, we take data from Example 4.8, in which tensions in wires supporting a traffic light were calculated. The left wire made an angle 30.0º below the horizontal with the top of its pole and carried a tension of 108 N. The 12.0 m tall hollow aluminum pole is equivalent in strength to a 4.50 cm diameter solid cylinder. (a) How far is it bent to the side? (b) By how much is it compressed? 41. A farmer making grape juice fills a glass bottle to the brim and caps it tightly. The juice expands more than the glass when it warms up, in such a way that the volume increases by 0.2% (that is, Δ / 0 = 2×10−3 available. Calculate the magnitude of the normal force exerted by the juice per square centimeter if its bulk modulus is 1.8×109 N/m2 , assuming the bottle does not break. In view of your answer, do you think the bottle will survive? ) relative to the space 42. (a) When water freezes, its volume increases by 9.05% (that is, Δ / 0 = 9.05×10−2 ). What force per unit area is water capable of exerting on a container when it freezes? (It is acceptable to use the bulk modulus of water in this problem.) (b) Is it surprising that such forces can fracture engine blocks, boulders, and the like? 43. This problem returns to the tightrope walker studied in Example 4.6, who created a tension of 3.94×103 N in a wire making an angle 5.0º below the horizontal with each supporting pole. Calculate how much this tension stretches the steel wire if it was originally 15 m long and 0.50 cm in diameter. 44. The pole in Figure 5.24 is at a 90.0º bend in a power line and is therefore subjected to more shear force than poles in straight parts of the line. The tension in each line is 4.00×104 N , at the angles shown. The pole is 15.0 m tall, has an 18.0 cm diameter, and can be considered to have half the strength of hardwood. (a) Calculate the compression of the pole. (b) Find how much it bends and in what direction. (c) Find the tension in a guy wire used to keep the pole straight if it is attached to the top of the pole at an angle of 30.0º with the vertical. (Clearly, the guy wire must be in the opposite direction of the bend.) 218 Chapter 5 \| Further Applications of Newton's Laws: Friction, Drag, and Elasticity Figure 5.24 This telephone pole is at a 90º bend in a power line. A guy wire is attached to the top of the pole at an angle of 30º with the vertical. Test Prep for AP® Courses 5.1 Friction 1. When a force of 20 N is applied to a stationary box weighing 40 N, the box does not move. This means the coefficient of static friction is equal to 0.5. is greater than 0.5. is less than 0.5. a. b. c. d. cannot be determined. 2. A 2-kg block slides down a ramp which is at an incline of 25º. If the frictional force is 4.86 N, what is the coefficient of friction? At what incline will the box slide at a constant velocity? Assume g = 10 m/s2. 3. A block is given a short push and then slides with constant friction across a horizontal floor. Which statement best explains the direction of the force that friction applies on the moving block? a. Friction will be in the same direction as the block's motion because molecular interactions between the block and the floor will deform the block in the direction of motion. b. Friction will be in the same direction as the block's motion because thermal energy generated at the interface between the block and the floor adds kinetic energy to the block. c. Friction will be in the opposite direction of the block's motion because molecular interactions between the block and the floor will deform the block in the opposite direction of motion. d. Friction will be in the opposite direction of the block's motion because thermal energy generated at the interface between the block and the floor converts some of the block's kinetic energy to potential energy. 4. A student pushes a cardboard box across a carpeted floor and afterwards notices that the bottom of the box feels warm. Explain how interactions between molecules in the cardboard and molecules in the carpet produced this heat. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 6 \| Gravitation and Uniform Circular Motion 219 6 GRAVITATION AND UNIFORM CIRCULAR MOTION Figure 6.1 This Australian Grand Prix Formula 1 race car moves in a circular path as it makes the turn. Its wheels also spin rapidly—the latter completing many revolutions, the former only part of one (a circular arc). The same physical principles are involved in each. (credit: Richard Munckton) Chapter Outline 6.1. Rotation Angle and Angular Velocity 6.2. Centripetal Acceleration 6.3. Centripetal Force 6.4. Fictitious Forces and Non-inertial Frames: The Coriolis Force 6.5. Newton's Universal Law of Gravitation 6.6. Satellites and Kepler's Laws: An Argument for Simplicity Connection for AP® Courses Many motions, such as the arc of a bird's flight or Earth's path around the Sun, are curved. Recall that Newton's first law tells us that motion is along a straight line at constant speed unless there is a net external force. We will therefore study not only motion along curves, but also the forces that cause it, including gravitational forces. This chapter supports Big Idea 3 that interactions between objects are described by forces, and thus change in motion is a result of a net force exerted on an object. In this chapter, this idea is applied to uniform circular motion. In some ways, this chapter is a continuation of Dynamics: Newton's Laws of Motion as we study more applications of Newton's laws of motion. This chapter deals with the simplest form of curved motion, uniform circu
lar motion, which is motion in a circular path at constant speed. As an object moves on a circular path, the magnitude of its velocity remains constant, but the direction of the velocity is changing. This means there is an acceleration that we will refer to as a “centripetal” acceleration caused by a net external force, also called the “centripetal” force (Enduring Understanding 3.B). The centripetal force is the net force totaling all 220 Chapter 6 \| Gravitation and Uniform Circular Motion external forces acting on the object (Essential Knowledge 3.B.1). In order to determine the net force, a free-body diagram may be useful (Essential Knowledge 3.B.2). Studying this topic illustrates most of the concepts associated with rotational motion and leads to many new topics we group under the name rotation. This motion can be described using kinematics variables (Essential Knowledge 3.A.1), but in addition to linear variables, we will introduce angular variables. We use various ways to describe motion, namely, verbally, algebraically and graphically (Learning Objective 3.A.1.1). Pure rotational motion occurs when points in an object move in circular paths centered on one point. Pure translational motion is motion with no rotation. Some motion combines both types, such as a rotating hockey puck moving over ice. Some combinations of both types of motion are conveniently described with fictitious forces which appear as a result of using a non-inertial frame of reference (Enduring Understanding 3.A). Furthermore, the properties of uniform circular motion can be applied to the motion of massive objects in a gravitational field. Thus, this chapter supports Big Idea 1 that gravitational mass is an important property of an object or a system. We have experimental evidence that gravitational and inertial masses are equal (Enduring Understanding 1.C), and that gravitational mass is a measure of the strength of the gravitational interaction (Essential Knowledge 1.C.2). Therefore, this chapter will support Big Idea 2 that fields existing in space can be used to explain interactions, because any massive object creates a gravitational field in space (Enduring Understanding 2.B). Mathematically, we use Newton's universal law of gravitation to provide a model for the gravitational interaction between two massive objects (Essential Knowledge 2.B.2). We will discover that this model describes the interaction of one object with mass with another object with mass (Essential Knowledge 3.C.1), and also that gravitational force is a long-range force (Enduring Understanding 3.C). The concepts in this chapter support: Big Idea 1 Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring Understanding 1.C Objects and systems have properties of inertial mass and gravitational mass that are experimentally verified to be the same and that satisfy conservation principles. Essential Knowledge 1.C.2 Gravitational mass is the property of an object or a system that determines the strength of the gravitational interaction with other objects, systems, or gravitational fields. Essential Knowledge 1.C.3 Objects and systems have properties of inertial mass and gravitational mass that are experimentally verified to be the same and that satisfy conservation principles. Big Idea 2 Fields existing in space can be used to explain interactions. Enduring Understanding 2.B A gravitational field is caused by an object with mass. Essential Knowledge 2.B.2. The gravitational field caused by a spherically symmetric object with mass is radial and, outside the object, varies as the inverse square of the radial distance from the center of that object. Big Idea 3 The interactions of an object with other objects can be described by forces. Enduring Understanding 3.A All forces share certain common characteristics when considered by observers in inertial reference frames. Essential Knowledge 3.A.1. An observer in a particular reference frame can describe the motion of an object using such quantities as position, displacement, distance, velocity, speed, and acceleration. Essential Knowledge 3.A.3. A force exerted on an object is always due to the interaction of that object with another object. Enduring Understanding 3.B Classically, the acceleration of an object interacting with other objects can be predicted by using = ∑ / . Essential Knowledge 3.B.1 If an object of interest interacts with several other objects, the net force is the vector sum of the individual forces. Essential Knowledge 3.B.2 Free-body diagrams are useful tools for visualizing forces being exerted on a single object and writing the equations that represent a physical situation. Enduring Understanding 3.C At the macroscopic level, forces can be categorized as either long-range (action-at-a-distance) forces or contact forces. Essential Knowledge 3.C.1. Gravitational force describes the interaction of one object that has mass with another object that has mass. 6.1 Rotation Angle and Angular Velocity Learning Objectives By the end of this section, you will be able to: • Define arc length, rotation angle, radius of curvature, and angular velocity. • Calculate the angular velocity of a car wheel spin. In Kinematics, we studied motion along a straight line and introduced such concepts as displacement, velocity, and acceleration. Two-Dimensional Kinematics dealt with motion in two dimensions. Projectile motion is a special case of two-dimensional This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 6 \| Gravitation and Uniform Circular Motion 221 kinematics in which the object is projected into the air, while being subject to the gravitational force, and lands a distance away. In this chapter, we consider situations where the object does not land but moves in a curve. We begin the study of uniform circular motion by defining two angular quantities needed to describe rotational motion. Rotation Angle When objects rotate about some axis—for example, when the CD (compact disc) in Figure 6.2 rotates about its center—each point in the object follows a circular arc. Consider a line from the center of the CD to its edge. Each pit used to record sound along this line moves through the same angle in the same amount of time. The rotation angle is the amount of rotation and is analogous to linear distance. We define the rotation angle Δ to be the ratio of the arc length to the radius of curvature: Δ = Δ . (6.1) Figure 6.2 All points on a CD travel in circular arcs. The pits along a line from the center to the edge all move through the same angle Δ in a time Δ . Figure 6.3 The radius of a circle is rotated through an angle Δ . The arc length Δs is described on the circumference. The arc length Δ is the distance traveled along a circular path as shown in Figure 6.3 Note that is the radius of curvature of the circular path. We know that for one complete revolution, the arc length is the circumference of a circle of radius . The circumference of a circle is 2π . Thus for one complete revolution the rotation angle is Δ = 2π = 2π. This result is the basis for defining the units used to measure rotation angles, Δ to be radians (rad), defined so that A comparison of some useful angles expressed in both degrees and radians is shown in Table 6.1. 2π rad = 1 revolution. (6.2) (6.3) 222 Chapter 6 \| Gravitation and Uniform Circular Motion Table 6.1 Comparison of Angular Units Degree Measures Radian Measure 30º 60º 90º 120º 135º 180º 6 3 2 2π 3 3π 4 Figure 6.4 Points 1 and 2 rotate through the same angle ( Δ ), but point 2 moves through a greater arc length (Δ) because it is at a greater distance from the center of rotation () . If Δ = 2 rad, then the CD has made one complete revolution, and every point on the CD is back at its original position. Because there are 360º in a circle or one revolution, the relationship between radians and degrees is thus so that Angular Velocity 2 rad = 360º 1 rad = 360º 2π ≈ 57.3º. How fast is an object rotating? We define angular velocity as the rate of change of an angle. In symbols, this is = Δ Δ , (6.4) (6.5) (6.6) where an angular rotation Δ takes place in a time Δ . The greater the rotation angle in a given amount of time, the greater the angular velocity. The units for angular velocity are radians per second (rad/s). Angular velocity is analogous to linear velocity . To get the precise relationship between angular and linear velocity, we again consider a pit on the rotating CD. This pit moves an arc length Δ in a time Δ , and so it has a linear velocity = Δ Δ . From Δ = Δ we see that Δ = Δ . Substituting this into the expression for gives = Δ Δ = . (6.7) (6.8) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 6 \| Gravitation and Uniform Circular Motion We write this relationship in two different ways and gain two different insights: = or = . 223 (6.9) states that the linear velocity is proportional to the distance from the center of The first relationship in = or = rotation, thus, it is largest for a point on the rim (largest ), as you might expect. We can also call this linear speed of a point on the rim the tangential speed. The second relationship in = or = moving car. Note that the speed of a point on the rim of the tire is the same as the speed of the car. See Figure 6.5. So the faster the car moves, the faster the tire spins—large means a large , because = . Similarly, a larger-radius tire rotating at the same angular velocity ( ) will produce a greater linear speed ( ) for the car. can be illustrated by considering the tire of a Figure 6.5 A car moving at a velocity to the right has a tire rotating with an angular velocity .The speed of the tread of the tire relative to the axle is , the same as if the car were jacked up. Thus the car moves forward at linear velocity = , where is the tire radius. A larger angular velocity for the tire means a greater velocity for the car. Example 6.
1 How Fast Does a Car Tire Spin? Calculate the angular velocity of a 0.300 m radius car tire when the car travels at 15.0 m/s (about 54 km/h ). See Figure 6.5. Strategy Because the linear speed of the tire rim is the same as the speed of the car, we have = 15.0 m/s. The radius of the tire is given to be = 0.300 m. Knowing and , we can use the second relationship in = , = to calculate the angular velocity. Solution To calculate the angular velocity, we will use the following relationship: Substituting the knowns, Discussion = . = 15.0 m/s 0.300 m = 50.0 rad/s. (6.10) (6.11) When we cancel units in the above calculation, we get 50.0/s. But the angular velocity must have units of rad/s. Because radians are actually unitless (radians are defined as a ratio of distance), we can simply insert them into the answer for the angular velocity. Also note that if an earth mover with much larger tires, say 1.20 m in radius, were moving at the same speed of 15.0 m/s, its tires would rotate more slowly. They would have an angular velocity = (15.0 m/s) / (1.20 m) = 12.5 rad/s. (6.12) 224 Chapter 6 \| Gravitation and Uniform Circular Motion Both and have directions (hence they are angular and linear velocities, respectively). Angular velocity has only two directions with respect to the axis of rotation—it is either clockwise or counterclockwise. Linear velocity is tangent to the path, as illustrated in Figure 6.6. Take-Home Experiment Tie an object to the end of a string and swing it around in a horizontal circle above your head (swing at your wrist). Maintain uniform speed as the object swings and measure the angular velocity of the motion. What is the approximate speed of the object? Identify a point close to your hand and take appropriate measurements to calculate the linear speed at this point. Identify other circular motions and measure their angular velocities. Figure 6.6 As an object moves in a circle, here a fly on the edge of an old-fashioned vinyl record, its instantaneous velocity is always tangent to the circle. The direction of the angular velocity is clockwise in this case. PhET Explorations: Ladybug Revolution Figure 6.7 Ladybug Revolution (http://cnx.org/content/m54992/1.2/rotation_en.jar) Join the ladybug in an exploration of rotational motion. Rotate the merry-go-round to change its angle, or choose a constant angular velocity or angular acceleration. Explore how circular motion relates to the bug's x,y position, velocity, and acceleration using vectors or graphs. 6.2 Centripetal Acceleration Learning Objectives By the end of this section, you will be able to: • Establish the expression for centripetal acceleration. • Explain the centrifuge. We know from kinematics that acceleration is a change in velocity, either in its magnitude or in its direction, or both. In uniform circular motion, the direction of the velocity changes constantly, so there is always an associated acceleration, even though the magnitude of the velocity might be constant. You experience this acceleration yourself when you turn a corner in your car. (If you hold the wheel steady during a turn and move at constant speed, you are in uniform circular motion.) What you notice is a sideways acceleration because you and the car are changing direction. The sharper the curve and the greater your speed, the more noticeable this acceleration will become. In this section we examine the direction and magnitude of that acceleration. Figure 6.8 shows an object moving in a circular path at constant speed. The direction of the instantaneous velocity is shown at two points along the path. Acceleration is in the direction of the change in velocity, which points directly toward the center of rotation (the center of the circular path). This pointing is shown with the vector diagram in the figure. We call the acceleration of This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 6 \| Gravitation and Uniform Circular Motion 225 an object moving in uniform circular motion (resulting from a net external force) the centripetal acceleration( c ); centripetal means “toward the center” or “center seeking.” Figure 6.8 The directions of the velocity of an object at two different points are shown, and the change in velocity Δv is seen to point directly toward the center of curvature. (See small inset.) Because ac = Δv / Δ , the acceleration is also toward the center; a is called centripetal acceleration. (Because Δ is very small, the arc length Δ is equal to the chord length Δ for small time differences.) The direction of centripetal acceleration is toward the center of curvature, but what is its magnitude? Note that the triangle formed by the velocity vectors and the one formed by the radii and Δ are similar. Both the triangles ABC and PQR are isosceles triangles (two equal sides). The two equal sides of the velocity vector triangle are the speeds 1 = 2 = . Using the properties of two similar triangles, we obtain Acceleration is Δ / Δ , and so we first solve this expression for Δ : Δ = Δ . Then we divide this by Δ , yielding Δ = Δ. Δ Δ = × Δ Δ . (6.13) (6.14) (6.15) Finally, noting that Δ / Δ = c and that Δ / Δ = , the linear or tangential speed, we see that the magnitude of the centripetal acceleration is c = 2 , which is the acceleration of an object in a circle of radius at a speed . So, centripetal acceleration is greater at high speeds and in sharp curves (smaller radius), as you have noticed when driving a car. But it is a bit surprising that c is proportional to speed squared, implying, for example, that it is four times as hard to take a curve at 100 km/h than at 50 km/h. A sharp corner has a small radius, so that c is greater for tighter turns, as you have probably noticed. (6.16) It is also useful to express c in terms of angular velocity. Substituting = into the above expression, we find c = ()2 / = 2 . We can express the magnitude of centripetal acceleration using either of two equations: c = 2 ; c = 2. (6.17) Recall that the direction of c is toward the center. You may use whichever expression is more convenient, as illustrated in examples below. 226 Chapter 6 \| Gravitation and Uniform Circular Motion A centrifuge (see Figure 6.9b) is a rotating device used to separate specimens of different densities. High centripetal acceleration significantly decreases the time it takes for separation to occur, and makes separation possible with small samples. Centrifuges are used in a variety of applications in science and medicine, including the separation of single cell suspensions such as bacteria, viruses, and blood cells from a liquid medium and the separation of macromolecules, such as DNA and protein, from a solution. Centrifuges are often rated in terms of their centripetal acceleration relative to acceleration due to gravity () ; maximum centripetal acceleration of several hundred thousand is possible in a vacuum. Human centrifuges, extremely large centrifuges, have been used to test the tolerance of astronauts to the effects of accelerations larger than that of Earth's gravity. Example 6.2 How Does the Centripetal Acceleration of a Car Around a Curve Compare with That Due to Gravity? What is the magnitude of the centripetal acceleration of a car following a curve of radius 500 m at a speed of 25.0 m/s (about 90 km/h)? Compare the acceleration with that due to gravity for this fairly gentle curve taken at highway speed. See Figure 6.9(a). Strategy Because and are given, the first expression in c = 2 ; c = 2 is the most convenient to use. Solution Entering the given values of = 25.0 m/s and = 500 m into the first expression for c gives c = 2 = (25.0 m/s)2 500 m = 1.25 m/s2. (6.18) Discussion To compare this with the acceleration due to gravity ( = 9.80 m/s2) , we take the ratio of c / = seat belt. 9.80 m/s2 1.25 m/s2 / = 0.128 . Thus, c = 0.128 g and is noticeable especially if you were not wearing a This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 6 \| Gravitation and Uniform Circular Motion 227 Figure 6.9 (a) The car following a circular path at constant speed is accelerated perpendicular to its velocity, as shown. The magnitude of this centripetal acceleration is found in Example 6.2. (b) A particle of mass in a centrifuge is rotating at constant angular velocity . It must be accelerated perpendicular to its velocity or it would continue in a straight line. The magnitude of the necessary acceleration is found in Example 6.3. Example 6.3 How Big Is the Centripetal Acceleration in an Ultracentrifuge? Calculate the centripetal acceleration of a point 7.50 cm from the axis of an ultracentrifuge spinning at 7.5 × 104 rev/min. Determine the ratio of this acceleration to that due to gravity. See Figure 6.9(b). Strategy The term rev/min stands for revolutions per minute. By converting this to radians per second, we obtain the angular velocity . Because is given, we can use the second expression in the equation c = 2 c = 2 to calculate the centripetal acceleration. Solution To convert 7.50×104 rev / min to radians per second, we use the facts that one revolution is 2πrad and one minute is 60.0 s. Thus, = 7.50×104 rev min × 2π rad 1 rev × 1 min 60.0 s = 7854 rad/s. Now the centripetal acceleration is given by the second expression in c = 2 ; c = 2 as Converting 7.50 cm to meters and substituting known values gives c = (0.0750 m)(7854 rad/s)2 = 4.63×106 m/s2. c = 2. (6.19) (6.20) (6.21) 228 Chapter 6 \| Gravitation and Uniform Circular Motion Note that the unitless radians are discarded in order to get the correct units for centripetal acceleration. Taking the ratio of c to yields = 4.63×106 c 9.80 = 4.72×105. (6.22) Discussion This last result means that the centripetal acceleration is 472,000 times as strong as . It is no wonder that such high centrifuges are called ultracentrifuges. The extremely large accelerations involved greatly decrease the time needed to cause the sedimentation of blood cells or other m
aterials. Of course, a net external force is needed to cause any acceleration, just as Newton proposed in his second law of motion. So a net external force is needed to cause a centripetal acceleration. In Centripetal Force, we will consider the forces involved in circular motion. PhET Explorations: Ladybug Motion 2D Learn about position, velocity and acceleration vectors. Move the ladybug by setting the position, velocity or acceleration, and see how the vectors change. Choose linear, circular or elliptical motion, and record and playback the motion to analyze the behavior. Figure 6.10 Ladybug Motion 2D (http://cnx.org/content/m54995/1.2/ladybug-motion-2d_en.jar) 6.3 Centripetal Force Learning Objectives By the end of this section, you will be able to: • Calculate coefficient of friction on a car tire. • Calculate ideal speed and angle of a car on a turn. Any force or combination of forces can cause a centripetal or radial acceleration. Just a few examples are the tension in the rope on a tether ball, the force of Earth's gravity on the Moon, friction between roller skates and a rink floor, a banked roadway's force on a car, and forces on the tube of a spinning centrifuge. Any net force causing uniform circular motion is called a centripetal force. The direction of a centripetal force is toward the center of curvature, the same as the direction of centripetal acceleration. According to Newton's second law of motion, net force is mass times acceleration: net F = . For uniform circular motion, the acceleration is the centripetal acceleration— = . Thus, the magnitude of centripetal force Fc is Fc = c. (6.23) By using the expressions for centripetal acceleration from = 2 force Fc in terms of mass, velocity, angular velocity, and radius of curvature: = 2 , we get two expressions for the centripetal = 2. You may use whichever expression for centripetal force is more convenient. Centripetal force c is always perpendicular to the path and pointing to the center of curvature, because a is perpendicular to the velocity and pointing to the center of curvature. = 2 (6.24) Note that if you solve the first expression for , you get This implies that for a given mass and velocity, a large centripetal force causes a small radius of curvature—that is, a tight curve. = 2 . (6.25) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 6 \| Gravitation and Uniform Circular Motion 229 Figure 6.11 The frictional force supplies the centripetal force and is numerically equal to it. Centripetal force is perpendicular to velocity and causes uniform circular motion. The larger the Fc , the smaller the radius of curvature and the sharper the curve. The second curve has the same , but a larger Fc produces a smaller ′ . Example 6.4 What Coefficient of Friction Do Car Tires Need on a Flat Curve? (a) Calculate the centripetal force exerted on a 900 kg car that negotiates a 500 m radius curve at 25.0 m/s. (b) Assuming an unbanked curve, find the minimum static coefficient of friction, between the tires and the road, static friction being the reason that keeps the car from slipping (see Figure 6.12). Strategy and Solution for (a) We know that c = 2 . Thus, Strategy for (b) c = 2 = (900 kg)(25.0 m/s)2 (500 m) = 1125 N. (6.26) Figure 6.12 shows the forces acting on the car on an unbanked (level ground) curve. Friction is to the left, keeping the car from slipping, and because it is the only horizontal force acting on the car, the friction is the centripetal force in this case. We know that the maximum static friction (at which the tires roll but do not slip) is s , where s is the static coefficient of friction and N is the normal force. The normal force equals the car's weight on level ground, so that = . Thus the centripetal force in this situation is c = = s = s. (6.27) Now we have a relationship between centripetal force and the coefficient of friction. Using the first expression for c from the equation c = 2 c = 2 2 = s. (6.28) (6.29) We solve this for s , noting that mass cancels, and obtain 230 Chapter 6 \| Gravitation and Uniform Circular Motion Solution for (b) Substituting the knowns, s = 2 . s = (25.0 m/s)2 (500 m)(9.80 m/s2) = 0.13. (6.30) (6.31) (Because coefficients of friction are approximate, the answer is given to only two digits.) Discussion We could also solve part (a) using the first expression in c = 2 c = 2 because and are given. The coefficient of friction found in part (b) is much smaller than is typically found between tires and roads. The car will still negotiate the curve if the coefficient is greater than 0.13, because static friction is a responsive force, being able to assume a value less than but no more than s . A higher coefficient would also allow the car to negotiate the curve at a higher speed, but if the coefficient of friction is less, the safe speed would be less than 25 m/s. Note that mass cancels, implying that in this example, it does not matter how heavily loaded the car is to negotiate the turn. Mass cancels because friction is assumed proportional to the normal force, which in turn is proportional to mass. If the surface of the road were banked, the normal force would be less as will be discussed below. Figure 6.12 This car on level ground is moving away and turning to the left. The centripetal force causing the car to turn in a circular path is due to friction between the tires and the road. A minimum coefficient of friction is needed, or the car will move in a larger-radius curve and leave the roadway. Let us now consider banked curves, where the slope of the road helps you negotiate the curve. See Figure 6.13. The greater the angle , the faster you can take the curve. Race tracks for bikes as well as cars, for example, often have steeply banked curves. In an “ideally banked curve,” the angle is such that you can negotiate the curve at a certain speed without the aid of friction between the tires and the road. We will derive an expression for for an ideally banked curve and consider an example related to it. For ideal banking, the net external force equals the horizontal centripetal force in the absence of friction. The components of the normal force N in the horizontal and vertical directions must equal the centripetal force and the weight of the car, respectively. In cases in which forces are not parallel, it is most convenient to consider components along perpendicular axes—in this case, the vertical and horizontal directions. Figure 6.13 shows a free body diagram for a car on a frictionless banked curve. If the angle is ideal for the speed and radius, then the net external force will equal the necessary centripetal force. The only two external forces acting on the car are its weight w and the normal force of the road N . (A frictionless surface can only exert a force perpendicular to the surface—that is, a normal force.) These two forces must add to give a net external force that is horizontal toward the center of curvature and has This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 6 \| Gravitation and Uniform Circular Motion 231 magnitude mv2 /r . Because this is the crucial force and it is horizontal, we use a coordinate system with vertical and horizontal axes. Only the normal force has a horizontal component, and so this must equal the centripetal force—that is, sin = 2 . (6.32) Because the car does not leave the surface of the road, the net vertical force must be zero, meaning that the vertical components of the two external forces must be equal in magnitude and opposite in direction. From the figure, we see that the vertical component of the normal force is cos , and the only other vertical force is the car's weight. These must be equal in magnitude; thus, Now we can combine the last two equations to eliminate and get an expression for , as desired. Solving the second equation for = / (cos ) , and substituting this into the first yields cos = . = 2 sin cos tan() = 2 2 tan = Taking the inverse tangent gives = tan−1 2 (ideally banked curve, no friction). (6.33) (6.34) (6.35) (6.36) This expression can be understood by considering how depends on and . A large will be obtained for a large and a small . That is, roads must be steeply banked for high speeds and sharp curves. Friction helps, because it allows you to take the curve at greater or lower speed than if the curve is frictionless. Note that does not depend on the mass of the vehicle. Figure 6.13 The car on this banked curve is moving away and turning to the left. Example 6.5 What Is the Ideal Speed to Take a Steeply Banked Tight Curve? Curves on some test tracks and race courses, such as the Daytona International Speedway in Florida, are very steeply banked. This banking, with the aid of tire friction and very stable car configurations, allows the curves to be taken at very high speed. To illustrate, calculate the speed at which a 100 m radius curve banked at 65.0° should be driven if the road is frictionless. Strategy We first note that all terms in the expression for the ideal angle of a banked curve except for speed are known; thus, we need only rearrange it so that speed appears on the left-hand side and then substitute known quantities. Solution Starting with tan = 2 (6.37) 232 we get Noting that tan 65.0º = 2.14, we obtain = ( tan )1 / 2. = (100 m)(9.80 m/s2)(2.14) Chapter 6 \| Gravitation and Uniform Circular Motion 1 / 2 (6.38) (6.39) Discussion = 45.8 m/s. This is just about 165 km/h, consistent with a very steeply banked and rather sharp curve. Tire friction enables a vehicle to take the curve at significantly higher speeds. Calculations similar to those in the preceding examples can be performed for a host of interesting situations in which centripetal force is involved—a number of these are presented in this chapter's Problems and Exercises. Take-Home Experiment Ask a friend or relative to swing a golf club or a tennis racquet. Take appropriate measurements to estimate
the centripetal acceleration of the end of the club or racquet. You may choose to do this in slow motion. PhET Explorations: Gravity and Orbits Move the sun, earth, moon and space station to see how it affects their gravitational forces and orbital paths. Visualize the sizes and distances between different heavenly bodies, and turn off gravity to see what would happen without it! Figure 6.14 Gravity and Orbits (http://cnx.org/content/m55002/1.2/gravity-and-orbits_en.jar) 6.4 Fictitious Forces and Non-inertial Frames: The Coriolis Force Learning Objectives By the end of this section, you will be able to: • Discuss the inertial frame of reference. • Discuss the non-inertial frame of reference. • Describe the effects of the Coriolis force. What do taking off in a jet airplane, turning a corner in a car, riding a merry-go-round, and the circular motion of a tropical cyclone have in common? Each exhibits fictitious forces—unreal forces that arise from motion and may seem real, because the observer’s frame of reference is accelerating or rotating. When taking off in a jet, most people would agree it feels as if you are being pushed back into the seat as the airplane accelerates down the runway. Yet a physicist would say that you tend to remain stationary while the seat pushes forward on you, and there is no real force backward on you. An even more common experience occurs when you make a tight curve in your car—say, to the right. You feel as if you are thrown (that is, forced) toward the left relative to the car. Again, a physicist would say that you are going in a straight line but the car moves to the right, and there is no real force on you to the left. Recall Newton’s first law. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 6 \| Gravitation and Uniform Circular Motion 233 Figure 6.15 (a) The car driver feels herself forced to the left relative to the car when she makes a right turn. This is a fictitious force arising from the use of the car as a frame of reference. (b) In the Earth’s frame of reference, the driver moves in a straight line, obeying Newton’s first law, and the car moves to the right. There is no real force to the left on the driver relative to Earth. There is a real force to the right on the car to make it turn. We can reconcile these points of view by examining the frames of reference used. Let us concentrate on people in a car. Passengers instinctively use the car as a frame of reference, while a physicist uses Earth. The physicist chooses Earth because it is very nearly an inertial frame of reference—one in which all forces are real (that is, in which all forces have an identifiable physical origin). In such a frame of reference, Newton’s laws of motion take the form given in Dynamics: Newton's Laws of Motion The car is a non-inertial frame of reference because it is accelerated to the side. The force to the left sensed by car passengers is a fictitious force having no physical origin. There is nothing real pushing them left—the car, as well as the driver, is actually accelerating to the right. Let us now take a mental ride on a merry-go-round—specifically, a rapidly rotating playground merry-go-round. You take the merry-go-round to be your frame of reference because you rotate together. In that non-inertial frame, you feel a fictitious force, named centrifugal force (not to be confused with centripetal force), trying to throw you off. You must hang on tightly to counteract the centrifugal force. In Earth’s frame of reference, there is no force trying to throw you off. Rather you must hang on to make yourself go in a circle because otherwise you would go in a straight line, right off the merry-go-round. Figure 6.16 (a) A rider on a merry-go-round feels as if he is being thrown off. This fictitious force is called the centrifugal force—it explains the rider’s motion in the rotating frame of reference. (b) In an inertial frame of reference and according to Newton’s laws, it is his inertia that carries him off and not a real force (the unshaded rider has net = 0 and heads in a straight line). A real force, centripetal , is needed to cause a circular path. This inertial effect, carrying you away from the center of rotation if there is no centripetal force to cause circular motion, is put to good use in centrifuges (see Figure 6.17). A centrifuge spins a sample very rapidly, as mentioned earlier in this chapter. Viewed from the rotating frame of reference, the fictitious centrifugal force throws particles outward, hastening their sedimentation. The greater the angular velocity, the greater the centrifugal force. But what really happens is that the inertia of the particles carries them along a line tangent to the circle while the test tube is forced in a circular path by a centripetal force. 234 Chapter 6 \| Gravitation and Uniform Circular Motion Figure 6.17 Centrifuges use inertia to perform their task. Particles in the fluid sediment come out because their inertia carries them away from the center of rotation. The large angular velocity of the centrifuge quickens the sedimentation. Ultimately, the particles will come into contact with the test tube walls, which will then supply the centripetal force needed to make them move in a circle of constant radius. Let us now consider what happens if something moves in a frame of reference that rotates. For example, what if you slide a ball directly away from the center of the merry-go-round, as shown in Figure 6.18? The ball follows a straight path relative to Earth (assuming negligible friction) and a path curved to the right on the merry-go-round’s surface. A person standing next to the merry-go-round sees the ball moving straight and the merry-go-round rotating underneath it. In the merry-go-round’s frame of reference, we explain the apparent curve to the right by using a fictitious force, called the Coriolis force, that causes the ball to curve to the right. The fictitious Coriolis force can be used by anyone in that frame of reference to explain why objects follow curved paths and allows us to apply Newton’s Laws in non-inertial frames of reference. Figure 6.18 Looking down on the counterclockwise rotation of a merry-go-round, we see that a ball slid straight toward the edge follows a path curved to the right. The person slides the ball toward point B, starting at point A. Both points rotate to the shaded positions (A’ and B’) shown in the time that the ball follows the curved path in the rotating frame and a straight path in Earth’s frame. Up until now, we have considered Earth to be an inertial frame of reference with little or no worry about effects due to its rotation. Yet such effects do exist—in the rotation of weather systems, for example. Most consequences of Earth’s rotation can be qualitatively understood by analogy with the merry-go-round. Viewed from above the North Pole, Earth rotates counterclockwise, as does the merry-go-round in Figure 6.18. As on the merry-go-round, any motion in Earth’s northern hemisphere experiences a This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 6 \| Gravitation and Uniform Circular Motion 235 Coriolis force to the right. Just the opposite occurs in the southern hemisphere; there, the force is to the left. Because Earth’s angular velocity is small, the Coriolis force is usually negligible, but for large-scale motions, such as wind patterns, it has substantial effects. The Coriolis force causes hurricanes in the northern hemisphere to rotate in the counterclockwise direction, while the tropical cyclones (what hurricanes are called below the equator) in the southern hemisphere rotate in the clockwise direction. The terms hurricane, typhoon, and tropical storm are regionally-specific names for tropical cyclones, storm systems characterized by low pressure centers, strong winds, and heavy rains. Figure 6.19 helps show how these rotations take place. Air flows toward any region of low pressure, and tropical cyclones contain particularly low pressures. Thus winds flow toward the center of a tropical cyclone or a low-pressure weather system at the surface. In the northern hemisphere, these inward winds are deflected to the right, as shown in the figure, producing a counterclockwise circulation at the surface for low-pressure zones of any type. Low pressure at the surface is associated with rising air, which also produces cooling and cloud formation, making low-pressure patterns quite visible from space. Conversely, wind circulation around high-pressure zones is clockwise in the northern hemisphere but is less visible because high pressure is associated with sinking air, producing clear skies. The rotation of tropical cyclones and the path of a ball on a merry-go-round can just as well be explained by inertia and the rotation of the system underneath. When non-inertial frames are used, fictitious forces, such as the Coriolis force, must be invented to explain the curved path. There is no identifiable physical source for these fictitious forces. In an inertial frame, inertia explains the path, and no force is found to be without an identifiable source. Either view allows us to describe nature, but a view in an inertial frame is the simplest and truest, in the sense that all forces have real origins and explanations. Figure 6.19 (a) The counterclockwise rotation of this northern hemisphere hurricane is a major consequence of the Coriolis force. (credit: NASA) (b) Without the Coriolis force, air would flow straight into a low-pressure zone, such as that found in tropical cyclones. (c) The Coriolis force deflects the winds to the right, producing a counterclockwise rotation. (d) Wind flowing away from a high-pressure zone is also deflected to the right, producing a clockwise rotation. (e) The opposite direction of rotation is produced by the Coriolis force in the southern hemisphere, leading to tropical cyclones. (credit: NASA) 6.5 Newton's Universal Law
of Gravitation Learning Objectives By the end of this section, you will be able to: • Explain Earth's gravitational force. • Describe the gravitational effect of the Moon on Earth. • Discuss weightlessness in space. • Understand the Cavendish experiment. The information presented in this section supports the following AP® learning objectives and science practices: • 2.B.2.1 The student is able to apply = 2 to calculate the gravitational field due to an object with mass M, where the field is a vector directed toward the center of the object of mass M. (S.P. 2.2) • 2.B.2.2 The student is able to approximate a numerical value of the gravitational field (g) near the surface of an object from its radius and mass relative to those of the Earth or other reference objects. (S.P. 2.2) • 3.A.3.4. The student is able to make claims about the force on an object due to the presence of other objects with the same property: mass, electric charge. (S.P. 6.1, 6.4) 236 Chapter 6 \| Gravitation and Uniform Circular Motion What do aching feet, a falling apple, and the orbit of the Moon have in common? Each is caused by the gravitational force. Our feet are strained by supporting our weight—the force of Earth's gravity on us. An apple falls from a tree because of the same force acting a few meters above Earth's surface. And the Moon orbits Earth because gravity is able to supply the necessary centripetal force at a distance of hundreds of millions of meters. In fact, the same force causes planets to orbit the Sun, stars to orbit the center of the galaxy, and galaxies to cluster together. Gravity is another example of underlying simplicity in nature. It is the weakest of the four basic forces found in nature, and in some ways the least understood. It is a force that acts at a distance, without physical contact, and is expressed by a formula that is valid everywhere in the universe, for masses and distances that vary from the tiny to the immense. Sir Isaac Newton was the first scientist to precisely define the gravitational force, and to show that it could explain both falling bodies and astronomical motions. See Figure 6.20. But Newton was not the first to suspect that the same force caused both our weight and the motion of planets. His forerunner Galileo Galilei had contended that falling bodies and planetary motions had the same cause. Some of Newton's contemporaries, such as Robert Hooke, Christopher Wren, and Edmund Halley, had also made some progress toward understanding gravitation. But Newton was the first to propose an exact mathematical form and to use that form to show that the motion of heavenly bodies should be conic sections—circles, ellipses, parabolas, and hyperbolas. This theoretical prediction was a major triumph—it had been known for some time that moons, planets, and comets follow such paths, but no one had been able to propose a mechanism that caused them to follow these paths and not others. This was one of the earliest examples of a theory derived from empirical evidence doing more than merely describing those empirical results; it made claims about the fundamental workings of the universe. Figure 6.20 According to early accounts, Newton was inspired to make the connection between falling bodies and astronomical motions when he saw an apple fall from a tree and realized that if the gravitational force could extend above the ground to a tree, it might also reach the Sun. The inspiration of Newton's apple is a part of worldwide folklore and may even be based in fact. Great importance is attached to it because Newton's universal law of gravitation and his laws of motion answered very old questions about nature and gave tremendous support to the notion of underlying simplicity and unity in nature. Scientists still expect underlying simplicity to emerge from their ongoing inquiries into nature. The gravitational force is relatively simple. It is always attractive, and it depends only on the masses involved and the distance between them. Stated in modern language, Newton's universal law of gravitation states that every particle in the universe attracts every other particle with a force along a line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 6 \| Gravitation and Uniform Circular Motion 237 Figure 6.21 Gravitational attraction is along a line joining the centers of mass of these two bodies. The magnitude of the force is the same on each, consistent with Newton's third law. Misconception Alert The magnitude of the force on each object (one has larger mass than the other) is the same, consistent with Newton's third law. The bodies we are dealing with tend to be large. To simplify the situation we assume that the body acts as if its entire mass is concentrated at one specific point called the center of mass (CM), which will be further explored in Linear Momentum and Collisions. For two bodies having masses and with a distance between their centers of mass, the equation for Newton's universal law of gravitation is = 2 , (6.40) where is the magnitude of the gravitational force and is a proportionality factor called the gravitational constant. is a universal gravitational constant—that is, it is thought to be the same everywhere in the universe. It has been measured experimentally to be = 6.673×10−11N ⋅ m2 kg2 (6.41) in SI units. Note that the units of are such that a force in newtons is obtained from = 2 , when considering masses in kilograms and distance in meters. For example, two 1.000 kg masses separated by 1.000 m will experience a gravitational attraction of 6.673×10−11 N . This is an extraordinarily small force. The small magnitude of the gravitational force is consistent with everyday experience. We are unaware that even large objects like mountains exert gravitational forces on us. In fact, our body weight is the force of attraction of the entire Earth on us with a mass of 6×1024 kg . The experiment to measure G was first performed by Cavendish, and is explained in more detail later. The fundamental concept it is based on is having a known mass on a spring with a known force (or spring) constant. Then, a second known mass is placed at multiple known distances from the first, and the amount of stretch in the spring resulting from the gravitational attraction of the two masses is measured. Recall that the acceleration due to gravity is about 9.80 m/s2 on Earth. We can now determine why this is so. The weight of an object mg is the gravitational force between it and Earth. Substituting mg for in Newton's universal law of gravitation gives = 2 where is the mass of the object, is the mass of Earth, and is the distance to the center of Earth (the distance between the centers of mass of the object and Earth). See Figure 6.22. The mass of the object cancels, leaving an equation for : , (6.42) 238 Chapter 6 \| Gravitation and Uniform Circular Motion = 2 . Substituting known values for Earth's mass and radius (to three significant figures), 5.98×1024 kg (6.38×106 m)2 6.67×10−11N ⋅ m2 kg2 × = and we obtain a value for the acceleration of a falling body: = 9.80 m/s2. , (6.43) (6.44) (6.45) Figure 6.22 The distance between the centers of mass of Earth and an object on its surface is very nearly the same as the radius of Earth, because Earth is so much larger than the object. This is the expected value and is independent of the body's mass. Newton's law of gravitation takes Galileo's observation that all masses fall with the same acceleration a step further, explaining the observation in terms of a force that causes objects to fall—in fact, in terms of a universally existing force of attraction between masses. Gravitational Mass and Inertial Mass Notice that, in Equation 6.40, the mass of the objects under consideration is directly proportional to the gravitational force. More mass means greater forces, and vice versa. However, we have already seen the concept of mass before in a different context. In Chapter 4, you read that mass is a measure of inertia. However, we normally measure the mass of an object by measuring the force of gravity (F) on it. How do we know that inertial mass is identical to gravitational mass? Assume that we compare the mass of two objects. The objects have inertial masses m1 and m2. If the objects balance each other on a pan balance, we can conclude that they have the same gravitational mass, that is, that they experience the same force due to gravity, F. Using Newton's second law of motion, F = ma, we can write m1 a1 = m2 a2. If we can show that the two objects experience the same acceleration due to gravity, we can conclude that m1 = m2, that is, that the objects' inertial masses are equal. In fact, Galileo and others conducted experiments to show that, when factors such as wind resistance are kept constant, all objects, regardless of their mass, experience the same acceleration due to gravity. Galileo is famously said to have dropped two balls of different masses off the leaning tower of Pisa to demonstrate this. The balls accelerated at the same rate. Since acceleration due to gravity is constant for all objects on Earth, regardless of their mass or composition, i.e., a1 = a2, then m1 = m2. Thus, we can conclude that inertial mass is identical to gravitational mass. This allows us to calculate the acceleration of free fall due to gravity, such as in the orbits of planets. Take-Home Experiment Take a marble, a ball, and a spoon and drop them from the same height. Do they hit the floor at the same time? If you drop a piece of paper as well, does it behave like the other objects? Explain your observations. Making Connections: Gravitation, Other Forces, and General Relativity Attempts are still being made to understand the gravitational force. As we shall see in Particle Physics, modern physics is exploring the conne
ctions of gravity to other forces, space, and time. General relativity alters our view of gravitation, leading us to think of gravitation as bending space and time. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 6 \| Gravitation and Uniform Circular Motion 239 Applying the Science Practices: All Objects Have Gravitational Fields We can use the formula developed above, = 2 , to calculate the gravitational fields of other objects. For example, the Moon has a radius of 1.7 × 106 m and a mass of 7.3 × 1022 kg. The gravitational field on the surface of the Moon can be expressed as = 2 = 6.67×10−11 N·m2 kg2 × = 1.685 m/s2 7.3×1022 kg 2 1.7×106 m This is about 1/6 of the gravity on Earth, which seems reasonable, since the Moon has a much smaller mass than Earth does. A person has a mass of 50 kg. The gravitational field 1.0 m from the person's center of mass can be expressed as = 2 = 6.67×10−11 N·m2 kg2 × 50 kg (1 m)2 = 3.34×10−9 m/s2 This is less than one millionth of the gravitational field at the surface of Earth. In the following example, we make a comparison similar to one made by Newton himself. He noted that if the gravitational force caused the Moon to orbit Earth, then the acceleration due to gravity should equal the centripetal acceleration of the Moon in its orbit. Newton found that the two accelerations agreed “pretty nearly.” Example 6.6 Earth's Gravitational Force Is the Centripetal Force Making the Moon Move in a Curved Path (a) Find the acceleration due to Earth's gravity at the distance of the Moon. (b) Calculate the centripetal acceleration needed to keep the Moon in its orbit (assuming a circular orbit about a fixed Earth), and compare it with the value of the acceleration due to Earth's gravity that you have just found. Strategy for (a) This calculation is the same as the one finding the acceleration due to gravity at Earth's surface, except that is the distance from the center of Earth to the center of the Moon. The radius of the Moon's nearly circular orbit is 3.84×108 m . Solution for (a) Substituting known values into the expression for found above, remembering that is the mass of Earth not the Moon, yields = = 2 6.67×10−11N ⋅ m2 kg2 = 2.70×10−3 m/s.2 Strategy for (b) Centripetal acceleration can be calculated using either form of We choose to use the second form: = 2 = 2 . × 5.98×1024 kg (3.84×108 m)2 (6.46) (6.47) 240 Chapter 6 \| Gravitation and Uniform Circular Motion where is the angular velocity of the Moon about Earth. Solution for (b) = 2, Given that the period (the time it takes to make one complete rotation) of the Moon's orbit is 27.3 days, (d) and using we see that The centripetal acceleration is 1 d×24hr d ×60min hr ×60 s min = 86,400 s = Δ Δ = 2π rad (27.3 d)(86,400 s/d) = 2.66×10−6rad s . = 2 = (3.84×108 m)(2.66×10−6 rad/s)2 = 2.72×10−3 m/s.2 (6.48) (6.49) (6.50) (6.51) The direction of the acceleration is toward the center of the Earth. Discussion The centripetal acceleration of the Moon found in (b) differs by less than 1% from the acceleration due to Earth's gravity found in (a). This agreement is approximate because the Moon's orbit is slightly elliptical, and Earth is not stationary (rather the Earth-Moon system rotates about its center of mass, which is located some 1700 km below Earth's surface). The clear implication is that Earth's gravitational force causes the Moon to orbit Earth. Why does Earth not remain stationary as the Moon orbits it? This is because, as expected from Newton's third law, if Earth exerts a force on the Moon, then the Moon should exert an equal and opposite force on Earth (see Figure 6.23). We do not sense the Moon's effect on Earth's motion, because the Moon's gravity moves our bodies right along with Earth but there are other signs on Earth that clearly show the effect of the Moon's gravitational force as discussed in Satellites and Kepler's Laws: An Argument for Simplicity. Figure 6.23 (a) Earth and the Moon rotate approximately once a month around their common center of mass. (b) Their center of mass orbits the Sun in an elliptical orbit, but Earth's path around the Sun has “wiggles” in it. Similar wiggles in the paths of stars have been observed and are considered direct evidence of planets orbiting those stars. This is important because the planets' reflected light is often too dim to be observed. Tides Ocean tides are one very observable result of the Moon's gravity acting on Earth. Figure 6.24 is a simplified drawing of the Moon's position relative to the tides. Because water easily flows on Earth's surface, a high tide is created on the side of Earth nearest to the Moon, where the Moon's gravitational pull is strongest. Why is there also a high tide on the opposite side of Earth? The answer is that Earth is pulled toward the Moon more than the water on the far side, because Earth is closer to the Moon. So the water on the side of Earth closest to the Moon is pulled away from Earth, and Earth is pulled away from water on the far side. As Earth rotates, the tidal bulge (an effect of the tidal forces between an orbiting natural satellite and the primary planet that it orbits) keeps its orientation with the Moon. Thus there are two tides per day (the actual tidal period is about 12 hours and 25.2 minutes), because the Moon moves in its orbit each day as well). This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 6 \| Gravitation and Uniform Circular Motion 241 Figure 6.24 The Moon causes ocean tides by attracting the water on the near side more than Earth, and by attracting Earth more than the water on the far side. The distances and sizes are not to scale. For this simplified representation of the Earth-Moon system, there are two high and two low tides per day at any location, because Earth rotates under the tidal bulge. The Sun also affects tides, although it has about half the effect of the Moon. However, the largest tides, called spring tides, occur when Earth, the Moon, and the Sun are aligned. The smallest tides, called neap tides, occur when the Sun is at a 90º angle to the Earth-Moon alignment. Figure 6.25 (a, b) Spring tides: The highest tides occur when Earth, the Moon, and the Sun are aligned. (c) Neap tide: The lowest tides occur when the Sun lies at 90º to the Earth-Moon alignment. Note that this figure is not drawn to scale. Tides are not unique to Earth but occur in many astronomical systems. The most extreme tides occur where the gravitational force is the strongest and varies most rapidly, such as near black holes (see Figure 6.26). A few likely candidates for black holes have been observed in our galaxy. These have masses greater than the Sun but have diameters only a few kilometers across. The tidal forces near them are so great that they can actually tear matter from a companion star. 242 Chapter 6 \| Gravitation and Uniform Circular Motion Figure 6.26 A black hole is an object with such strong gravity that not even light can escape it. This black hole was created by the supernova of one star in a two-star system. The tidal forces created by the black hole are so great that it tears matter from the companion star. This matter is compressed and heated as it is sucked into the black hole, creating light and X-rays observable from Earth. ”Weightlessness” and Microgravity In contrast to the tremendous gravitational force near black holes is the apparent gravitational field experienced by astronauts orbiting Earth. What is the effect of “weightlessness” upon an astronaut who is in orbit for months? Or what about the effect of weightlessness upon plant growth? Weightlessness doesn't mean that an astronaut is not being acted upon by the gravitational force. There is no “zero gravity” in an astronaut's orbit. The term just means that the astronaut is in free-fall, accelerating with the acceleration due to gravity. If an elevator cable breaks, the passengers inside will be in free fall and will experience weightlessness. You can experience short periods of weightlessness in some rides in amusement parks. Figure 6.27 Astronauts experiencing weightlessness on board the International Space Station. (credit: NASA) Microgravity refers to an environment in which the apparent net acceleration of a body is small compared with that produced by Earth at its surface. Many interesting biology and physics topics have been studied over the past three decades in the presence of microgravity. Of immediate concern is the effect on astronauts of extended times in outer space, such as at the International Space Station. Researchers have observed that muscles will atrophy (waste away) in this environment. There is also a corresponding loss of bone mass. Study continues on cardiovascular adaptation to space flight. On Earth, blood pressure is usually higher in the feet than in the head, because the higher column of blood exerts a downward force on it, due to gravity. When standing, 70% of your blood is below the level of the heart, while in a horizontal position, just the opposite occurs. What difference does the absence of this pressure differential have upon the heart? Some findings in human physiology in space can be clinically important to the management of diseases back on Earth. On a somewhat negative note, spaceflight is known to affect the human immune system, possibly making the crew members more vulnerable to infectious diseases. Experiments flown in space also have shown that some bacteria grow faster in microgravity than they do on Earth. However, on a positive note, studies indicate that microbial antibiotic production can increase by a factor of two in space-grown cultures. One hopes to be able to understand these mechanisms so that similar successes can be achieved on the ground. In another area of physics space research, inorganic crystals and protein crystals have been grown in outer space that have much higher quality than any grown on Earth, so crystallography studies on t
heir structure can yield much better results. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 6 \| Gravitation and Uniform Circular Motion 243 Plants have evolved with the stimulus of gravity and with gravity sensors. Roots grow downward and shoots grow upward. Plants might be able to provide a life support system for long duration space missions by regenerating the atmosphere, purifying water, and producing food. Some studies have indicated that plant growth and development are not affected by gravity, but there is still uncertainty about structural changes in plants grown in a microgravity environment. The Cavendish Experiment: Then and Now As previously noted, the universal gravitational constant is determined experimentally. This definition was first done accurately by Henry Cavendish (1731–1810), an English scientist, in 1798, more than 100 years after Newton published his universal law of gravitation. The measurement of is very basic and important because it determines the strength of one of the four forces in nature. Cavendish's experiment was very difficult because he measured the tiny gravitational attraction between two ordinary-sized masses (tens of kilograms at most), using apparatus like that in Figure 6.28. Remarkably, his value for differs by less than 1% from the best modern value. One important consequence of knowing was that an accurate value for Earth's mass could finally be obtained. This was done by measuring the acceleration due to gravity as accurately as possible and then calculating the mass of Earth from the relationship Newton's universal law of gravitation gives = 2 where is the mass of the object, is the mass of Earth, and is the distance to the center of Earth (the distance between the centers of mass of the object and Earth). See Figure 6.21. The mass of the object cancels, leaving an equation for : , (6.52) Rearranging to solve for yields = 2 . = 2 . (6.53) (6.54) So can be calculated because all quantities on the right, including the radius of Earth , are known from direct measurements. We shall see in Satellites and Kepler's Laws: An Argument for Simplicity that knowing also allows for the determination of astronomical masses. Interestingly, of all the fundamental constants in physics, is by far the least well determined. The Cavendish experiment is also used to explore other aspects of gravity. One of the most interesting questions is whether the gravitational force depends on substance as well as mass—for example, whether one kilogram of lead exerts the same gravitational pull as one kilogram of water. A Hungarian scientist named Roland von Eötvös pioneered this inquiry early in the 20th century. He found, with an accuracy of five parts per billion, that the gravitational force does not depend on the substance. Such experiments continue today, and have improved upon Eötvös' measurements. Cavendish-type experiments such as those of Eric Adelberger and others at the University of Washington, have also put severe limits on the possibility of a fifth force and have verified a major prediction of general relativity—that gravitational energy contributes to rest mass. Ongoing measurements there use a torsion balance and a parallel plate (not spheres, as Cavendish used) to examine how Newton's law of gravitation works over sub-millimeter distances. On this small-scale, do gravitational effects depart from the inverse square law? So far, no deviation has been observed. Figure 6.28 Cavendish used an apparatus like this to measure the gravitational attraction between the two suspended spheres ( ) and the two on the stand ( ) by observing the amount of torsion (twisting) created in the fiber. Distance between the masses can be varied to check the dependence of the force on distance. Modern experiments of this type continue to explore gravity. Chapter 6 \| Gravitation and Uniform Circular Motion 249 Figure 6.31(b) represents the modern or Copernican model. In this model, a small set of rules and a single underlying force explain not only all motions in the solar system, but all other situations involving gravity. The breadth and simplicity of the laws of physics are compelling. As our knowledge of nature has grown, the basic simplicity of its laws has become ever more evident. Figure 6.31 (a) The Ptolemaic model of the universe has Earth at the center with the Moon, the planets, the Sun, and the stars revolving about it in complex superpositions of circular paths. This geocentric model, which can be made progressively more accurate by adding more circles, is purely descriptive, containing no hints as to what are the causes of these motions. (b) The Copernican model has the Sun at the center of the solar system. It is fully explained by a small number of laws of physics, including Newton's universal law of gravitation. Glossary angular velocity: , the rate of change of the angle with which an object moves on a circular path arc length: Δ , the distance traveled by an object along a circular path banked curve: the curve in a road that is sloping in a manner that helps a vehicle negotiate the curve center of mass: the point where the entire mass of an object can be thought to be concentrated centrifugal force: a fictitious force that tends to throw an object off when the object is rotating in a non-inertial frame of reference centripetal acceleration: the acceleration of an object moving in a circle, directed toward the center centripetal force: any net force causing uniform circular motion Coriolis force: reference the fictitious force causing the apparent deflection of moving objects when viewed in a rotating frame of fictitious force: a force having no physical origin gravitational constant, G: a proportionality factor used in the equation for Newton's universal law of gravitation; it is a universal constant—that is, it is thought to be the same everywhere in the universe ideal angle: the angle at which a car can turn safely on a steep curve, which is in proportion to the ideal speed ideal banking: the sloping of a curve in a road, where the angle of the slope allows the vehicle to negotiate the curve at a certain speed without the aid of friction between the tires and the road; the net external force on the vehicle equals the horizontal centripetal force in the absence of friction ideal speed: the maximum safe speed at which a vehicle can turn on a curve without the aid of friction between the tire and the road microgravity: an environment in which the apparent net acceleration of a body is small compared with that produced by Earth at its surface Newton's universal law of gravitation: every particle in the universe attracts every other particle with a force along a line joining them; the force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them non-inertial frame of reference: an accelerated frame of reference pit: a tiny indentation on the spiral track moulded into the top of the polycarbonate layer of CD 250 Chapter 6 \| Gravitation and Uniform Circular Motion radians: a unit of angle measurement radius of curvature: radius of a circular path the ratio of the arc length to the radius of curvature on a circular path: rotation angle: Δ = Δ ultracentrifuge: a centrifuge optimized for spinning a rotor at very high speeds uniform circular motion: the motion of an object in a circular path at constant speed Section Summary 6.1 Rotation Angle and Angular Velocity • Uniform circular motion is motion in a circle at constant speed. The rotation angle Δ is defined as the ratio of the arc length to the radius of curvature: Δ = Δ , where arc length Δ is distance traveled along a circular path and is the radius of curvature of the circular path. The quantity Δ is measured in units of radians (rad), for which • The conversion between radians and degrees is 1 rad = 57.3º . • Angular velocity is the rate of change of an angle, 2π rad = 360º= 1 revolution. = Δ Δ , where a rotation Δ takes place in a time Δ . The units of angular velocity are radians per second (rad/s). Linear velocity and angular velocity are related by = or = . 6.2 Centripetal Acceleration • Centripetal acceleration c is the acceleration experienced while in uniform circular motion. It always points toward the center of rotation. It is perpendicular to the linear velocity and has the magnitude • The unit of centripetal acceleration is m / s2 . 6.3 Centripetal Force c = 2 c = 2. • Centripetal force Fc is any force causing uniform circular motion. It is a “center-seeking” force that always points toward the center of rotation. It is perpendicular to linear velocity and has magnitude which can also be expressed as c = c, c = 2 or c = 2 6.4 Fictitious Forces and Non-inertial Frames: The Coriolis Force • Rotating and accelerated frames of reference are non-inertial. • Fictitious forces, such as the Coriolis force, are needed to explain motion in such frames. 6.5 Newton's Universal Law of Gravitation • Newton's universal law of gravitation: Every particle in the universe attracts every other particle with a force along a line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. In equation form, this is = 2 , This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 6 \| Gravitation and Uniform Circular Motion 251 where F is the magnitude of the gravitational force. is the gravitational constant, given by = 6.673×10–11 N ⋅ m2/kg2 . • Newton's law of gravitation applies universally. 6.6 Satellites and Kepler's Laws: An Argument for Simplicity • Kepler's laws are stated for a small mass orbiting a larger mass in near-isolation. Kepler's laws of planetary motion are then as follows: Kepler's first law The orbit of each planet about the Sun is an ellipse with the Sun at one focus. Kepler's second law Each planet moves
so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times. Kepler's third law The ratio of the squares of the periods of any two planets about the Sun is equal to the ratio of the cubes of their average distances from the Sun: where is the period (time for one orbit) and is the average radius of the orbit. • The period and radius of a satellite's orbit about a larger body are related by 2 1 2 2 = 3 1 3 2 , or Conceptual Questions 6.1 Rotation Angle and Angular Velocity 2 = 4π2 3 3 2 = 4π2. 1. There is an analogy between rotational and linear physical quantities. What rotational quantities are analogous to distance and velocity? 6.2 Centripetal Acceleration 2. Can centripetal acceleration change the speed of circular motion? Explain. 6.3 Centripetal Force 3. If you wish to reduce the stress (which is related to centripetal force) on high-speed tires, would you use large- or smalldiameter tires? Explain. 4. Define centripetal force. Can any type of force (for example, tension, gravitational force, friction, and so on) be a centripetal force? Can any combination of forces be a centripetal force? 5. If centripetal force is directed toward the center, why do you feel that you are ‘thrown' away from the center as a car goes around a curve? Explain. 6. Race car drivers routinely cut corners as shown in Figure 6.32. Explain how this allows the curve to be taken at the greatest speed. 252 Chapter 6 \| Gravitation and Uniform Circular Motion Figure 6.32 Two paths around a race track curve are shown. Race car drivers will take the inside path (called cutting the corner) whenever possible because it allows them to take the curve at the highest speed. 7. A number of amusement parks have rides that make vertical loops like the one shown in Figure 6.33. For safety, the cars are attached to the rails in such a way that they cannot fall off. If the car goes over the top at just the right speed, gravity alone will supply the centripetal force. What other force acts and what is its direction if: (a) The car goes over the top at faster than this speed? (b)The car goes over the top at slower than this speed? Figure 6.33 Amusement rides with a vertical loop are an example of a form of curved motion. 8. What is the direction of the force exerted by the car on the passenger as the car goes over the top of the amusement ride pictured in Figure 6.33 under the following circumstances: (a) The car goes over the top at such a speed that the gravitational force is the only force acting? (b) The car goes over the top faster than this speed? (c) The car goes over the top slower than this speed? 9. As a skater forms a circle, what force is responsible for making her turn? Use a free body diagram in your answer. 10. Suppose a child is riding on a merry-go-round at a distance about halfway between its center and edge. She has a lunch box resting on wax paper, so that there is very little friction between it and the merry-go-round. Which path shown in Figure 6.34 will the lunch box take when she lets go? The lunch box leaves a trail in the dust on the merry-go-round. Is that trail straight, curved to the left, or curved to the right? Explain your answer. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 6 \| Gravitation and Uniform Circular Motion 253 Figure 6.34 A child riding on a merry-go-round releases her lunch box at point P. This is a view from above the clockwise rotation. Assuming it slides with negligible friction, will it follow path A, B, or C, as viewed from Earth's frame of reference? What will be the shape of the path it leaves in the dust on the merry-go-round? 11. Do you feel yourself thrown to either side when you negotiate a curve that is ideally banked for your car's speed? What is the direction of the force exerted on you by the car seat? 12. Suppose a mass is moving in a circular path on a frictionless table as shown in figure. In the Earth's frame of reference, there is no centrifugal force pulling the mass away from the centre of rotation, yet there is a very real force stretching the string attaching the mass to the nail. Using concepts related to centripetal force and Newton's third law, explain what force stretches the string, identifying its physical origin. Figure 6.35 A mass attached to a nail on a frictionless table moves in a circular path. The force stretching the string is real and not fictional. What is the physical origin of the force on the string? 6.4 Fictitious Forces and Non-inertial Frames: The Coriolis Force 13. When a toilet is flushed or a sink is drained, the water (and other material) begins to rotate about the drain on the way down. Assuming no initial rotation and a flow initially directly straight toward the drain, explain what causes the rotation and which direction it has in the northern hemisphere. (Note that this is a small effect and in most toilets the rotation is caused by directional water jets.) Would the direction of rotation reverse if water were forced up the drain? 14. Is there a real force that throws water from clothes during the spin cycle of a washing machine? Explain how the water is removed. 15. In one amusement park ride, riders enter a large vertical barrel and stand against the wall on its horizontal floor. The barrel is spun up and the floor drops away. Riders feel as if they are pinned to the wall by a force something like the gravitational force. This is a fictitious force sensed and used by the riders to explain events in the rotating frame of reference of the barrel. Explain in an inertial frame of reference (Earth is nearly one) what pins the riders to the wall, and identify all of the real forces acting on them. 254 Chapter 6 \| Gravitation and Uniform Circular Motion 16. Action at a distance, such as is the case for gravity, was once thought to be illogical and therefore untrue. What is the ultimate determinant of the truth in physics, and why was this action ultimately accepted? 17. Two friends are having a conversation. Anna says a satellite in orbit is in freefall because the satellite keeps falling toward Earth. Tom says a satellite in orbit is not in freefall because the acceleration due to gravity is not 9.80 m/s2 . Who do you agree with and why? 18. A non-rotating frame of reference placed at the center of the Sun is very nearly an inertial one. Why is it not exactly an inertial frame? 6.5 Newton's Universal Law of Gravitation 19. Action at a distance, such as is the case for gravity, was once thought to be illogical and therefore untrue. What is the ultimate determinant of the truth in physics, and why was this action ultimately accepted? 20. Two friends are having a conversation. Anna says a satellite in orbit is in freefall because the satellite keeps falling toward Earth. Tom says a satellite in orbit is not in freefall because the acceleration due to gravity is not 9.80 m/s2 . Who do you agree with and why? 21. Draw a free body diagram for a satellite in an elliptical orbit showing why its speed increases as it approaches its parent body and decreases as it moves away. 22. Newton's laws of motion and gravity were among the first to convincingly demonstrate the underlying simplicity and unity in nature. Many other examples have since been discovered, and we now expect to find such underlying order in complex situations. Is there proof that such order will always be found in new explorations? 6.6 Satellites and Kepler's Laws: An Argument for Simplicity 23. In what frame(s) of reference are Kepler's laws valid? Are Kepler's laws purely descriptive, or do they contain causal information? This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 6 \| Gravitation and Uniform Circular Motion 255 Problems & Exercises 6.1 Rotation Angle and Angular Velocity 1. Semi-trailer trucks have an odometer on one hub of a trailer wheel. The hub is weighted so that it does not rotate, but it contains gears to count the number of wheel revolutions—it then calculates the distance traveled. If the wheel has a 1.15 m diameter and goes through 200,000 rotations, how many kilometers should the odometer read? 2. Microwave ovens rotate at a rate of about 6 rev/min. What is this in revolutions per second? What is the angular velocity in radians per second? 3. An automobile with 0.260 m radius tires travels 80,000 km before wearing them out. How many revolutions do the tires make, neglecting any backing up and any change in radius due to wear? 4. (a) What is the period of rotation of Earth in seconds? (b) What is the angular velocity of Earth? (c) Given that Earth has a radius of 6.4×106 m at its equator, what is the linear velocity at Earth's surface? 5. A baseball pitcher brings his arm forward during a pitch, rotating the forearm about the elbow. If the velocity of the ball in the pitcher's hand is 35.0 m/s and the ball is 0.300 m from the elbow joint, what is the angular velocity of the forearm? 6. In lacrosse, a ball is thrown from a net on the end of a stick by rotating the stick and forearm about the elbow. If the angular velocity of the ball about the elbow joint is 30.0 rad/s and the ball is 1.30 m from the elbow joint, what is the velocity of the ball? 7. A truck with 0.420-m-radius tires travels at 32.0 m/s. What is the angular velocity of the rotating tires in radians per second? What is this in rev/min? 8. Integrated Concepts When kicking a football, the kicker rotates his leg about the hip joint. (a) If the velocity of the tip of the kicker's shoe is 35.0 m/s and the hip joint is 1.05 m from the tip of the shoe, what is the shoe tip's angular velocity? (b) The shoe is in contact with the initially stationary 0.500 kg football for 20.0 ms. What average force is exerted on the football to give it a velocity of 20.0 m/s? (c) Find the maximum range of the football, neglecting air resistance. 9. Construct Your Own Problem Consider an amusement park ride in which participants are rotat
ed about a vertical axis in a cylinder with vertical walls. Once the angular velocity reaches its full value, the floor drops away and friction between the walls and the riders prevents them from sliding down. Construct a problem in which you calculate the necessary angular velocity that assures the riders will not slide down the wall. Include a free body diagram of a single rider. Among the variables to consider are the radius of the cylinder and the coefficients of friction between the riders' clothing and the wall. 6.2 Centripetal Acceleration 10. A fairground ride spins its occupants inside a flying saucer-shaped container. If the horizontal circular path the riders follow has an 8.00 m radius, at how many revolutions per minute will the riders be subjected to a centripetal acceleration whose magnitude is 1.50 times that due to gravity? 11. A runner taking part in the 200 m dash must run around the end of a track that has a circular arc with a radius of curvature of 30 m. If he completes the 200 m dash in 23.2 s and runs at constant speed throughout the race, what is the magnitude of his centripetal acceleration as he runs the curved portion of the track? 12. Taking the age of Earth to be about 4×109 years and assuming its orbital radius of 1.5 ×1011 has not changed and is circular, calculate the approximate total distance Earth has traveled since its birth (in a frame of reference stationary with respect to the Sun). 13. The propeller of a World War II fighter plane is 2.30 m in diameter. (a) What is its angular velocity in radians per second if it spins at 1200 rev/min? (b) What is the linear speed of its tip at this angular velocity if the plane is stationary on the tarmac? (c) What is the centripetal acceleration of the propeller tip under these conditions? Calculate it in meters per second squared and convert to multiples of . 14. An ordinary workshop grindstone has a radius of 7.50 cm and rotates at 6500 rev/min. (a) Calculate the magnitude of the centripetal acceleration at its edge in meters per second squared and convert it to multiples of . (b) What is the linear speed of a point on its edge? 15. Helicopter blades withstand tremendous stresses. In addition to supporting the weight of a helicopter, they are spun at rapid rates and experience large centripetal accelerations, especially at the tip. (a) Calculate the magnitude of the centripetal acceleration at the tip of a 4.00 m long helicopter blade that rotates at 300 rev/min. (b) Compare the linear speed of the tip with the speed of sound (taken to be 340 m/s). 16. Olympic ice skaters are able to spin at about 5 rev/s. (a) What is their angular velocity in radians per second? (b) What is the centripetal acceleration of the skater's nose if it is 0.120 m from the axis of rotation? (c) An exceptional skater named Dick Button was able to spin much faster in the 1950s than anyone since—at about 9 rev/ s. What was the centripetal acceleration of the tip of his nose, assuming it is at 0.120 m radius? (d) Comment on the magnitudes of the accelerations found. It is reputed that Button ruptured small blood vessels during his spins. 17. What percentage of the acceleration at Earth's surface is the acceleration due to gravity at the position of a satellite located 300 km above Earth? 18. Verify that the linear speed of an ultracentrifuge is about 0.50 km/s, and Earth in its orbit is about 30 km/s by calculating: (a) The linear speed of a point on an ultracentrifuge 0.100 m from its center, rotating at 50,000 rev/min. (b) The linear speed of Earth in its orbit about the Sun (use data from the text on the radius of Earth's orbit and approximate it as being circular). 256 Chapter 6 \| Gravitation and Uniform Circular Motion 19. A rotating space station is said to create “artificial gravity”—a loosely-defined term used for an acceleration that would be crudely similar to gravity. The outer wall of the rotating space station would become a floor for the astronauts, and centripetal acceleration supplied by the floor would allow astronauts to exercise and maintain muscle and bone strength more naturally than in non-rotating space environments. If the space station is 200 m in diameter, what angular velocity would produce an “artificial gravity” of 9.80 m/s2 at the rim? 20. At takeoff, a commercial jet has a 60.0 m/s speed. Its tires have a diameter of 0.850 m. (a) At how many rev/min are the tires rotating? (b) What is the centripetal acceleration at the edge of the tire? (c) With what force must a determined 1.00×10−15 kg bacterium cling to the rim? (d) Take the ratio of this force to the bacterium's weight. 21. Integrated Concepts Riders in an amusement park ride shaped like a Viking ship hung from a large pivot are rotated back and forth like a rigid pendulum. Sometime near the middle of the ride, the ship is momentarily motionless at the top of its circular arc. The ship then swings down under the influence of gravity. (a) Assuming negligible friction, find the speed of the riders at the bottom of its arc, given the system's center of mass travels in an arc having a radius of 14.0 m and the riders are near the center of mass. (b) What is the centripetal acceleration at the bottom of the arc? (c) Draw a free body diagram of the forces acting on a rider at the bottom of the arc. (d) Find the force exerted by the ride on a 60.0 kg rider and compare it to her weight. (e) Discuss whether the answer seems reasonable. 22. Unreasonable Results A mother pushes her child on a swing so that his speed is 9.00 m/s at the lowest point of his path. The swing is suspended 2.00 m above the child's center of mass. (a) What is the magnitude of the centripetal acceleration of the child at the low point? (b) What is the magnitude of the force the child exerts on the seat if his mass is 18.0 kg? (c) What is unreasonable about these results? (d) Which premises are unreasonable or inconsistent? 6.3 Centripetal Force 23. (a) A 22.0 kg child is riding a playground merry-go-round that is rotating at 40.0 rev/min. What centripetal force must she exert to stay on if she is 1.25 m from its center? (b) What centripetal force does she need to stay on an amusement park merry-go-round that rotates at 3.00 rev/min if she is 8.00 m from its center? (c) Compare each force with her weight. 24. Calculate the centripetal force on the end of a 100 m (radius) wind turbine blade that is rotating at 0.5 rev/s. Assume the mass is 4 kg. This content is available for free at http://cnx.org/content/col11844/1.13 25. What is the ideal banking angle for a gentle turn of 1.20 km radius on a highway with a 105 km/h speed limit (about 65 mi/h), assuming everyone travels at the limit? 26. What is the ideal speed to take a 100 m radius curve banked at a 20.0° angle? 27. (a) What is the radius of a bobsled turn banked at 75.0° and taken at 30.0 m/s, assuming it is ideally banked? (b) Calculate the centripetal acceleration. (c) Does this acceleration seem large to you? 28. Part of riding a bicycle involves leaning at the correct angle when making a turn, as seen in Figure 6.36. To be stable, the force exerted by the ground must be on a line going through the center of gravity. The force on the bicycle wheel can be resolved into two perpendicular components—friction parallel to the road (this must supply the centripetal force), and the vertical normal force (which must equal the system's weight). (a) Show that (as defined in the figure) is related to the speed and radius of curvature of the turn in the same way as for an ideally banked roadway—that is, = tan–1 2/ (b) Calculate for a 12.0 m/s turn of radius 30.0 m (as in a race). Figure 6.36 A bicyclist negotiating a turn on level ground must lean at the correct angle—the ability to do this becomes instinctive. The force of the ground on the wheel needs to be on a line through the center of gravity. The net external force on the system is the centripetal force. The vertical component of the force on the wheel cancels the weight of the system while its horizontal component must supply the centripetal force. This process produces a relationship among the angle , the speed , and the radius of curvature of the turn similar to that for the ideal banking of roadways. 29. A large centrifuge, like the one shown in Figure 6.37(a), is used to expose aspiring astronauts to accelerations similar to those experienced in rocket launches and atmospheric reentries. (a) At what angular velocity is the centripetal acceleration 10 if the rider is 15.0 m from the center of rotation? Chapter 6 \| Gravitation and Uniform Circular Motion 257 (b) The rider's cage hangs on a pivot at the end of the arm, allowing it to swing outward during rotation as shown in Figure 6.37(b). At what angle below the horizontal will the cage hang when the centripetal acceleration is 10 ? (Hint: The arm supplies centripetal force and supports the weight of the cage. Draw a free body diagram of the forces to see what the angle should be.) Figure 6.38 Teardrop-shaped loops are used in the latest roller coasters so that the radius of curvature gradually decreases to a minimum at the top. This means that the centripetal acceleration builds from zero to a maximum at the top and gradually decreases again. A circular loop would cause a jolting change in acceleration at entry, a disadvantage discovered long ago in railroad curve design. With a small radius of curvature at the top, the centripetal acceleration can more easily be kept greater than so that the passengers do not lose contact with their seats nor do they need seat belts to keep them in place. 32. Unreasonable Results (a) Calculate the minimum coefficient of friction needed for a car to negotiate an unbanked 50.0 m radius curve at 30.0 m/ s. (b) What is unreasonable about the result? (c) Which premises are unreasonable or inconsistent? 6.5 Newton's Universal Law of Gravitation 33. (a) Calculate Earth's mass given the acceleration due to gravity at the North P
ole is 9.830 m/s2 and the radius of the Earth is 6371 km from pole to pole. (b) Compare this with the accepted value of 5.979×1024 kg . 34. (a) Calculate the magnitude of the acceleration due to gravity on the surface of Earth due to the Moon. (b) Calculate the magnitude of the acceleration due to gravity at Earth due to the Sun. (c) Take the ratio of the Moon's acceleration to the Sun's and comment on why the tides are predominantly due to the Moon in spite of this number. 35. (a) What is the acceleration due to gravity on the surface of the Moon? (b) On the surface of Mars? The mass of Mars is 6.418×1023 kg and its radius is 3.38×106 m . 36. (a) Calculate the acceleration due to gravity on the surface of the Sun. (b) By what factor would your weight increase if you could stand on the Sun? (Never mind that you cannot.) Figure 6.37 (a) NASA centrifuge used to subject trainees to accelerations similar to those experienced in rocket launches and reentries. (credit: NASA) (b) Rider in cage showing how the cage pivots outward during rotation. This allows the total force exerted on the rider by the cage to be along its axis at all times. 30. Integrated Concepts If a car takes a banked curve at less than the ideal speed, friction is needed to keep it from sliding toward the inside of the curve (a real problem on icy mountain roads). (a) Calculate the ideal speed to take a 100 m radius curve banked at 15.0º. (b) What is the minimum coefficient of friction needed for a frightened driver to take the same curve at 20.0 km/h? 31. Modern roller coasters have vertical loops like the one shown in Figure 6.38. The radius of curvature is smaller at the top than on the sides so that the downward centripetal acceleration at the top will be greater than the acceleration due to gravity, keeping the passengers pressed firmly into their seats. What is the speed of the roller coaster at the top of the loop if the radius of curvature there is 15.0 m and the downward acceleration of the car is 1.50 g? 258 Chapter 6 \| Gravitation and Uniform Circular Motion 37. The Moon and Earth rotate about their common center of mass, which is located about 4700 km from the center of Earth. (This is 1690 km below the surface.) (a) Calculate the magnitude of the acceleration due to the Moon's gravity at that point. (b) Calculate the magnitude of the centripetal acceleration of the center of Earth as it rotates about that point once each lunar month (about 27.3 d) and compare it with the acceleration found in part (a). Comment on whether or not they are equal and why they should or should not be. 38. Solve part (b) of Example 6.6 using = 2 / . 39. Astrology, that unlikely and vague pseudoscience, makes much of the position of the planets at the moment of one's birth. The only known force a planet exerts on Earth is gravitational. (a) Calculate the magnitude of the gravitational force exerted on a 4.20 kg baby by a 100 kg father 0.200 m away at birth (he is assisting, so he is close to the child). (b) Calculate the magnitude of the force on the baby due to Jupiter if it is at its closest distance to Earth, some 6.29×1011 m away. How does the force of Jupiter on the baby compare to the force of the father on the baby? Other objects in the room and the hospital building also exert similar gravitational forces. (Of course, there could be an unknown force acting, but scientists first need to be convinced that there is even an effect, much less that an unknown force causes it.) 40. The existence of the dwarf planet Pluto was proposed based on irregularities in Neptune's orbit. Pluto was subsequently discovered near its predicted position. But it now appears that the discovery was fortuitous, because Pluto is small and the irregularities in Neptune's orbit were not well known. To illustrate that Pluto has a minor effect on the orbit of Neptune compared with the closest planet to Neptune: (a) Calculate the acceleration due to gravity at Neptune due to Pluto when they are 4.50×1012 m apart, as they are at present. The mass of Pluto is 1.4×1022 kg . (b) Calculate the acceleration due to gravity at Neptune due to Uranus, presently about 2.50×1012 m apart, and compare it with that due to Pluto. The mass of Uranus is 8.62×1025 kg . 41. (a) The Sun orbits the Milky Way galaxy once each 2.60 x 108 y , with a roughly circular orbit averaging 3.00 x 104 light years in radius. (A light year is the distance traveled by light in 1 y.) Calculate the centripetal acceleration of the Sun in its galactic orbit. Does your result support the contention that a nearly inertial frame of reference can be located at the Sun? (b) Calculate the average speed of the Sun in its galactic orbit. Does the answer surprise you? 42. Unreasonable Result A mountain 10.0 km from a person exerts a gravitational force on him equal to 2.00% of his weight. (a) Calculate the mass of the mountain. (b) Compare the mountain's mass with that of Earth. (c) What is unreasonable about these results? This content is available for free at http://cnx.org/content/col11844/1.13 (d) Which premises are unreasonable or inconsistent? (Note that accurate gravitational measurements can easily detect the effect of nearby mountains and variations in local geology.) 6.6 Satellites and Kepler's Laws: An Argument for Simplicity 43. A geosynchronous Earth satellite is one that has an orbital period of precisely 1 day. Such orbits are useful for communication and weather observation because the satellite remains above the same point on Earth (provided it orbits in the equatorial plane in the same direction as Earth's rotation). Calculate the radius of such an orbit based on the data for the moon in Table 6.2. 44. Calculate the mass of the Sun based on data for Earth's orbit and compare the value obtained with the Sun's actual mass. 45. Find the mass of Jupiter based on data for the orbit of one of its moons, and compare your result with its actual mass. 46. Find the ratio of the mass of Jupiter to that of Earth based on data in Table 6.2. 47. Astronomical observations of our Milky Way galaxy indicate that it has a mass of about 8.0×1011 solar masses. A star orbiting on the galaxy's periphery is about 6.0×104 light years from its center. (a) What should the orbital period of that star be? (b) If its period is 6.0×107 instead, what is the mass of the galaxy? Such calculations are used to imply the existence of “dark matter” in the universe and have indicated, for example, the existence of very massive black holes at the centers of some galaxies. 48. Integrated Concepts Space debris left from old satellites and their launchers is becoming a hazard to other satellites. (a) Calculate the speed of a satellite in an orbit 900 km above Earth's surface. (b) Suppose a loose rivet is in an orbit of the same radius that intersects the satellite's orbit at an angle of 90º relative to Earth. What is the velocity of the rivet relative to the satellite just before striking it? (c) Given the rivet is 3.00 mm in size, how long will its collision with the satellite last? (d) If its mass is 0.500 g, what is the average force it exerts on the satellite? (e) How much energy in joules is generated by the collision? (The satellite's velocity does not change appreciably, because its mass is much greater than the rivet's.) 49. Unreasonable Results (a) Based on Kepler's laws and information on the orbital characteristics of the Moon, calculate the orbital radius for an Earth satellite having a period of 1.00 h. (b) What is unreasonable about this result? (c) What is unreasonable or inconsistent about the premise of a 1.00 h orbit? 50. Construct Your Own Problem On February 14, 2000, the NEAR spacecraft was successfully inserted into orbit around Eros, becoming the first artificial satellite of an asteroid. Construct a problem in which you determine the orbital speed for a satellite near Eros. You will need to find the mass of the asteroid and consider such things as a safe distance for the orbit. Although Eros is not spherical, calculate the acceleration due to gravity on its surface at a point an average distance from its center of mass. Your instructor may also wish to have you calculate the escape velocity from this point on Eros. Chapter 6 \| Gravitation and Uniform Circular Motion 259 Test Prep for AP® Courses 6.5 Newton's Universal Law of Gravitation 1. Jupiter has a mass approximately 300 times greater than Earth's and a radius about 11 times greater. How will the gravitational acceleration at the surface of Jupiter compare to that at the surface of the Earth? a. Greater b. Less c. About the same d. Not enough information 2. Given Newton's universal law of gravitation (Equation 6.40), under what circumstances is the force due to gravity maximized? 3. In the formula = 2 , what does G represent? a. The acceleration due to gravity b. A gravitational constant that is the same everywhere in the universe c. A gravitational constant that is inversely proportional to the radius d. The factor by which you multiply the inertial mass to obtain the gravitational mass 4. Saturn's moon Titan has a radius of 2.58 × 106 m and a measured gravitational field of 1.35 m/s2. What is its mass? 5. A recently discovered planet has a mass twice as great as Earth's and a radius twice as large as Earth's. What will be the approximate size of its gravitational field? a. 19 m/s2 b. 4.9 m/s2 c. 2.5 m/s2 d. 9.8 m/s2 6. 4. Earth is 1.5 × 1011 m from the Sun. Mercury is 5.7 × 1010 m from the Sun. How does the gravitational field of the Sun on Mercury (gSM) compare to the gravitational field of the Sun on Earth (gSE)? 260 Chapter 6 \| Gravitation and Uniform Circular Motion This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources 261 7 WORK, ENERGY, AND ENERGY RESOURCES Figure 7.1 How many forms of energy can you identify in this photograph of a wind farm in Iowa? (credit: Jürgen from Sandesneben, Germany, Wikimedi
a Commons) Chapter Outline 7.1. Work: The Scientific Definition 7.2. Kinetic Energy and the Work-Energy Theorem 7.3. Gravitational Potential Energy 7.4. Conservative Forces and Potential Energy 7.5. Nonconservative Forces 7.6. Conservation of Energy 7.7. Power 7.8. Work, Energy, and Power in Humans 7.9. World Energy Use Connection for AP® Courses Energy plays an essential role both in everyday events and in scientific phenomena. You can no doubt name many forms of energy, from that provided by our foods to the energy we use to run our cars and the sunlight that warms us on the beach. You can also cite examples of what people call “energy” that may not be scientific, such as someone having an energetic personality. Not only does energy have many interesting forms, it is involved in almost all phenomena, and is one of the most important concepts of physics. There is no simple and accurate scientific definition for energy. Energy is characterized by its many forms and the fact that it is conserved. We can loosely define energy as the ability to do work, admitting that in some circumstances not all energy is available to do work. Because of the association of energy with work, we begin the chapter with a discussion of work. Work is intimately related to energy and how energy moves from one system to another or changes form. The work-energy theorem supports Big Idea 3, that interactions between objects are described by forces. In particular, exerting a force on an object may do work on it, changing it's energy (Enduring Understanding 3.E). The work-energy theorem, introduced in this chapter, establishes the relationship between work done on an object by an external force and changes in the object’s kinetic energy (Essential Knowledge 3.E.1). Similarly, systems can do work on each other, supporting Big Idea 4, that interactions between systems can result in changes in those systems—in this case, changes in the total energy of the system (Enduring Understanding 4.C). The total energy of the system is the sum of its kinetic energy, potential energy, and microscopic internal energy (Essential Knowledge 4.C.1). In this chapter students learn how to calculate kinetic, gravitational, and elastic potential energy in order to determine the total mechanical energy of a system. The transfer of mechanical energy into or out of a system is equal to the work done on the system by an external force with a nonzero component parallel to the displacement (Essential Knowledge 4.C.2). An important aspect of energy is that the total amount of energy in the universe is constant. Energy can change forms, but it cannot appear from nothing or disappear without a trace. Energy is thus one of a handful of physical quantities that we say is 262 Chapter 7 \| Work, Energy, and Energy Resources “conserved.” Conservation of energy (as physicists call the principle that energy can neither be created nor destroyed) is based on experiment. Even as scientists discovered new forms of energy, conservation of energy has always been found to apply. Perhaps the most dramatic example of this was supplied by Einstein when he suggested that mass is equivalent to energy (his famous equation E = mc2). This is one of the most important applications of Big Idea 5, that changes that occur as a result of interactions are constrained by conservation laws. Specifically, there are many situations where conservation of energy (Enduring Understanding 5.B) is both a useful concept and starting point for calculations related to the system. Note, however, that conservation doesn’t necessarily mean that energy in a system doesn’t change. Energy may be transferred into or out of the system, and the change must be equal to the amount transferred (Enduring Understanding 5.A). This may occur if there is an external force or a transfer between external objects and the system (Essential Knowledge 5.A.3). Energy is one of the fundamental quantities that are conserved for all systems (Essential Knowledge 5.A.2). The chapter introduces concepts of kinetic energy and potential energy. Kinetic energy is introduced as an energy of motion that can be changed by the amount of work done by an external force. Potential energy can only exist when objects interact with each other via conservative forces according to classical physics (Essential Knowledge 5.B.3). Because of this, a single object can only have kinetic energy and no potential energy (Essential Knowledge 5.B.1). The chapter also introduces the idea that the energy transfer is equal to the work done on the system by external forces and the rate of energy transfer is defined as power (Essential Knowledge 5.B.5). From a societal viewpoint, energy is one of the major building blocks of modern civilization. Energy resources are key limiting factors to economic growth. The world use of energy resources, especially oil, continues to grow, with ominous consequences economically, socially, politically, and environmentally. We will briefly examine the world’s energy use patterns at the end of this chapter. The concepts in this chapter support: Big Idea 3 The interactions of an object with other objects can be described by forces. Enduring Understanding 3.E A force exerted on an object can change the kinetic energy of the object. Essential Knowledge 3.E.1 The change in the kinetic energy of an object depends on the force exerted on the object and on the displacement of the object during the interval that the force is exerted. Big Idea 4 Interactions between systems can result in changes in those systems. Enduring Understanding 4.C Interactions with other objects or systems can change the total energy of a system. Essential Knowledge 4.C.1 The energy of a system includes its kinetic energy, potential energy, and microscopic internal energy. Examples should include gravitational potential energy, elastic potential energy, and kinetic energy. Essential Knowledge 4.C.2 Mechanical energy (the sum of kinetic and potential energy) is transferred into or out of a system when an external force is exerted on a system such that a component of the force is parallel to its displacement. The process through which the energy is transferred is called work. Big Idea 5 Changes that occur as a result of interactions are constrained by conservation laws. Enduring Understanding 5.A Certain quantities are conserved, in the sense that the changes of those quantities in a given system are always equal to the transfer of that quantity to or from the system by all possible interactions with other systems. Essential Knowledge 5.A.2 For all systems under all circumstances, energy, charge, linear momentum, and angular momentum are conserved. Essential Knowledge 5.A.3 An interaction can be either a force exerted by objects outside the system or the transfer of some quantity with objects outside the system. Enduring Understanding 5.B The energy of a system is conserved. Essential Knowledge 5.B.1 Classically, an object can only have kinetic energy since potential energy requires an interaction between two or more objects. Essential Knowledge 5.B.3 A system with internal structure can have potential energy. Potential energy exists within a system if the objects within that system interact with conservative forces. Essential Knowledge 5.B.5 Energy can be transferred by an external force exerted on an object or system that moves the object or system through a distance; this energy transfer is called work. Energy transfer in mechanical or electrical systems may occur at different rates. Power is deﬁned as the rate of energy transfer into, out of, or within a system. 7.1 Work: The Scientific Definition By the end of this section, you will be able to: Learning Objectives • Explain how an object must be displaced for a force on it to do work. • Explain how relative directions of force and displacement of an object determine whether the work done on the object is positive, negative, or zero. The information presented in this section supports the following AP® learning objectives and science practices: • 5.B.5.1 The student is able to design an experiment and analyze data to examine how a force exerted on an object or system does work on the object or system as it moves through a distance. (S.P. 4.2, 5.1) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources 263 • 5.B.5.2 The student is able to design an experiment and analyze graphical data in which interpretations of the area under a force-distance curve are needed to determine the work done on or by the object or system. (S.P. 4.5, 5.1) • 5.B.5.3 The student is able to predict and calculate from graphical data the energy transfer to or work done on an object or system from information about a force exerted on the object or system through a distance. (S.P. 1.5, 2.2, 6.4) What It Means to Do Work The scientific definition of work differs in some ways from its everyday meaning. Certain things we think of as hard work, such as writing an exam or carrying a heavy load on level ground, are not work as defined by a scientist. The scientific definition of work reveals its relationship to energy—whenever work is done, energy is transferred. For work, in the scientific sense, to be done on an object, a force must be exerted on that object and there must be motion or displacement of that object in the direction of the force. Formally, the work done on a system by a constant force is defined to be the product of the component of the force in the direction of motion and the distance through which the force acts. For a constant force, this is expressed in equation form as = ∣ F ∣ (cos ) ∣ d ∣ , (7.1) where is work, d is the displacement of the system, and is the angle between the force vector F and the displacement vector d , as in Figure 7.2. We can also write this as To find the work done on a system that undergoes motion that is not one-way or that is in two or three di
mensions, we divide the motion into one-way one-dimensional segments and add up the work done over each segment. = cos . (7.2) What is Work? The work done on a system by a constant force is the product of the component of the force in the direction of motion times the distance through which the force acts. For one-way motion in one dimension, this is expressed in equation form as = cos , (7.3) where is work, is the magnitude of the force on the system, is the magnitude of the displacement of the system, and is the angle between the force vector F and the displacement vector d . 264 Chapter 7 \| Work, Energy, and Energy Resources Figure 7.2 Examples of work. (a) The work done by the force F on this lawn mower is cos . Note that cos is the component of the force in the direction of motion. (b) A person holding a briefcase does no work on it, because there is no motion. No energy is transferred to or from the briefcase. (c) The person moving the briefcase horizontally at a constant speed does no work on it, and transfers no energy to it. (d) Work is done on the briefcase by carrying it up stairs at constant speed, because there is necessarily a component of force F in the direction of the motion. Energy is transferred to the briefcase and could in turn be used to do work. (e) When the briefcase is lowered, energy is transferred out of the briefcase and into an electric generator. Here the work done on the briefcase by the generator is negative, removing energy from the briefcase, because F and d are in opposite directions. To examine what the definition of work means, let us consider the other situations shown in Figure 7.2. The person holding the briefcase in Figure 7.2(b) does no work, for example. Here = 0 , so = 0 . Why is it you get tired just holding a load? The answer is that your muscles are doing work against one another, but they are doing no work on the system of interest (the “briefcase-Earth system”—see Gravitational Potential Energy for more details). There must be displacement for work to be done, and there must be a component of the force in the direction of the motion. For example, the person carrying the briefcase This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources 265 on level ground in Figure 7.2(c) does no work on it, because the force is perpendicular to the motion. That is, cos 90º = 0 , and so = 0 . In contrast, when a force exerted on the system has a component in the direction of motion, such as in Figure 7.2(d), work is done—energy is transferred to the briefcase. Finally, in Figure 7.2(e), energy is transferred from the briefcase to a generator. There are two good ways to interpret this energy transfer. One interpretation is that the briefcase’s weight does work on the generator, giving it energy. The other interpretation is that the generator does negative work on the briefcase, thus removing energy from it. The drawing shows the latter, with the force from the generator upward on the briefcase, and the displacement downward. This makes = 180º , and cos 180º = –1 ; therefore, is negative. Real World Connections: When Work Happens Note that work as we define it is not the same as effort. You can push against a concrete wall all you want, but you won’t move it. While the pushing represents effort on your part, the fact that you have not changed the wall’s state in any way indicates that you haven’t done work. If you did somehow push the wall over, this would indicate a change in the wall’s state, and therefore you would have done work. This can also be shown with Figure 7.2(a): as you push a lawnmower against friction, both you and friction are changing the lawnmower’s state. However, only the component of the force parallel to the movement is changing the lawnmower’s state. The component perpendicular to the motion is trying to push the lawnmower straight into Earth; the lawnmower does not move into Earth, and therefore the lawnmower’s state is not changing in the direction of Earth. Similarly, in Figure 7.2(c), both your hand and gravity are exerting force on the briefcase. However, they are both acting perpendicular to the direction of motion, hence they are not changing the condition of the briefcase and do no work. However, if the briefcase were dropped, then its displacement would be parallel to the force of gravity, which would do work on it, changing its state (it would fall to the ground). Calculating Work Work and energy have the same units. From the definition of work, we see that those units are force times distance. Thus, in SI units, work and energy are measured in newton-meters. A newton-meter is given the special name joule (J), and 1 J = 1 N ⋅ m = 1 kg ⋅ m2/s2 . One joule is not a large amount of energy; it would lift a small 100-gram apple a distance of about 1 meter. Example 7.1 Calculating the Work You Do to Push a Lawn Mower Across a Large Lawn How much work is done on the lawn mower by the person in Figure 7.2(a) if he exerts a constant force of 75.0 N at an angle 35º below the horizontal and pushes the mower 25.0 m on level ground? Convert the amount of work from joules to kilocalories and compare it with this person’s average daily intake of 10000 kJ (about 2400 kcal ) of food energy. One calorie (1 cal) of heat is the amount required to warm 1 g of water by 1ºC , and is equivalent to 4.184 J , while one food calorie (1 kcal) is equivalent to 4184 J . Strategy We can solve this problem by substituting the given values into the definition of work done on a system, stated in the equation = cos . The force, angle, and displacement are given, so that only the work is unknown. Solution The equation for the work is Substituting the known values gives = cos . = (75.0 N)(25.0 m) cos (35.0º) = 1536 J = 1.54×103 J. (7.4) (7.5) Converting the work in joules to kilocalories yields = (1536 J)(1 kcal / 4184 J) = 0.367 kcal . The ratio of the work done to the daily consumption is 2400 kcal = 1.53×10−4. (7.6) Discussion This ratio is a tiny fraction of what the person consumes, but it is typical. Very little of the energy released in the consumption of food is used to do work. Even when we “work” all day long, less than 10% of our food energy intake is used to do work and more than 90% is converted to thermal energy or stored as chemical energy in fat. 266 Chapter 7 \| Work, Energy, and Energy Resources Applying the Science Practices: Boxes on Floors Plan and design an experiment to determine how much work you do on a box when you are pushing it over different floor surfaces. Make sure your experiment can help you answer the following questions: What happens on different surfaces? What happens if you take different routes across the same surface? Do you get different results with two people pushing on perpendicular surfaces of the box? What if you vary the mass in the box? Remember to think about both your effort in any given instant (a proxy for force exerted) and the total work you do. Also, when planning your experiments, remember that in any given set of trials you should only change one variable. You should find that you have to exert more effort on surfaces that will create more friction with the box, though you might be surprised by which surfaces the box slides across easily. Longer routes result in your doing more work, even though the box ends up in the same place. Two people pushing on perpendicular sides do less work for their total effort, due to the forces and displacement not being parallel. A more massive box will take more effort to move. Applying the Science Practices: Force-Displacement Diagrams Suppose you are given two carts and a track to run them on, a motion detector, a force sensor, and a computer that can record the data from the two sensors. Plan and design an experiment to measure the work done on one of the carts, and compare your results to the work-energy theorem. Note that the motion detector can measure both displacement and velocity versus time, while the force sensor measures force over time, and the carts have known masses. Recall that the work-energy theorem states that the work done on a system (force over displacement) should equal the change in kinetic energy. In your experimental design, describe and compare two possible ways to calculate the work done. Sample Response: One possible technique is to set up the motion detector at one end of the track, and have the computer record both displacement and velocity over time. Then attach the force sensor to one of the carts, and use this cart, through the force sensor, to push the second cart toward the motion detector. Calculate the difference between the final and initial kinetic energies (the kinetic energies after and before the push), and compare this to the area of a graph of force versus displacement for the duration of the push. They should be the same. 7.2 Kinetic Energy and the Work-Energy Theorem Learning Objectives By the end of this section, you will be able to: • Explain work as a transfer of energy and net work as the work done by the net force. • Explain and apply the work-energy theorem. The information presented in this section supports the following AP® learning objectives and science practices: • 3.E.1.1 The student is able to make predictions about the changes in kinetic energy of an object based on considerations of the direction of the net force on the object as the object moves. (S.P. 6.4, 7.2) • 3.E.1.2 The student is able to use net force and velocity vectors to determine qualitatively whether kinetic energy of an object would increase, decrease, or remain unchanged. (S.P. 1.4) • 3.E.1.3 The student is able to use force and velocity vectors to determine qualitatively or quantitatively the net force exerted on an object and qualitatively whether kinetic energy of that object would increase, decrease, or remain unchanged. (S.P. 1.4, 2.2) • 3.E.1.4 The student is able to apply mathematical routines to det
ermine the change in kinetic energy of an object given the forces on the object and the displacement of the object. (S.P. 2.2) • 4.C.1.1 The student is able to calculate the total energy of a system and justify the mathematical routines used in the calculation of component types of energy within the system whose sum is the total energy. (S.P. 1.4, 2.1, 2.2) • 4.C.2.1 The student is able to make predictions about the changes in the mechanical energy of a system when a component of an external force acts parallel or antiparallel to the direction of the displacement of the center of mass. (S.P. 6.4) • 4.C.2.2 The student is able to apply the concepts of conservation of energy and the work-energy theorem to determine qualitatively and/or quantitatively that work done on a two-object system in linear motion will change the kinetic energy of the center of mass of the system, the potential energy of the systems, and/or the internal energy of the system. (S.P. 1.4, 2.2, 7.2) • 5.B.5.3 The student is able to predict and calculate from graphical data the energy transfer to or work done on an object or system from information about a force exerted on the object or system through a distance. (S.P. 1.5, 2.2, 6.4) Work Transfers Energy What happens to the work done on a system? Energy is transferred into the system, but in what form? Does it remain in the system or move on? The answers depend on the situation. For example, if the lawn mower in Figure 7.2(a) is pushed just hard enough to keep it going at a constant speed, then energy put into the mower by the person is removed continuously by friction, and eventually leaves the system in the form of heat transfer. In contrast, work done on the briefcase by the person carrying it up stairs in Figure 7.2(d) is stored in the briefcase-Earth system and can be recovered at any time, as shown in Figure 7.2(e). In fact, the building of the pyramids in ancient Egypt is an example of storing energy in a system by doing work on the system. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources 267 Some of the energy imparted to the stone blocks in lifting them during construction of the pyramids remains in the stone-Earth system and has the potential to do work. In this section we begin the study of various types of work and forms of energy. We will find that some types of work leave the energy of a system constant, for example, whereas others change the system in some way, such as making it move. We will also develop definitions of important forms of energy, such as the energy of motion. Net Work and the Work-Energy Theorem We know from the study of Newton’s laws in Dynamics: Force and Newton's Laws of Motion that net force causes acceleration. We will see in this section that work done by the net force gives a system energy of motion, and in the process we will also find an expression for the energy of motion. Let us start by considering the total, or net, work done on a system. Net work is defined to be the sum of work done by all external forces—that is, net work is the work done by the net external force Fnet . In equation form, this is net = net cos where is the angle between the force vector and the displacement vector. Figure 7.3(a) shows a graph of force versus displacement for the component of the force in the direction of the displacement—that is, an cos vs. graph. In this case, cos is constant. You can see that the area under the graph is cos , or the work done. Figure 7.3(b) shows a more general process where the force varies. The area under the curve is divided into strips, each having an average force ( cos ) (ave) for each strip, and the (ave) total work done is the sum of the . Thus the total work done is the total area under the curve, a useful property to which we shall refer later. . The work done is ( cos ) Figure 7.3 (a) A graph of cos vs. , when cos is constant. The area under the curve represents the work done by the force. (b) A graph of cos vs. in which the force varies. The work done for each interval is the area of each strip; thus, the total area under the curve equals the total work done. Real World Connections: Work and Direction Consider driving in a car. While moving, you have forward velocity and therefore kinetic energy. When you hit the brakes, they exert a force opposite to your direction of motion (acting through the wheels). The brakes do work on your car and reduce the kinetic energy. Similarly, when you accelerate, the engine (acting through the wheels) exerts a force in the direction of motion. The engine does work on your car, and increases the kinetic energy. Finally, if you go around a corner at a constant speed, you have the same kinetic energy both before and after the corner. The force exerted by the engine was perpendicular to the direction of motion, and therefore did no work and did not change the kinetic energy. 268 Chapter 7 \| Work, Energy, and Energy Resources Net work will be simpler to examine if we consider a one-dimensional situation where a force is used to accelerate an object in a direction parallel to its initial velocity. Such a situation occurs for the package on the roller belt conveyor system shown in Figure 7.4. Figure 7.4 A package on a roller belt is pushed horizontally through a distance d . The force of gravity and the normal force acting on the package are perpendicular to the displacement and do no work. Moreover, they are also equal in magnitude and opposite in direction so they cancel in calculating the net force. The net force arises solely from the horizontal applied force Fapp and the horizontal friction force f . Thus, as expected, the net force is parallel to the displacement, so that = 0º and cos = 1 , and the net work is given by net = net. (7.7) The effect of the net force Fnet is to accelerate the package from 0 to . The kinetic energy of the package increases, indicating that the net work done on the system is positive. (See Example 7.2.) By using Newton’s second law, and doing some algebra, we can reach an interesting conclusion. Substituting net = from Newton’s second law gives net = (7.8) To get a relationship between net work and the speed given to a system by the net force acting on it, we take = − 0 and use the equation studied in Motion Equations for Constant Acceleration in One Dimension for the change in speed over a 2 + 2 (note that appears in the expression for distance if the acceleration has the constant value ; namely, 2 = 0 the net work). Solving for acceleration gives = 2 2 − 0 2 obtain . When is substituted into the preceding expression for net , we The cancels, and we rearrange this to obtain net = 2 2 − 0 2 . = 1 22 − 1 2 . 20 (7.9) (7.10) This expression is called the work-energy theorem, and it actually applies in general (even for forces that vary in direction and magnitude), although we have derived it for the special case of a constant force parallel to the displacement. The theorem implies that the net work on a system equals the change in the quantity 1 22 . This quantity is our first example of a form of energy. The Work-Energy Theorem The net work on a system equals the change in the quantity 1 22 . 22 − 1 2 20 net = 1 (7.11) The quantity 1 22 in the work-energy theorem is defined to be the translational kinetic energy (KE) of a mass moving at a speed . (Translational kinetic energy is distinct from rotational kinetic energy, which is considered later.) In equation form, the translational kinetic energy, KE = 1 22 (7.12) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources 269 is the energy associated with translational motion. Kinetic energy is a form of energy associated with the motion of a particle, single body, or system of objects moving together. We are aware that it takes energy to get an object, like a car or the package in Figure 7.4, up to speed, but it may be a bit surprising that kinetic energy is proportional to speed squared. This proportionality means, for example, that a car traveling at 100 km/h has four times the kinetic energy it has at 50 km/h, helping to explain why high-speed collisions are so devastating. We will now consider a series of examples to illustrate various aspects of work and energy. Applying the Science Practices: Cars on a Hill Assemble a ramp suitable for rolling some toy cars up or down. Then plan a series of experiments to determine how the direction of a force relative to the velocity of an object alters the kinetic energy of the object. Note that gravity will be pointing down in all cases. What happens if you start the car at the top? How about at the bottom, with an initial velocity that is increasing? If your ramp is wide enough, what happens if you send the toy car straight across? Does varying the surface of the ramp change your results? Sample Response: When the toy car is going down the ramp, with a component of gravity in the same direction, the kinetic energy increases. Sending the car up the ramp decreases the kinetic energy, as gravity is opposing the motion. Sending the car sideways should result in little to no change. If you have a surface that generates more friction than a smooth surface (carpet), note that the friction always opposed the motion, and hence decreases the kinetic energy. Example 7.2 Calculating the Kinetic Energy of a Package Suppose a 30.0-kg package on the roller belt conveyor system in Figure 7.4 is moving at 0.500 m/s. What is its kinetic energy? Strategy Because the mass and speed are given, the kinetic energy can be calculated from its definition as given in the equation KE = 1 22 . Solution The kinetic energy is given by Entering known values gives which yields Discussion KE = 1 22. KE = 0.5(30.0 kg)(0.500 m/s)2 KE = 3.75 kg ⋅ m2/s2 = 3.75 J. (7.13) (7.14) (7.15) Note that the unit of kinetic energy is the joule, the same as the unit of work, as mentioned when
work was first defined. It is also interesting that, although this is a fairly massive package, its kinetic energy is not large at this relatively low speed. This fact is consistent with the observation that people can move packages like this without exhausting themselves. Real World Connections: Center of Mass Suppose we have two experimental carts, of equal mass, latched together on a track with a compressed spring between them. When the latch is released, the spring does 10 J of work on the carts (we’ll see how in a couple of sections). The carts move relative to the spring, which is the center of mass of the system. However, the center of mass stays fixed. How can we consider the kinetic energy of this system? By the work-energy theorem, the work done by the spring on the carts must turn into kinetic energy. So this system has 10 J of kinetic energy. The total kinetic energy of the system is the kinetic energy of the center of mass of the system relative to the fixed origin plus the kinetic energy of each cart relative to the center of mass. We know that the center of mass relative to the fixed origin does not move, and therefore all of the kinetic energy must be distributed among the carts relative to the center of mass. Since the carts have equal mass, they each receive an equal amount of kinetic energy, so each cart has 5.0 J of kinetic energy. In our example, the forces between the spring and each cart are internal to the system. According to Newton’s third law, these internal forces will cancel since they are equal and opposite in direction. However, this does not imply that these internal forces will not do work. Thus, the change in kinetic energy of the system is caused by work done by the force of the spring, and results in the motion of the two carts relative to the center of mass. 270 Chapter 7 \| Work, Energy, and Energy Resources Example 7.3 Determining the Work to Accelerate a Package Suppose that you push on the 30.0-kg package in Figure 7.4 with a constant force of 120 N through a distance of 0.800 m, and that the opposing friction force averages 5.00 N. (a) Calculate the net work done on the package. (b) Solve the same problem as in part (a), this time by finding the work done by each force that contributes to the net force. Strategy and Concept for (a) This is a motion in one dimension problem, because the downward force (from the weight of the package) and the normal force have equal magnitude and opposite direction, so that they cancel in calculating the net force, while the applied force, friction, and the displacement are all horizontal. (See Figure 7.4.) As expected, the net work is the net force times distance. Solution for (a) The net force is the push force minus friction, or net = 120 N – 5.00 N = 115 N . Thus the net work is net = net = (115 N)(0.800 m) = 92.0 N ⋅ m = 92.0 J. (7.16) Discussion for (a) This value is the net work done on the package. The person actually does more work than this, because friction opposes the motion. Friction does negative work and removes some of the energy the person expends and converts it to thermal energy. The net work equals the sum of the work done by each individual force. Strategy and Concept for (b) The forces acting on the package are gravity, the normal force, the force of friction, and the applied force. The normal force and force of gravity are each perpendicular to the displacement, and therefore do no work. Solution for (b) The applied force does work. app = app cos(0º) = app = (120 N)(0.800 m) = 96.0 J The friction force and displacement are in opposite directions, so that = 180º , and the work done by friction is fr = fr cos(180º) = −fr = −(5.00 N)(0.800 m) = −4.00 J. So the amounts of work done by gravity, by the normal force, by the applied force, and by friction are, respectively, gr = 0, N = 0, app = 96.0 J, fr = − 4.00 J. The total work done as the sum of the work done by each force is then seen to be total = gr + N + app + fr = 92.0 J. Discussion for (b) (7.17) (7.18) (7.19) (7.20) The calculated total work total as the sum of the work by each force agrees, as expected, with the work net done by the net force. The work done by a collection of forces acting on an object can be calculated by either approach. Example 7.4 Determining Speed from Work and Energy Find the speed of the package in Figure 7.4 at the end of the push, using work and energy concepts. Strategy Here the work-energy theorem can be used, because we have just calculated the net work, net , and the initial kinetic energy, 1 2 . These calculations allow us to find the final kinetic energy, 1 22 , and thus the final speed . 20 This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources 271 Solution The work-energy theorem in equation form is net = 1 22 − 1 20 2. Solving for 1 22 gives Thus, 1 22 = net + 1 20 2. 1 22 = 92.0 J+3.75 J = 95.75 J. Solving for the final speed as requested and entering known values gives = 2(95.75 J) = 191.5 kg ⋅ m2/s2 30.0 kg = 2.53 m/s. (7.21) (7.22) (7.23) (7.24) Discussion Using work and energy, we not only arrive at an answer, we see that the final kinetic energy is the sum of the initial kinetic energy and the net work done on the package. This means that the work indeed adds to the energy of the package. Example 7.5 Work and Energy Can Reveal Distance, Too How far does the package in Figure 7.4 coast after the push, assuming friction remains constant? Use work and energy considerations. Strategy We know that once the person stops pushing, friction will bring the package to rest. In terms of energy, friction does negative work until it has removed all of the package’s kinetic energy. The work done by friction is the force of friction times the distance traveled times the cosine of the angle between the friction force and displacement; hence, this gives us a way of finding the distance traveled after the person stops pushing. Solution The normal force and force of gravity cancel in calculating the net force. The horizontal friction force is then the net force, and it acts opposite to the displacement, so = 180º . To reduce the kinetic energy of the package to zero, the work fr by friction must be minus the kinetic energy that the package started with plus what the package accumulated due to the pushing. Thus fr = −95.75 J . Furthermore, fr = ′ cos = – ′ , where ′ is the distance it takes to stop. Thus, and so Discussion ′ = − fr = − −95.75 J 5.00 N , ′ = 19.2 m. (7.25) (7.26) This is a reasonable distance for a package to coast on a relatively friction-free conveyor system. Note that the work done by friction is negative (the force is in the opposite direction of motion), so it removes the kinetic energy. Some of the examples in this section can be solved without considering energy, but at the expense of missing out on gaining insights about what work and energy are doing in this situation. On the whole, solutions involving energy are generally shorter and easier than those using kinematics and dynamics alone. 7.3 Gravitational Potential Energy By the end of this section, you will be able to: • Explain gravitational potential energy in terms of work done against gravity. Learning Objectives 272 Chapter 7 \| Work, Energy, and Energy Resources • Show that the gravitational potential energy of an object of mass m at height h on Earth is given by PEg = mgh. • Show how knowledge of potential energy as a function of position can be used to simplify calculations and explain physical phenomena. The information presented in this section supports the following AP® learning objectives and science practices: • 4.C.1.1 The student is able to calculate the total energy of a system and justify the mathematical routines used in the calculation of component types of energy within the system whose sum is the total energy. (S.P. 1.4, 2.1, 2.2) • 5.B.1.1 The student is able to set up a representation or model showing that a single object can only have kinetic energy and use information about that object to calculate its kinetic energy. (S.P. 1.4, 2.2) • 5.B.1.2 The student is able to translate between a representation of a single object, which can only have kinetic energy, and a system that includes the object, which may have both kinetic and potential energies. (S.P. 1.5) Work Done Against Gravity Climbing stairs and lifting objects is work in both the scientific and everyday sense—it is work done against the gravitational force. When there is work, there is a transformation of energy. The work done against the gravitational force goes into an important form of stored energy that we will explore in this section. Let us calculate the work done in lifting an object of mass through a height , such as in Figure 7.5. If the object is lifted straight up at constant speed, then the force needed to lift it is equal to its weight . The work done on the mass is then . We define this to be the gravitational potential energy (PEg) put into (or gained by) the object-Earth system. This energy is associated with the state of separation between two objects that attract each other by the gravitational force. For convenience, we refer to this as the PEg gained by the object, recognizing that this is energy stored in the gravitational field of Earth. Why do we use the word “system”? Potential energy is a property of a system rather than of a single object—due to its physical position. An object’s gravitational potential is due to its position relative to the surroundings within the Earth-object system. The force applied to the object is an external force, from outside the system. When it does positive work it increases the gravitational potential energy of the system. Because gravitational potential energy depends on relative position, we need a reference level at which to set the potential energy equal to 0. We usually choose this point to be Earth’s surface, but this point is arbitrary; what is important is
the difference in gravitational potential energy, because this difference is what relates to the work done. The difference in gravitational potential energy of an object (in the Earth-object system) between two rungs of a ladder will be the same for the first two rungs as for the last two rungs. Converting Between Potential Energy and Kinetic Energy Gravitational potential energy may be converted to other forms of energy, such as kinetic energy. If we release the mass, gravitational force will do an amount of work equal to on it, thereby increasing its kinetic energy by that same amount (by the work-energy theorem). We will find it more useful to consider just the conversion of PEg to KE without explicitly considering the intermediate step of work. (See Example 7.7.) This shortcut makes it is easier to solve problems using energy (if possible) rather than explicitly using forces. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources 273 Figure 7.5 (a) The work done to lift the weight is stored in the mass-Earth system as gravitational potential energy. (b) As the weight moves downward, this gravitational potential energy is transferred to the cuckoo clock. More precisely, we define the change in gravitational potential energy ΔPEg to be ΔPEg = , (7.27) where, for simplicity, we denote the change in height by rather than the usual Δ . Note that is positive when the final height is greater than the initial height, and vice versa. For example, if a 0.500-kg mass hung from a cuckoo clock is raised 1.00 m, then its change in gravitational potential energy is = 0.500 kg 9.80 m/s2 (1.00 m) (7.28) = 4.90 kg ⋅ m2/s2 = 4.90 J. Note that the units of gravitational potential energy turn out to be joules, the same as for work and other forms of energy. As the clock runs, the mass is lowered. We can think of the mass as gradually giving up its 4.90 J of gravitational potential energy, without directly considering the force of gravity that does the work. Using Potential Energy to Simplify Calculations The equation ΔPEg = applies for any path that has a change in height of , not just when the mass is lifted straight up. (See Figure 7.6.) It is much easier to calculate (a simple multiplication) than it is to calculate the work done along a complicated path. The idea of gravitational potential energy has the double advantage that it is very broadly applicable and it makes calculations easier. From now on, we will consider that any change in vertical position of a mass is accompanied by a change in gravitational potential energy , and we will avoid the equivalent but more difficult task of calculating work done by or against the gravitational force. 274 Chapter 7 \| Work, Energy, and Energy Resources Figure 7.6 The change in gravitational potential energy (ΔPEg) between points A and B is independent of the path. ΔPEg = for any path between the two points. Gravity is one of a small class of forces where the work done by or against the force depends only on the starting and ending points, not on the path between them. Example 7.6 The Force to Stop Falling A 60.0-kg person jumps onto the floor from a height of 3.00 m. If he lands stiffly (with his knee joints compressing by 0.500 cm), calculate the force on the knee joints. Strategy This person’s energy is brought to zero in this situation by the work done on him by the floor as he stops. The initial PEg is transformed into KE as he falls. The work done by the floor reduces this kinetic energy to zero. Solution The work done on the person by the floor as he stops is given by = cos = −, (7.29) with a minus sign because the displacement while stopping and the force from floor are in opposite directions (cos = cos 180º = − 1) . The floor removes energy from the system, so it does negative work. The kinetic energy the person has upon reaching the floor is the amount of potential energy lost by falling through height : KE = −ΔPEg = −, (7.30) The distance that the person’s knees bend is much smaller than the height of the fall, so the additional change in gravitational potential energy during the knee bend is ignored. The work done by the floor on the person stops the person and brings the person’s kinetic energy to zero: Combining this equation with the expression for gives − = . = −KE = . Recalling that is negative because the person fell down, the force on the knee joints is given by (7.31) (7.32) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources = − = − Discussion 275 (7.33) = 3.53×105 N. 60.0 kg 9.80 m/s2 5.00×10−3 m (−3.00 m) Such a large force (500 times more than the person’s weight) over the short impact time is enough to break bones. A much better way to cushion the shock is by bending the legs or rolling on the ground, increasing the time over which the force acts. A bending motion of 0.5 m this way yields a force 100 times smaller than in the example. A kangaroo's hopping shows this method in action. The kangaroo is the only large animal to use hopping for locomotion, but the shock in hopping is cushioned by the bending of its hind legs in each jump.(See Figure 7.7.) Figure 7.7 The work done by the ground upon the kangaroo reduces its kinetic energy to zero as it lands. However, by applying the force of the ground on the hind legs over a longer distance, the impact on the bones is reduced. (credit: Chris Samuel, Flickr) Example 7.7 Finding the Speed of a Roller Coaster from its Height (a) What is the final speed of the roller coaster shown in Figure 7.8 if it starts from rest at the top of the 20.0 m hill and work done by frictional forces is negligible? (b) What is its final speed (again assuming negligible friction) if its initial speed is 5.00 m/s? Figure 7.8 The speed of a roller coaster increases as gravity pulls it downhill and is greatest at its lowest point. Viewed in terms of energy, the roller-coaster-Earth system’s gravitational potential energy is converted to kinetic energy. If work done by friction is negligible, all ΔPEg is converted to KE . Strategy The roller coaster loses potential energy as it goes downhill. We neglect friction, so that the remaining force exerted by the track is the normal force, which is perpendicular to the direction of motion and does no work. The net work on the roller 276 Chapter 7 \| Work, Energy, and Energy Resources coaster is then done by gravity alone. The loss of gravitational potential energy from moving downward through a distance equals the gain in kinetic energy. This can be written in equation form as −ΔPEg = ΔKE . Using the equations for PEg and KE , we can solve for the final speed , which is the desired quantity. Solution for (a) Here the initial kinetic energy is zero, so that ΔKE = 1 22 . The equation for change in potential energy states that ΔPEg = . Since is negative in this case, we will rewrite this as ΔPEg = − ∣ ∣ clearly. Thus, to show the minus sign becomes −ΔPEg = ΔKE ∣ ∣ = 1 22. Solving for , we find that mass cancels and that = 2 ∣ ∣ . Substituting known values, = 2 9.80 m/s2 (20.0 m) = 19.8 m/s. Solution for (b) Again − ΔPEg = ΔKE . In this case there is initial kinetic energy, so ΔKE = 1 22 − 1 20 ∣ ∣ = 1 22 − 1 20 2. Rearranging gives 22 = ∣ ∣ + 1 1 20 2. (7.34) (7.35) (7.36) (7.37) (7.38) (7.39) 2 . Thus, This means that the final kinetic energy is the sum of the initial kinetic energy and the gravitational potential energy. Mass again cancels, and = 2 ∣ ∣ + 0 2. (7.40) This equation is very similar to the kinematics equation = 0 valid only for constant acceleration, whereas our equation above is valid for any path regardless of whether the object moves with a constant acceleration. Now, substituting known values gives 2 + 2 , but it is more general—the kinematics equation is = 2(9.80 m/s2)(20.0 m) + (5.00 m/s)2 = 20.4 m/s. (7.41) Discussion and Implications First, note that mass cancels. This is quite consistent with observations made in Falling Objects that all objects fall at the same rate if friction is negligible. Second, only the speed of the roller coaster is considered; there is no information about its direction at any point. This reveals another general truth. When friction is negligible, the speed of a falling body depends only on its initial speed and height, and not on its mass or the path taken. For example, the roller coaster will have the same final speed whether it falls 20.0 m straight down or takes a more complicated path like the one in the figure. Third, and perhaps unexpectedly, the final speed in part (b) is greater than in part (a), but by far less than 5.00 m/s. Finally, note that speed can be found at any height along the way by simply using the appropriate value of at the point of interest. We have seen that work done by or against the gravitational force depends only on the starting and ending points, and not on the path between, allowing us to define the simplifying concept of gravitational potential energy. We can do the same thing for a few other forces, and we will see that this leads to a formal definition of the law of conservation of energy. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources 277 Making Connections: Take-Home Investigation—Converting Potential to Kinetic Energy One can study the conversion of gravitational potential energy into kinetic energy in this experiment. On a smooth, level surface, use a ruler of the kind that has a groove running along its length and a book to make an incline (see Figure 7.9). Place a marble at the 10-cm position on the ruler and let it roll down the ruler. When it hits the level surface, measure the time it takes to roll one meter. Now place the marble at the 20-cm and the 30-cm positions and again measure the times it takes to roll 1 m on the level surface. Find the velocity of the marble on the level surface for all thr
ee positions. Plot velocity squared versus the distance traveled by the marble. What is the shape of each plot? If the shape is a straight line, the plot shows that the marble’s kinetic energy at the bottom is proportional to its potential energy at the release point. Figure 7.9 A marble rolls down a ruler, and its speed on the level surface is measured. 7.4 Conservative Forces and Potential Energy Learning Objectives By the end of this section, you will be able to: • Define conservative force, potential energy, and mechanical energy. • Explain the potential energy of a spring in terms of its compression when Hooke’s law applies. • Use the work-energy theorem to show how having only conservative forces leads to conservation of mechanical energy. The information presented in this section supports the following AP® learning objectives and science practices: • 4.C.1.1 The student is able to calculate the total energy of a system and justify the mathematical routines used in the calculation of component types of energy within the system whose sum is the total energy. (S.P. 1.4, 2.1, 2.2) • 4.C.2.1 The student is able to make predictions about the changes in the mechanical energy of a system when a component of an external force acts parallel or antiparallel to the direction of the displacement of the center of mass. (S.P. 6.4) • 5.B.1.1 The student is able to set up a representation or model showing that a single object can only have kinetic energy and use information about that object to calculate its kinetic energy. (S.P. 1.4, 2.2) • 5.B.1.2 The student is able to translate between a representation of a single object, which can only have kinetic energy, and a system that includes the object, which may have both kinetic and potential energies. (S.P. 1.5) • 5.B.3.1 The student is able to describe and make qualitative and/or quantitative predictions about everyday examples of systems with internal potential energy. (S.P. 2.2, 6.4, 7.2) • 5.B.3.2 The student is able to make quantitative calculations of the internal potential energy of a system from a description or diagram of that system. (S.P. 1.4, 2.2) • 5.B.3.3 The student is able to apply mathematical reasoning to create a description of the internal potential energy of a system from a description or diagram of the objects and interactions in that system. (S.P. 1.4, 2.2) Potential Energy and Conservative Forces Work is done by a force, and some forces, such as weight, have special characteristics. A conservative force is one, like the gravitational force, for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken. We can define a potential energy (PE) for any conservative force, just as we did for the gravitational force. For example, when you wind up a toy, an egg timer, or an old-fashioned watch, you do work against its spring and store energy in it. (We treat these springs as ideal, in that we assume there is no friction and no production of thermal energy.) This stored energy is recoverable as work, and it is useful to think of it as potential energy contained in the spring. Indeed, the reason that the spring has this characteristic is that its force is conservative. That is, a conservative force results in stored or potential energy. Gravitational potential energy is one example, as is the energy stored in a spring. We will also see how conservative forces are related to the conservation of energy. Potential Energy and Conservative Forces Potential energy is the energy a system has due to position, shape, or configuration. It is stored energy that is completely recoverable. 278 Chapter 7 \| Work, Energy, and Energy Resources A conservative force is one for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken. We can define a potential energy (PE) for any conservative force. The work done against a conservative force to reach a final configuration depends on the configuration, not the path followed, and is the potential energy added. Real World Connections: Energy of a Bowling Ball How much energy does a bowling ball have? (Just think about it for a minute.) If you are thinking that you need more information, you’re right. If we can measure the ball’s velocity, then determining its kinetic energy is simple. Note that this does require defining a reference frame in which to measure the velocity. Determining the ball’s potential energy also requires more information. You need to know its height above the ground, which requires a reference frame of the ground. Without the ground—in other words, Earth—the ball does not classically have potential energy. Potential energy comes from the interaction between the ball and the ground. Another way of thinking about this is to compare the ball’s potential energy on Earth and on the Moon. A bowling ball a certain height above Earth is going to have more potential energy than the same bowling ball the same height above the surface of the Moon, because Earth has greater mass than the Moon and therefore exerts more gravity on the ball. Thus, potential energy requires a system of at least two objects, or an object with an internal structure of at least two parts. Potential Energy of a Spring First, let us obtain an expression for the potential energy stored in a spring ( PEs ). We calculate the work done to stretch or compress a spring that obeys Hooke’s law. (Hooke’s law was examined in Elasticity: Stress and Strain, and states that the magnitude of force on the spring and the resulting deformation Δ are proportional, = Δ .) (See Figure 7.10.) For our spring, we will replace Δ (the amount of deformation produced by a force ) by the distance that the spring is stretched or compressed along its length. So the force needed to stretch the spring has magnitude , where is the spring’s force constant. The force increases linearly from 0 at the start to in the fully stretched position. The average force is / 2 . Thus the work done in stretching or compressing the spring is s = = 22 . Alternatively, we noted in Kinetic Energy = 1 2 and the Work-Energy Theorem that the area under a graph of vs. is the work done by the force. In Figure 7.10(c) we see that this area is also 1 22 . We therefore define the potential energy of a spring, PEs , to be 22, where is the spring’s force constant and is the displacement from its undeformed position. The potential energy represents the work done on the spring and the energy stored in it as a result of stretching or compressing it a distance . The potential energy of the spring PEs does not depend on the path taken; it depends only on the stretch or squeeze in the final configuration. PEs = 1 (7.42) Figure 7.10 (a) An undeformed spring has no PEs stored in it. (b) The force needed to stretch (or compress) the spring a distance has a magnitude = , and the work done to stretch (or compress) it is 1 energy (PEs) in the spring, and it can be fully recovered. (c) A graph of vs. has a slope of , and the area under the graph is 1 the work done or potential energy stored is 1 . Because the force is conservative, this work is stored as potential 22 22 . . Thus 22 This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources 279 The equation PEs = 1 22 has general validity beyond the special case for which it was derived. Potential energy can be stored in any elastic medium by deforming it. Indeed, the general definition of potential energy is energy due to position, shape, or configuration. For shape or position deformations, stored energy is PEs = 1 system and is its deformation. Another example is seen in Figure 7.11 for a guitar string. 22 , where is the force constant of the particular Figure 7.11 Work is done to deform the guitar string, giving it potential energy. When released, the potential energy is converted to kinetic energy and back to potential as the string oscillates back and forth. A very small fraction is dissipated as sound energy, slowly removing energy from the string. Conservation of Mechanical Energy Let us now consider what form the work-energy theorem takes when only conservative forces are involved. This will lead us to the conservation of energy principle. The work-energy theorem states that the net work done by all forces acting on a system equals its change in kinetic energy. In equation form, this is net = 1 22 − 1 20 If only conservative forces act, then where c is the total work done by all conservative forces. Thus, c = ΔKE. net = c, 2 = ΔKE. (7.43) (7.44) (7.45) Now, if the conservative force, such as the gravitational force or a spring force, does work, the system loses potential energy. That is, c = −ΔPE . Therefore, or −ΔPE = ΔKE ΔKE + ΔPE = 0. (7.46) (7.47) This equation means that the total kinetic and potential energy is constant for any process involving only conservative forces. That is, KE + PE = constant or KEi + PEi = KEf + PEf (conservative forces only), (7.48) where i and f denote initial and final values. This equation is a form of the work-energy theorem for conservative forces; it is known as the conservation of mechanical energy principle. Remember that this applies to the extent that all the forces are conservative, so that friction is negligible. The total kinetic plus potential energy of a system is defined to be its mechanical 280 Chapter 7 \| Work, Energy, and Energy Resources energy, (KE + PE) . In a system that experiences only conservative forces, there is a potential energy associated with each force, and the energy only changes form between KE and the various types of PE , with the total energy remaining constant. The internal energy of a system is the sum of the kinetic energies of all of its elements, plus the potential energy due to all of the interactions due to conservative forces between all of the elements. Real World Connections Consider a wind-up toy, such
as a car. It uses a spring system to store energy. The amount of energy stored depends only on how many times it is wound, not how quickly or slowly the winding happens. Similarly, a dart gun using compressed air stores energy in its internal structure. In this case, the energy stored inside depends only on how many times it is pumped, not how quickly or slowly the pumping is done. The total energy put into the system, whether through winding or pumping, is equal to the total energy conserved in the system (minus any energy loss in the system due to interactions between its parts, such as air leaks in the dart gun). Since the internal energy of the system is conserved, you can calculate the amount of stored energy by measuring the kinetic energy of the system (the moving car or dart) when the potential energy is released. Example 7.8 Using Conservation of Mechanical Energy to Calculate the Speed of a Toy Car A 0.100-kg toy car is propelled by a compressed spring, as shown in Figure 7.12. The car follows a track that rises 0.180 m above the starting point. The spring is compressed 4.00 cm and has a force constant of 250.0 N/m. Assuming work done by friction to be negligible, find (a) how fast the car is going before it starts up the slope and (b) how fast it is going at the top of the slope. Figure 7.12 A toy car is pushed by a compressed spring and coasts up a slope. Assuming negligible friction, the potential energy in the spring is first completely converted to kinetic energy, and then to a combination of kinetic and gravitational potential energy as the car rises. The details of the path are unimportant because all forces are conservative—the car would have the same final speed if it took the alternate path shown. Strategy The spring force and the gravitational force are conservative forces, so conservation of mechanical energy can be used. Thus, or KEi +PEi = KEf + PEf 1 2i 2 + i + 1 2i 2 = 1 2f 2 + f + 1 2f 2, (7.49) (7.50) where is the height (vertical position) and is the compression of the spring. This general statement looks complex but becomes much simpler when we start considering specific situations. First, we must identify the initial and final conditions in a problem; then, we enter them into the last equation to solve for an unknown. Solution for (a) This part of the problem is limited to conditions just before the car is released and just after it leaves the spring. Take the initial height to be zero, so that both i and f are zero. Furthermore, the initial speed i is zero and the final compression of the spring f is zero, and so several terms in the conservation of mechanical energy equation are zero and it simplifies to 1 2i 2 = 1 2f 2. (7.51) In other words, the initial potential energy in the spring is converted completely to kinetic energy in the absence of friction. Solving for the final speed and entering known values yields This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources f = i = 250.0 N/m 0.100 kg = 2.00 m/s. (0.0400 m) 281 (7.52) Solution for (b) One method of finding the speed at the top of the slope is to consider conditions just before the car is released and just after it reaches the top of the slope, completely ignoring everything in between. Doing the same type of analysis to find which terms are zero, the conservation of mechanical energy becomes 2 = 1 1 2i 2f 2 + f. (7.53) This form of the equation means that the spring’s initial potential energy is converted partly to gravitational potential energy and partly to kinetic energy. The final speed at the top of the slope will be less than at the bottom. Solving for f and substituting known values gives f = 2 i − 2f 250.0 N/m 0.100 kg (0.0400 m)2 − 2(9.80 m/s2)(0.180 m) (7.54) = Discussion = 0.687 m/s. Another way to solve this problem is to realize that the car’s kinetic energy before it goes up the slope is converted partly to potential energy—that is, to take the final conditions in part (a) to be the initial conditions in part (b). Applying the Science Practices: Potential Energy in a Spring Suppose you are running an experiment in which two 250 g carts connected by a spring (with spring constant 120 N/m) are run into a solid block, and the compression of the spring is measured. In one run of this experiment, the spring was measured to compress from its rest length of 5.0 cm to a minimum length of 2.0 cm. What was the potential energy stored in this system? Answer Note that the change in length of the spring is 3.0 cm. Hence we can apply Equation 7.42 to find that the potential energy is PE = (1/2)(120 N/m)(0.030 m)2 = 0.0541 J. Note that, for conservative forces, we do not directly calculate the work they do; rather, we consider their effects through their corresponding potential energies, just as we did in Example 7.8. Note also that we do not consider details of the path taken—only the starting and ending points are important (as long as the path is not impossible). This assumption is usually a tremendous simplification, because the path may be complicated and forces may vary along the way. PhET Explorations: Energy Skate Park Learn about conservation of energy with a skater dude! Build tracks, ramps and jumps for the skater and view the kinetic energy, potential energy and friction as he moves. You can also take the skater to different planets or even space! Figure 7.13 Energy Skate Park (http://cnx.org/content/m55076/1.4/energy-skate-park_en.jar) 282 Chapter 7 \| Work, Energy, and Energy Resources 7.5 Nonconservative Forces By the end of this section, you will be able to: Learning Objectives • Define nonconservative forces and explain how they affect mechanical energy. • Show how the principle of conservation of energy can be applied by treating the conservative forces in terms of their potential energies and any nonconservative forces in terms of the work they do. The information presented in this section supports the following AP® learning objectives and science practices: • 4.C.1.2 The student is able to predict changes in the total energy of a system due to changes in position and speed of objects or frictional interactions within the system. (S.P. 6.4) • 4.C.2.1 The student is able to make predictions about the changes in the mechanical energy of a system when a component of an external force acts parallel or antiparallel to the direction of the displacement of the center of mass. (S.P. 6.4) Nonconservative Forces and Friction Forces are either conservative or nonconservative. Conservative forces were discussed in Conservative Forces and Potential Energy. A nonconservative force is one for which work depends on the path taken. Friction is a good example of a nonconservative force. As illustrated in Figure 7.14, work done against friction depends on the length of the path between the starting and ending points. Because of this dependence on path, there is no potential energy associated with nonconservative forces. An important characteristic is that the work done by a nonconservative force adds or removes mechanical energy from a system. Friction, for example, creates thermal energy that dissipates, removing energy from the system. Furthermore, even if the thermal energy is retained or captured, it cannot be fully converted back to work, so it is lost or not recoverable in that sense as well. Figure 7.14 The amount of the happy face erased depends on the path taken by the eraser between points A and B, as does the work done against friction. Less work is done and less of the face is erased for the path in (a) than for the path in (b). The force here is friction, and most of the work goes into thermal energy that subsequently leaves the system (the happy face plus the eraser). The energy expended cannot be fully recovered. How Nonconservative Forces Affect Mechanical Energy Mechanical energy may not be conserved when nonconservative forces act. For example, when a car is brought to a stop by friction on level ground, it loses kinetic energy, which is dissipated as thermal energy, reducing its mechanical energy. Figure 7.15 compares the effects of conservative and nonconservative forces. We often choose to understand simpler systems such as that described in Figure 7.15(a) first before studying more complicated systems as in Figure 7.15(b). This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources 283 Figure 7.15 Comparison of the effects of conservative and nonconservative forces on the mechanical energy of a system. (a) A system with only conservative forces. When a rock is dropped onto a spring, its mechanical energy remains constant (neglecting air resistance) because the force in the spring is conservative. The spring can propel the rock back to its original height, where it once again has only potential energy due to gravity. (b) A system with nonconservative forces. When the same rock is dropped onto the ground, it is stopped by nonconservative forces that dissipate its mechanical energy as thermal energy, sound, and surface distortion. The rock has lost mechanical energy. How the Work-Energy Theorem Applies Now let us consider what form the work-energy theorem takes when both conservative and nonconservative forces act. We will see that the work done by nonconservative forces equals the change in the mechanical energy of a system. As noted in Kinetic Energy and the Work-Energy Theorem, the work-energy theorem states that the net work on a system equals the change in its kinetic energy, or net = ΔKE . The net work is the sum of the work by nonconservative forces plus the work by conservative forces. That is, so that net = nc + c, nc + c = ΔKE, (7.55) (7.56) where nc is the total work done by all nonconservative forces and c is the total work done by all conservative forces. Figure 7.16 A person pushes a crate up a ramp, doing work on the crate. Friction and gravitational force (
not shown) also do work on the crate; both forces oppose the person’s push. As the crate is pushed up the ramp, it gains mechanical energy, implying that the work done by the person is greater than the work done by friction. Consider Figure 7.16, in which a person pushes a crate up a ramp and is opposed by friction. As in the previous section, we note that work done by a conservative force comes from a loss of gravitational potential energy, so that c = −ΔPE . Substituting this equation into the previous one and solving for nc gives nc = ΔKE + ΔPE. (7.57) This equation means that the total mechanical energy (KE + PE) changes by exactly the amount of work done by nonconservative forces. In Figure 7.16, this is the work done by the person minus the work done by friction. So even if energy is not conserved for the system of interest (such as the crate), we know that an equal amount of work was done to cause the change in total mechanical energy. We rearrange nc = ΔKE + ΔPE to obtain KEi +PEi + nc = KEf + PEf . (7.58) 284 Chapter 7 \| Work, Energy, and Energy Resources This means that the amount of work done by nonconservative forces adds to the mechanical energy of a system. If nc is positive, then mechanical energy is increased, such as when the person pushes the crate up the ramp in Figure 7.16. If nc is negative, then mechanical energy is decreased, such as when the rock hits the ground in Figure 7.15(b). If nc is zero, then mechanical energy is conserved, and nonconservative forces are balanced. For example, when you push a lawn mower at constant speed on level ground, your work done is removed by the work of friction, and the mower has a constant energy. Applying Energy Conservation with Nonconservative Forces When no change in potential energy occurs, applying KEi +PEi + nc = KEf + PEf amounts to applying the work-energy theorem by setting the change in kinetic energy to be equal to the net work done on the system, which in the most general case includes both conservative and nonconservative forces. But when seeking instead to find a change in total mechanical energy in situations that involve changes in both potential and kinetic energy, the previous equation KEi + PEi + nc = KEf + PEf says that you can start by finding the change in mechanical energy that would have resulted from just the conservative forces, including the potential energy changes, and add to it the work done, with the proper sign, by any nonconservative forces involved. Example 7.9 Calculating Distance Traveled: How Far a Baseball Player Slides Consider the situation shown in Figure 7.17, where a baseball player slides to a stop on level ground. Using energy considerations, calculate the distance the 65.0-kg baseball player slides, given that his initial speed is 6.00 m/s and the force of friction against him is a constant 450 N. Figure 7.17 The baseball player slides to a stop in a distance . In the process, friction removes the player’s kinetic energy by doing an amount of work equal to the initial kinetic energy. Strategy Friction stops the player by converting his kinetic energy into other forms, including thermal energy. In terms of the workenergy theorem, the work done by friction, which is negative, is added to the initial kinetic energy to reduce it to zero. The work done by friction is negative, because f is in the opposite direction of the motion (that is, = 180º , and so cos = −1 ). Thus nc = − . The equation simplifies to or 1 2i 2 − = 0 = 1 2i 2. This equation can now be solved for the distance . Solution Solving the previous equation for and substituting known values yields (7.59) (7.60) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources = = 2 i 2 (65.0 kg)(6.00 m/s)2 (2)(450 N) = 2.60 m. 285 (7.61) Discussion The most important point of this example is that the amount of nonconservative work equals the change in mechanical energy. For example, you must work harder to stop a truck, with its large mechanical energy, than to stop a mosquito. Example 7.10 Calculating Distance Traveled: Sliding Up an Incline Suppose that the player from Example 7.9 is running up a hill having a 5.00º incline upward with a surface similar to that in the baseball stadium. The player slides with the same initial speed. Determine how far he slides. Figure 7.18 The same baseball player slides to a stop on a 5.00º slope. Strategy In this case, the work done by the nonconservative friction force on the player reduces the mechanical energy he has from his kinetic energy at zero height, to the final mechanical energy he has by moving through distance to reach height along the hill, with = sin 5.00º . This is expressed by the equation KE + PEi + nc = KEf + PEf . (7.62) Solution The work done by friction is again nc = − ; initially the potential energy is PEi = ⋅ 0 = 0 and the kinetic energy is KEi = 1 2 ; the final energy contributions are KEf = 0 for the kinetic energy and PEf = = sin for the potential energy. 2i Substituting these values gives Solve this for to obtain 1 2i 2 + 0 + − = 0 + sin = 2 1 2 i + sin (7.63) (7.64) = (0.5)(65.0 kg)(6.00 m/s)2 450 N+(65.0 kg)(9.80 m/s2) sin (5.00º) = 2.31 m. Discussion As might have been expected, the player slides a shorter distance by sliding uphill. Note that the problem could also have been solved in terms of the forces directly and the work energy theorem, instead of using the potential energy. This method would have required combining the normal force and force of gravity vectors, which no longer cancel each other because they point in different directions, and friction, to find the net force. You could then use the net force and the net work to find the distance that reduces the kinetic energy to zero. By applying conservation of energy and using the potential energy 286 Chapter 7 \| Work, Energy, and Energy Resources instead, we need only consider the gravitational potential energy , without combining and resolving force vectors. This simplifies the solution considerably. Making Connections: Take-Home Investigation—Determining Friction from the Stopping Distance This experiment involves the conversion of gravitational potential energy into thermal energy. Use the ruler, book, and marble from Making Connections: Take-Home Investigation—Converting Potential to Kinetic Energy. In addition, you will need a foam cup with a small hole in the side, as shown in Figure 7.19. From the 10-cm position on the ruler, let the marble roll into the cup positioned at the bottom of the ruler. Measure the distance the cup moves before stopping. What forces caused it to stop? What happened to the kinetic energy of the marble at the bottom of the ruler? Next, place the marble at the 20-cm and the 30-cm positions and again measure the distance the cup moves after the marble enters it. Plot the distance the cup moves versus the initial marble position on the ruler. Is this relationship linear? With some simple assumptions, you can use these data to find the coefficient of kinetic friction k of the cup on the table. The force of friction on the cup is k , where the normal force is just the weight of the cup plus the marble. The normal force and force of gravity do no work because they are perpendicular to the displacement of the cup, which moves horizontally. The work done by friction is . You will need the mass of the marble as well to calculate its initial kinetic energy. It is interesting to do the above experiment also with a steel marble (or ball bearing). Releasing it from the same positions on the ruler as you did with the glass marble, is the velocity of this steel marble the same as the velocity of the marble at the bottom of the ruler? Is the distance the cup moves proportional to the mass of the steel and glass marbles? Figure 7.19 Rolling a marble down a ruler into a foam cup. PhET Explorations: The Ramp Explore forces, energy and work as you push household objects up and down a ramp. Lower and raise the ramp to see how the angle of inclination affects the parallel forces acting on the file cabinet. Graphs show forces, energy and work. Figure 7.20 The Ramp (http://cnx.org/content/m55047/1.4/the-ramp_en.jar) 7.6 Conservation of Energy By the end of this section, you will be able to: Learning Objectives • Explain the law of the conservation of energy. • Describe some of the many forms of energy. • Define efficiency of an energy conversion process as the fraction left as useful energy or work, rather than being transformed, for example, into thermal energy. The information presented in this section supports the following AP® learning objectives and science practices: • 4.C.1.2 The student is able to predict changes in the total energy of a system due to changes in position and speed of objects or frictional interactions within the system. (S.P. 6.4) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources 287 • 4.C.2.1 The student is able to make predictions about the changes in the mechanical energy of a system when a component of an external force acts parallel or antiparallel to the direction of the displacement of the center of mass. (S.P. 6.4) • 4.C.2.2 The student is able to apply the concepts of conservation of energy and the work-energy theorem to determine qualitatively and/or quantitatively that work done on a two-object system in linear motion will change the kinetic energy of the center of mass of the system, the potential energy of the systems, and/or the internal energy of the system. (S.P. 1.4, 2.2, 7.2) • 5.A.2.1 The student is able to define open and closed systems for everyday situations and apply conservation concepts for energy, charge, and linear momentum to those situations. (S.P. 6.4, 7.2) • 5.B.5.4 The student is able to make claims about the interaction between a system and its environment in which the environment exerts a force on the system, thus doing work on t
he system and changing the energy of the system (kinetic energy plus potential energy). (S.P. 6.4, 7.2) • 5.B.5.5 The student is able to predict and calculate the energy transfer to (i.e., the work done on) an object or system from information about a force exerted on the object or system through a distance. (S.P. 2.2, 6.4) Law of Conservation of Energy Energy, as we have noted, is conserved, making it one of the most important physical quantities in nature. The law of conservation of energy can be stated as follows: Total energy is constant in any process. It may change in form or be transferred from one system to another, but the total remains the same. We have explored some forms of energy and some ways it can be transferred from one system to another. This exploration led to the definition of two major types of energy—mechanical energy (KE + PE) and energy transferred via work done by nonconservative forces (nc) . But energy takes many other forms, manifesting itself in many different ways, and we need to be able to deal with all of these before we can write an equation for the above general statement of the conservation of energy. Other Forms of Energy than Mechanical Energy At this point, we deal with all other forms of energy by lumping them into a single group called other energy ( OE ). Then we can state the conservation of energy in equation form as KEi + PEi + nc + OEi = KEf + PEf + OEf. (7.65) All types of energy and work can be included in this very general statement of conservation of energy. Kinetic energy is KE , work done by a conservative force is represented by PE , work done by nonconservative forces is nc , and all other energies are included as OE . This equation applies to all previous examples; in those situations OE was constant, and so it subtracted out and was not directly considered. Making Connections: Usefulness of the Energy Conservation Principle The fact that energy is conserved and has many forms makes it very important. You will find that energy is discussed in many contexts, because it is involved in all processes. It will also become apparent that many situations are best understood in terms of energy and that problems are often most easily conceptualized and solved by considering energy. When does OE play a role? One example occurs when a person eats. Food is oxidized with the release of carbon dioxide, water, and energy. Some of this chemical energy is converted to kinetic energy when the person moves, to potential energy when the person changes altitude, and to thermal energy (another form of OE ). Some of the Many Forms of Energy What are some other forms of energy? You can probably name a number of forms of energy not yet discussed. Many of these will be covered in later chapters, but let us detail a few here. Electrical energy is a common form that is converted to many other forms and does work in a wide range of practical situations. Fuels, such as gasoline and food, carry chemical energy that can be transferred to a system through oxidation. Chemical fuel can also produce electrical energy, such as in batteries. Batteries can in turn produce light, which is a very pure form of energy. Most energy sources on Earth are in fact stored energy from the energy we receive from the Sun. We sometimes refer to this as radiant energy, or electromagnetic radiation, which includes visible light, infrared, and ultraviolet radiation. Nuclear energy comes from processes that convert measurable amounts of mass into energy. Nuclear energy is transformed into the energy of sunlight, into electrical energy in power plants, and into the energy of the heat transfer and blast in weapons. Atoms and molecules inside all objects are in random motion. This internal mechanical energy from the random motions is called thermal energy, because it is related to the temperature of the object. These and all other forms of energy can be converted into one another and can do work. 288 Chapter 7 \| Work, Energy, and Energy Resources Real World Connections: Open or Closed System? Consider whether the following systems are open or closed: a car, a spring-operated dart gun, and the system shown in Figure 7.15(a). A car is not a closed system. You add energy in the form of more gas in the tank (or charging the batteries), and energy is lost due to air resistance and friction. A spring-operated dart gun is not a closed system. You have to initially compress the spring. Once that has been done, however, the dart gun and dart can be treated as a closed system. All of the energy remains in the system consisting of these two objects. Figure 7.15(a) is an example of a closed system, once it has been started. All of the energy in the system remains there; none is brought in from outside or leaves. Table 7.1 gives the amount of energy stored, used, or released from various objects and in various phenomena. The range of energies and the variety of types and situations is impressive. Problem-Solving Strategies for Energy You will find the following problem-solving strategies useful whenever you deal with energy. The strategies help in organizing and reinforcing energy concepts. In fact, they are used in the examples presented in this chapter. The familiar general problem-solving strategies presented earlier—involving identifying physical principles, knowns, and unknowns, checking units, and so on—continue to be relevant here. Step 1. Determine the system of interest and identify what information is given and what quantity is to be calculated. A sketch will help. Step 2. Examine all the forces involved and determine whether you know or are given the potential energy from the work done by the forces. Then use step 3 or step 4. Step 3. If you know the potential energies for the forces that enter into the problem, then forces are all conservative, and you can apply conservation of mechanical energy simply in terms of potential and kinetic energy. The equation expressing conservation of energy is KEi + PEi = KEf + PEf. (7.66) Step 4. If you know the potential energy for only some of the forces, possibly because some of them are nonconservative and do not have a potential energy, or if there are other energies that are not easily treated in terms of force and work, then the conservation of energy law in its most general form must be used. KEi + PEi + nc + OEi = KEf + PEf + OEf. (7.67) In most problems, one or more of the terms is zero, simplifying its solution. Do not calculate c , the work done by conservative forces; it is already incorporated in the PE terms. Step 5. You have already identified the types of work and energy involved (in step 2). Before solving for the unknown, eliminate terms wherever possible to simplify the algebra. For example, choose = 0 at either the initial or final point, so that PEg is zero there. Then solve for the unknown in the customary manner. Step 6. Check the answer to see if it is reasonable. Once you have solved a problem, reexamine the forms of work and energy to see if you have set up the conservation of energy equation correctly. For example, work done against friction should be negative, potential energy at the bottom of a hill should be less than that at the top, and so on. Also check to see that the numerical value obtained is reasonable. For example, the final speed of a skateboarder who coasts down a 3-mhigh ramp could reasonably be 20 km/h, but not 80 km/h. Transformation of Energy The transformation of energy from one form into others is happening all the time. The chemical energy in food is converted into thermal energy through metabolism; light energy is converted into chemical energy through photosynthesis. In a larger example, the chemical energy contained in coal is converted into thermal energy as it burns to turn water into steam in a boiler. This thermal energy in the steam in turn is converted to mechanical energy as it spins a turbine, which is connected to a generator to produce electrical energy. (In all of these examples, not all of the initial energy is converted into the forms mentioned. This important point is discussed later in this section.) Another example of energy conversion occurs in a solar cell. Sunlight impinging on a solar cell (see Figure 7.21) produces electricity, which in turn can be used to run an electric motor. Energy is converted from the primary source of solar energy into electrical energy and then into mechanical energy. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources 289 Figure 7.21 Solar energy is converted into electrical energy by solar cells, which is used to run a motor in this solar-power aircraft. (credit: NASA) 290 Chapter 7 \| Work, Energy, and Energy Resources Table 7.1 Energy of Various Objects and Phenomena Object/phenomenon Energy in joules Big Bang Energy released in a supernova Fusion of all the hydrogen in Earth’s oceans Annual world energy use Large fusion bomb (9 megaton) 1 kg hydrogen (fusion to helium) 1 kg uranium (nuclear fission) Hiroshima-size fission bomb (10 kiloton) 90,000-ton aircraft carrier at 30 knots 1 barrel crude oil 1 ton TNT 1 gallon of gasoline 1068 1044 1034 41020 3.81016 6.41014 8.01013 4.21013 1.11010 5.9109 4.2109 1.2108 Daily home electricity use (developed countries) 7107 Daily adult food intake (recommended) 1000-kg car at 90 km/h 1 g fat (9.3 kcal) ATP hydrolysis reaction 1 g carbohydrate (4.1 kcal) 1 g protein (4.1 kcal) Tennis ball at 100 km/h Mosquito 10–2 g at 0.5 m/s Single electron in a TV tube beam Energy to break one DNA strand 1.2107 3.1105 3.9104 3.2104 1.7104 1.7104 22 1.310−6 4.010−15 10−19 Efficiency Even though energy is conserved in an energy conversion process, the output of useful energy or work will be less than the energy input. The efficiency of an energy conversion process is defined as Efficiency( ) = useful energy or work output total energy input = out in . (7.68) Table 7.2 lists some efficiencies of
mechanical devices and human activities. In a coal-fired power plant, for example, about 40% of the chemical energy in the coal becomes useful electrical energy. The other 60% transforms into other (perhaps less useful) energy forms, such as thermal energy, which is then released to the environment through combustion gases and cooling towers. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources 291 Table 7.2 Efficiency of the Human Body and Mechanical Devices Activity/device Efficiency (%)[1] Cycling and climbing Swimming, surface Swimming, submerged Shoveling Weightlifting Steam engine Gasoline engine Diesel engine Nuclear power plant Coal power plant Electric motor Compact fluorescent light Gas heater (residential) Solar cell 20 2 4 3 9 17 30 35 35 42 98 20 90 10 PhET Explorations: Masses and Springs A realistic mass and spring laboratory. Hang masses from springs and adjust the spring stiffness and damping. You can even slow time. Transport the lab to different planets. A chart shows the kinetic, potential, and thermal energies for each spring. Figure 7.22 Masses and Springs (http://cnx.org/content/m55049/1.3/mass-spring-lab_en.jar) 7.7 Power What is Power? Learning Objectives By the end of this section, you will be able to: • Calculate power by calculating changes in energy over time. • Examine power consumption and calculations of the cost of energy consumed. Power—the word conjures up many images: a professional football player muscling aside his opponent, a dragster roaring away from the starting line, a volcano blowing its lava into the atmosphere, or a rocket blasting off, as in Figure 7.23. 1. Representative values 292 Chapter 7 \| Work, Energy, and Energy Resources Figure 7.23 This powerful rocket on the Space Shuttle Endeavor did work and consumed energy at a very high rate. (credit: NASA) These images of power have in common the rapid performance of work, consistent with the scientific definition of power ( ) as the rate at which work is done. Power Power is the rate at which work is done. The SI unit for power is the watt ( W ), where 1 watt equals 1 joule/second (1 W = 1 J/s). = (7.69) Because work is energy transfer, power is also the rate at which energy is expended. A 60-W light bulb, for example, expends 60 J of energy per second. Great power means a large amount of work or energy developed in a short time. For example, when a powerful car accelerates rapidly, it does a large amount of work and consumes a large amount of fuel in a short time. Calculating Power from Energy Example 7.11 Calculating the Power to Climb Stairs What is the power output for a 60.0-kg woman who runs up a 3.00 m high flight of stairs in 3.50 s, starting from rest but having a final speed of 2.00 m/s? (See Figure 7.24.) Figure 7.24 When this woman runs upstairs starting from rest, she converts the chemical energy originally from food into kinetic energy and gravitational potential energy. Her power output depends on how fast she does this. Strategy and Concept This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources 293 The work going into mechanical energy is = KE + PE . At the bottom of the stairs, we take both KE and PEg as initially zero; thus, = KEf + PEg = 1 given, we can calculate and then divide it by time to get power. 2 + , where is the vertical height of the stairs. Because all terms are 2f Solution Substituting the expression for into the definition of power given in the previous equation, = / yields = = 1 2f 2 + . Entering known values yields = 0.5 60.0 kg (2.00 m/s)2 + 60.0 kg 9.80 m/s2 (3.00 m) 3.50 s (7.70) (7.71) = 120 J + 1764 J 3.50 s = 538 W. Discussion The woman does 1764 J of work to move up the stairs compared with only 120 J to increase her kinetic energy; thus, most of her power output is required for climbing rather than accelerating. It is impressive that this woman’s useful power output is slightly less than 1 horsepower (1 hp = 746 W) ! People can generate more than a horsepower with their leg muscles for short periods of time by rapidly converting available blood sugar and oxygen into work output. (A horse can put out 1 hp for hours on end.) Once oxygen is depleted, power output decreases and the person begins to breathe rapidly to obtain oxygen to metabolize more food—this is known as the aerobic stage of exercise. If the woman climbed the stairs slowly, then her power output would be much less, although the amount of work done would be the same. Making Connections: Take-Home Investigation—Measure Your Power Rating Determine your own power rating by measuring the time it takes you to climb a flight of stairs. We will ignore the gain in kinetic energy, as the above example showed that it was a small portion of the energy gain. Don’t expect that your output will be more than about 0.5 hp. Examples of Power Examples of power are limited only by the imagination, because there are as many types as there are forms of work and energy. (See Table 7.3 for some examples.) Sunlight reaching Earth’s surface carries a maximum power of about 1.3 kilowatts per square meter (kW/m2). A tiny fraction of this is retained by Earth over the long term. Our consumption rate of fossil fuels is far greater than the rate at which they are stored, so it is inevitable that they will be depleted. Power implies that energy is transferred, perhaps changing form. It is never possible to change one form completely into another without losing some of it as thermal energy. For example, a 60-W incandescent bulb converts only 5 W of electrical power to light, with 55 W dissipating into thermal energy. Furthermore, the typical electric power plant converts only 35 to 40% of its fuel into electricity. The remainder becomes a huge amount of thermal energy that must be dispersed as heat transfer, as rapidly as it is created. A coal-fired power plant may produce 1000 megawatts; 1 megawatt (MW) is 106 W of electric power. But the power plant consumes chemical energy at a rate of about 2500 MW, creating heat transfer to the surroundings at a rate of 1500 MW. (See Figure 7.25.) 294 Chapter 7 \| Work, Energy, and Energy Resources Figure 7.25 Tremendous amounts of electric power are generated by coal-fired power plants such as this one in China, but an even larger amount of power goes into heat transfer to the surroundings. The large cooling towers here are needed to transfer heat as rapidly as it is produced. The transfer of heat is not unique to coal plants but is an unavoidable consequence of generating electric power from any fuel—nuclear, coal, oil, natural gas, or the like. (credit: Kleinolive, Wikimedia Commons) Table 7.3 Power Output or Consumption Object or Phenomenon Power in Watts Supernova (at peak) Milky Way galaxy Crab Nebula pulsar The Sun Volcanic eruption (maximum) Lightning bolt Nuclear power plant (total electric and heat transfer) Aircraft carrier (total useful and heat transfer) Dragster (total useful and heat transfer) Car (total useful and heat transfer) Football player (total useful and heat transfer) Clothes dryer Person at rest (all heat transfer) 51037 1037 1028 41026 41015 21012 3109 108 2106 8104 5103 4103 100 Typical incandescent light bulb (total useful and heat transfer) 60 Heart, person at rest (total useful and heat transfer) Electric clock Pocket calculator 8 3 10−3 Power and Energy Consumption We usually have to pay for the energy we use. It is interesting and easy to estimate the cost of energy for an electrical appliance if its power consumption rate and time used are known. The higher the power consumption rate and the longer the appliance is used, the greater the cost of that appliance. The power consumption rate is = / = / , where is the energy supplied by the electricity company. So the energy consumed over a time is This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources = 295 (7.72) Electricity bills state the energy used in units of kilowatt-hours (kW ⋅ h) which is the product of power in kilowatts and time in hours. This unit is convenient because electrical power consumption at the kilowatt level for hours at a time is typical. Example 7.12 Calculating Energy Costs What is the cost of running a 0.200-kW computer 6.00 h per day for 30.0 d if the cost of electricity is $0.120 per kW ⋅ h ? Strategy Cost is based on energy consumed; thus, we must find from = and then calculate the cost. Because electrical energy is expressed in kW ⋅ h , at the start of a problem such as this it is convenient to convert the units into kW and hours. Solution The energy consumed in kW ⋅ h is = = (0.200 kW)(6.00 h/d)(30.0 d) = 36.0 kW ⋅ h, and the cost is simply given by cost = (36.0 kW ⋅ h)($0.120 per kW ⋅ h) = $4.32 per month. Discussion (7.73) (7.74) The cost of using the computer in this example is neither exorbitant nor negligible. It is clear that the cost is a combination of power and time. When both are high, such as for an air conditioner in the summer, the cost is high. The motivation to save energy has become more compelling with its ever-increasing price. Armed with the knowledge that energy consumed is the product of power and time, you can estimate costs for yourself and make the necessary value judgments about where to save energy. Either power or time must be reduced. It is most cost-effective to limit the use of highpower devices that normally operate for long periods of time, such as water heaters and air conditioners. This would not include relatively high power devices like toasters, because they are on only a few minutes per day. It would also not include electric clocks, in spite of their 24-hour-per-day usage, because they are very low power devices. It is sometimes possible to use devices that have greater efficiencies—that is, devices that consume less power to accomplish the same task. One
example is the compact fluorescent light bulb, which produces over four times more light per watt of power consumed than its incandescent cousin. Modern civilization depends on energy, but current levels of energy consumption and production are not sustainable. The likelihood of a link between global warming and fossil fuel use (with its concomitant production of carbon dioxide), has made reduction in energy use as well as a shift to non-fossil fuels of the utmost importance. Even though energy in an isolated system is a conserved quantity, the final result of most energy transformations is waste heat transfer to the environment, which is no longer useful for doing work. As we will discuss in more detail in Thermodynamics, the potential for energy to produce useful work has been “degraded” in the energy transformation. 7.8 Work, Energy, and Power in Humans By the end of this section, you will be able to: Learning Objectives • Explain the human body’s consumption of energy when at rest versus when engaged in activities that do useful work. • Calculate the conversion of chemical energy in food into useful work. Energy Conversion in Humans Our own bodies, like all living organisms, are energy conversion machines. Conservation of energy implies that the chemical energy stored in food is converted into work, thermal energy, and/or stored as chemical energy in fatty tissue. (See Figure 7.26.) The fraction going into each form depends both on how much we eat and on our level of physical activity. If we eat more than is needed to do work and stay warm, the remainder goes into body fat. Chapter 7 \| Work, Energy, and Energy Resources 301 bookkeeping device, and no exceptions have ever been found. It states that the total amount of energy in an isolated system will always remain constant. Related to this principle, but remarkably different from it, is the important philosophy of energy conservation. This concept has to do with seeking to decrease the amount of energy used by an individual or group through (1) reduced activities (e.g., turning down thermostats, driving fewer kilometers) and/or (2) increasing conversion efficiencies in the performance of a particular task—such as developing and using more efficient room heaters, cars that have greater miles-pergallon ratings, energy-efficient compact fluorescent lights, etc. Since energy in an isolated system is not destroyed or created or generated, one might wonder why we need to be concerned about our energy resources, since energy is a conserved quantity. The problem is that the final result of most energy transformations is waste heat transfer to the environment and conversion to energy forms no longer useful for doing work. To state it in another way, the potential for energy to produce useful work has been “degraded” in the energy transformation. (This will be discussed in more detail in Thermodynamics.) Glossary basal metabolic rate: the total energy conversion rate of a person at rest chemical energy: the energy in a substance stored in the bonds between atoms and molecules that can be released in a chemical reaction conservation of mechanical energy: the rule that the sum of the kinetic energies and potential energies remains constant if only conservative forces act on and within a system conservative force: followed a force that does the same work for any given initial and final configuration, regardless of the path efficiency: a measure of the effectiveness of the input of energy to do work; useful energy or work divided by the total input of energy electrical energy: the energy carried by a flow of charge energy: the ability to do work fossil fuels: oil, natural gas, and coal friction: the force between surfaces that opposes one sliding on the other; friction changes mechanical energy into thermal energy gravitational potential energy: the energy an object has due to its position in a gravitational field horsepower: an older non-SI unit of power, with 1 hp = 746 W joule: SI unit of work and energy, equal to one newton-meter kilowatt-hour: (kW ⋅ h) unit used primarily for electrical energy provided by electric utility companies kinetic energy: the energy an object has by reason of its motion, equal to 1 22 for the translational (i.e., non-rotational) motion of an object of mass moving at speed law of conservation of energy: the general law that total energy is constant in any process; energy may change in form or be transferred from one system to another, but the total remains the same mechanical energy: the sum of kinetic energy and potential energy metabolic rate: the rate at which the body uses food energy to sustain life and to do different activities net work: work done by the net force, or vector sum of all the forces, acting on an object nonconservative force: a force whose work depends on the path followed between the given initial and final configurations nuclear energy: energy released by changes within atomic nuclei, such as the fusion of two light nuclei or the fission of a heavy nucleus potential energy: energy due to position, shape, or configuration potential energy of a spring: the stored energy of a spring as a function of its displacement; when Hooke’s law applies, it is given by the expression 1 22 where is the distance the spring is compressed or extended and is the spring constant power: the rate at which work is done 302 Chapter 7 \| Work, Energy, and Energy Resources radiant energy: the energy carried by electromagnetic waves renewable forms of energy: those sources that cannot be used up, such as water, wind, solar, and biomass thermal energy: the energy within an object due to the random motion of its atoms and molecules that accounts for the object's temperature useful work: work done on an external system watt: (W) SI unit of power, with 1 W = 1 J/s work: the transfer of energy by a force that causes an object to be displaced; the product of the component of the force in the direction of the displacement and the magnitude of the displacement work-energy theorem: the result, based on Newton’s laws, that the net work done on an object is equal to its change in kinetic energy Section Summary 7.1 Work: The Scientific Definition • Work is the transfer of energy by a force acting on an object as it is displaced. • The work that a force F does on an object is the product of the magnitude of the force, times the magnitude of the displacement, times the cosine of the angle between them. In symbols, • The SI unit for work and energy is the joule (J), where 1 J = 1 N ⋅ m = 1 kg ⋅ m2/s2 . • The work done by a force is zero if the displacement is either zero or perpendicular to the force. • The work done is positive if the force and displacement have the same direction, and negative if they have opposite direction. = cos . 7.2 Kinetic Energy and the Work-Energy Theorem • The net work net is the work done by the net force acting on an object. • Work done on an object transfers energy to the object. • The translational kinetic energy of an object of mass moving at speed is KE = 1 22 . • The work-energy theorem states that the net work net on a system changes its kinetic energy, net = 1 22 − 1 20 2 . 7.3 Gravitational Potential Energy • Work done against gravity in lifting an object becomes potential energy of the object-Earth system. • The change in gravitational potential energy, ΔPEg , is ΔPEg = , with being the increase in height and g the acceleration due to gravity. • The gravitational potential energy of an object near Earth’s surface is due to its position in the mass-Earth system. Only differences in gravitational potential energy, ΔPEg , have physical significance. • As an object descends without friction, its gravitational potential energy changes into kinetic energy corresponding to increasing speed, so that ΔKE= −ΔPEg . 7.4 Conservative Forces and Potential Energy • A conservative force is one for which work depends only on the starting and ending points of a motion, not on the path taken. • We can define potential energy (PE) for any conservative force, just as we defined PEg for the gravitational force. • The potential energy of a spring is PEs = 1 22 , where is the spring’s force constant and is the displacement from its undeformed position. • Mechanical energy is defined to be KE + PE for a conservative force. • When only conservative forces act on and within a system, the total mechanical energy is constant. In equation form, This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources 303 KE + PE = constant or KEi + PEi = KEf + PEf where i and f denote initial and final values. This is known as the conservation of mechanical energy. 7.5 Nonconservative Forces • A nonconservative force is one for which work depends on the path. • Friction is an example of a nonconservative force that changes mechanical energy into thermal energy. • Work nc done by a nonconservative force changes the mechanical energy of a system. In equation form, nc = ΔKE + ΔPE or, equivalently, KEi + PEi + nc = KEf + PEf . • When both conservative and nonconservative forces act, energy conservation can be applied and used to calculate motion in terms of the known potential energies of the conservative forces and the work done by nonconservative forces, instead of finding the net work from the net force, or having to directly apply Newton’s laws. 7.6 Conservation of Energy • The law of conservation of energy states that the total energy is constant in any process. Energy may change in form or be transferred from one system to another, but the total remains the same. • When all forms of energy are considered, conservation of energy is written in equation form as KEi + PEi + nc + OEi = KEf + PEf + OEf , where OE is all other forms of energy besides mechanical energy. • Commonly encountered forms of energy include electric energy, chemical energy, radiant energy, nuclear energy, and thermal energy. • Energy
is often utilized to do work, but it is not possible to convert all the energy of a system to work. • The efficiency of a machine or human is defined to be = out in , where out is useful work output and in is the energy consumed. 7.7 Power • Power is the rate at which work is done, or in equation form, for the average power for work done over a time , = / . • The SI unit for power is the watt (W), where 1 W = 1 J/s . • The power of many devices such as electric motors is also often expressed in horsepower (hp), where 1 hp = 746 W . 7.8 Work, Energy, and Power in Humans • The human body converts energy stored in food into work, thermal energy, and/or chemical energy that is stored in fatty tissue. • The rate at which the body uses food energy to sustain life and to do different activities is called the metabolic rate, and the corresponding rate when at rest is called the basal metabolic rate (BMR) • The energy included in the basal metabolic rate is divided among various systems in the body, with the largest fraction going to the liver and spleen, and the brain coming next. • About 75% of food calories are used to sustain basic body functions included in the basal metabolic rate. • The energy consumption of people during various activities can be determined by measuring their oxygen use, because the digestive process is basically one of oxidizing food. 7.9 World Energy Use • The relative use of different fuels to provide energy has changed over the years, but fuel use is currently dominated by oil, although natural gas and solar contributions are increasing. • Although non-renewable sources dominate, some countries meet a sizeable percentage of their electricity needs from renewable resources. • The United States obtains only about 10% of its energy from renewable sources, mostly hydroelectric power. • Economic well-being is dependent upon energy use, and in most countries higher standards of living, as measured by GDP (Gross Domestic Product) per capita, are matched by higher levels of energy consumption per capita. • Even though, in accordance with the law of conservation of energy, energy can never be created or destroyed, energy that can be used to do work is always partly converted to less useful forms, such as waste heat to the environment, in all of our uses of energy for practical purposes. Conceptual Questions 7.1 Work: The Scientific Definition 1. Give an example of something we think of as work in everyday circumstances that is not work in the scientific sense. Is energy transferred or changed in form in your example? If so, explain how this is accomplished without doing work. 304 Chapter 7 \| Work, Energy, and Energy Resources 2. Give an example of a situation in which there is a force and a displacement, but the force does no work. Explain why it does no work. 3. Describe a situation in which a force is exerted for a long time but does no work. Explain. 7.2 Kinetic Energy and the Work-Energy Theorem 4. The person in Figure 7.33 does work on the lawn mower. Under what conditions would the mower gain energy? Under what conditions would it lose energy? Figure 7.33 5. Work done on a system puts energy into it. Work done by a system removes energy from it. Give an example for each statement. 6. When solving for speed in Example 7.4, we kept only the positive root. Why? 7.3 Gravitational Potential Energy 7. In Example 7.7, we calculated the final speed of a roller coaster that descended 20 m in height and had an initial speed of 5 m/s downhill. Suppose the roller coaster had had an initial speed of 5 m/s uphill instead, and it coasted uphill, stopped, and then rolled back down to a final point 20 m below the start. We would find in that case that it had the same final speed. Explain in terms of conservation of energy. 8. Does the work you do on a book when you lift it onto a shelf depend on the path taken? On the time taken? On the height of the shelf? On the mass of the book? 7.4 Conservative Forces and Potential Energy 9. What is a conservative force? 10. The force exerted by a diving board is conservative, provided the internal friction is negligible. Assuming friction is negligible, describe changes in the potential energy of a diving board as a swimmer dives from it, starting just before the swimmer steps on the board until just after his feet leave it. 11. Define mechanical energy. What is the relationship of mechanical energy to nonconservative forces? What happens to mechanical energy if only conservative forces act? 12. What is the relationship of potential energy to conservative force? 7.6 Conservation of Energy 13. Consider the following scenario. A car for which friction is not negligible accelerates from rest down a hill, running out of gasoline after a short distance. The driver lets the car coast farther down the hill, then up and over a small crest. He then coasts down that hill into a gas station, where he brakes to a stop and fills the tank with gasoline. Identify the forms of energy the car has, and how they are changed and transferred in this series of events. (See Figure 7.34.) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources 305 Figure 7.34 A car experiencing non-negligible friction coasts down a hill, over a small crest, then downhill again, and comes to a stop at a gas station. 14. Describe the energy transfers and transformations for a javelin, starting from the point at which an athlete picks up the javelin and ending when the javelin is stuck into the ground after being thrown. 15. Do devices with efficiencies of less than one violate the law of conservation of energy? Explain. 16. List four different forms or types of energy. Give one example of a conversion from each of these forms to another form. 17. List the energy conversions that occur when riding a bicycle. 7.7 Power 18. Most electrical appliances are rated in watts. Does this rating depend on how long the appliance is on? (When off, it is a zerowatt device.) Explain in terms of the definition of power. 19. Explain, in terms of the definition of power, why energy consumption is sometimes listed in kilowatt-hours rather than joules. What is the relationship between these two energy units? 20. A spark of static electricity, such as that you might receive from a doorknob on a cold dry day, may carry a few hundred watts of power. Explain why you are not injured by such a spark. 7.8 Work, Energy, and Power in Humans 21. Explain why it is easier to climb a mountain on a zigzag path rather than one straight up the side. Is your increase in gravitational potential energy the same in both cases? Is your energy consumption the same in both? 22. Do you do work on the outside world when you rub your hands together to warm them? What is the efficiency of this activity? 23. Shivering is an involuntary response to lowered body temperature. What is the efficiency of the body when shivering, and is this a desirable value? 24. Discuss the relative effectiveness of dieting and exercise in losing weight, noting that most athletic activities consume food energy at a rate of 400 to 500 W, while a single cup of yogurt can contain 1360 kJ (325 kcal). Specifically, is it likely that exercise alone will be sufficient to lose weight? You may wish to consider that regular exercise may increase the metabolic rate, whereas protracted dieting may reduce it. 7.9 World Energy Use 25. What is the difference between energy conservation and the law of conservation of energy? Give some examples of each. 26. If the efficiency of a coal-fired electrical generating plant is 35%, then what do we mean when we say that energy is a conserved quantity? 306 Chapter 7 \| Work, Energy, and Energy Resources Problems & Exercises 7.1 Work: The Scientific Definition 1. How much work does a supermarket checkout attendant do on a can of soup he pushes 0.600 m horizontally with a force of 5.00 N? Express your answer in joules and kilocalories. 2. A 75.0-kg person climbs stairs, gaining 2.50 meters in height. Find the work done to accomplish this task. 3. (a) Calculate the work done on a 1500-kg elevator car by its cable to lift it 40.0 m at constant speed, assuming friction averages 100 N. (b) What is the work done on the lift by the gravitational force in this process? (c) What is the total work done on the lift? 4. Suppose a car travels 108 km at a speed of 30.0 m/s, and uses 2.0 gal of gasoline. Only 30% of the gasoline goes into useful work by the force that keeps the car moving at constant speed despite friction. (See Table 7.1 for the energy content of gasoline.) (a) What is the magnitude of the force exerted to keep the car moving at constant speed? (b) If the required force is directly proportional to speed, how many gallons will be used to drive 108 km at a speed of 28.0 m/s? 5. Calculate the work done by an 85.0-kg man who pushes a crate 4.00 m up along a ramp that makes an angle of 20.0º with the horizontal. (See Figure 7.35.) He exerts a force of 500 N on the crate parallel to the ramp and moves at a constant speed. Be certain to include the work he does on the crate and on his body to get up the ramp. Figure 7.35 A man pushes a crate up a ramp. shopper exerts, using energy considerations. (e) What is the total work done on the cart? 8. Suppose the ski patrol lowers a rescue sled and victim, having a total mass of 90.0 kg, down a 60.0º slope at constant speed, as shown in Figure 7.37. The coefficient of friction between the sled and the snow is 0.100. (a) How much work is done by friction as the sled moves 30.0 m along the hill? (b) How much work is done by the rope on the sled in this distance? (c) What is the work done by the gravitational force on the sled? (d) What is the total work done? Figure 7.37 A rescue sled and victim are lowered down a steep slope. 7.2 Kinetic Energy and the Work-Energy Theorem 9. Compare the kinetic energy of a 20,000-kg truck moving at 110 km/h with that of an
80.0-kg astronaut in orbit moving at 27,500 km/h. 10. (a) How fast must a 3000-kg elephant move to have the same kinetic energy as a 65.0-kg sprinter running at 10.0 m/ s? (b) Discuss how the larger energies needed for the movement of larger animals would relate to metabolic rates. 6. How much work is done by the boy pulling his sister 30.0 m in a wagon as shown in Figure 7.36? Assume no friction acts on the wagon. 11. Confirm the value given for the kinetic energy of an aircraft carrier in Table 7.1. You will need to look up the definition of a nautical mile (1 knot = 1 nautical mile/h). 12. (a) Calculate the force needed to bring a 950-kg car to rest from a speed of 90.0 km/h in a distance of 120 m (a fairly typical distance for a non-panic stop). (b) Suppose instead the car hits a concrete abutment at full speed and is brought to a stop in 2.00 m. Calculate the force exerted on the car and compare it with the force found in part (a). 13. A car’s bumper is designed to withstand a 4.0-km/h (1.1-m/s) collision with an immovable object without damage to the body of the car. The bumper cushions the shock by absorbing the force over a distance. Calculate the magnitude of the average force on a bumper that collapses 0.200 m while bringing a 900-kg car to rest from an initial speed of 1.1 m/s. 14. Boxing gloves are padded to lessen the force of a blow. (a) Calculate the force exerted by a boxing glove on an opponent’s face, if the glove and face compress 7.50 cm during a blow in which the 7.00-kg arm and glove are brought to rest from an initial speed of 10.0 m/s. (b) Calculate the force exerted by an identical blow in the gory old days when no gloves were used and the knuckles and face would compress only 2.00 cm. (c) Discuss the magnitude of the Figure 7.36 The boy does work on the system of the wagon and the child when he pulls them as shown. 7. A shopper pushes a grocery cart 20.0 m at constant speed on level ground, against a 35.0 N frictional force. He pushes in a direction 25.0º below the horizontal. (a) What is the work done on the cart by friction? (b) What is the work done on the cart by the gravitational force? (c) What is the work done on the cart by the shopper? (d) Find the force the This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources 307 force with glove on. Does it seem high enough to cause damage even though it is lower than the force with no glove? 15. Using energy considerations, calculate the average force a 60.0-kg sprinter exerts backward on the track to accelerate from 2.00 to 8.00 m/s in a distance of 25.0 m, if he encounters a headwind that exerts an average force of 30.0 N against him. 7.3 Gravitational Potential Energy 16. A hydroelectric power facility (see Figure 7.38) converts the gravitational potential energy of water behind a dam to electric energy. (a) What is the gravitational potential energy relative to the generators of a lake of volume 50.0 km3 mass = 5.00×1013 kg) , given that the lake has an average height of 40.0 m above the generators? (b) Compare this with the energy stored in a 9-megaton fusion bomb. ( Figure 7.38 Hydroelectric facility (credit: Denis Belevich, Wikimedia Commons) 17. (a) How much gravitational potential energy (relative to the ground on which it is built) is stored in the Great Pyramid of Cheops, given that its mass is about 7 × 109 kg and its center of mass is 36.5 m above the surrounding ground? (b) How does this energy compare with the daily food intake of a person? 18. Suppose a 350-g kookaburra (a large kingfisher bird) picks up a 75-g snake and raises it 2.5 m from the ground to a branch. (a) How much work did the bird do on the snake? (b) How much work did it do to raise its own center of mass to the branch? 19. In Example 7.7, we found that the speed of a roller coaster that had descended 20.0 m was only slightly greater when it had an initial speed of 5.00 m/s than when it started from rest. This implies that ΔPE >> KEi . Confirm this statement by taking the ratio of ΔPE to KEi . (Note that mass cancels.) 20. A 100-g toy car is propelled by a compressed spring that starts it moving. The car follows the curved track in Figure 7.39. Show that the final speed of the toy car is 0.687 m/s if its initial speed is 2.00 m/s and it coasts up the frictionless slope, gaining 0.180 m in altitude. Figure 7.39 A toy car moves up a sloped track. (credit: Leszek Leszczynski, Flickr) 21. In a downhill ski race, surprisingly, little advantage is gained by getting a running start. (This is because the initial kinetic energy is small compared with the gain in gravitational potential energy on even small hills.) To demonstrate this, find the final speed and the time taken for a skier who skies 70.0 m along a 30º slope neglecting friction: (a) Starting from rest. (b) Starting with an initial speed of 2.50 m/s. (c) Does the answer surprise you? Discuss why it is still advantageous to get a running start in very competitive events. 7.4 Conservative Forces and Potential Energy 22. A 5.00×105-kg subway train is brought to a stop from a speed of 0.500 m/s in 0.400 m by a large spring bumper at the end of its track. What is the force constant of the spring? 23. A pogo stick has a spring with a force constant of 2.50104 N/m , which can be compressed 12.0 cm. To what maximum height can a child jump on the stick using only the energy in the spring, if the child and stick have a total mass of 40.0 kg? Explicitly show how you follow the steps in the Problem-Solving Strategies for Energy. 7.5 Nonconservative Forces 24. A 60.0-kg skier with an initial speed of 12.0 m/s coasts up a 2.50-m-high rise as shown in Figure 7.40. Find her final speed at the top, given that the coefficient of friction between her skis and the snow is 0.0800. (Hint: Find the distance traveled up the incline assuming a straight-line path as shown in the figure.) Figure 7.40 The skier’s initial kinetic energy is partially used in coasting to the top of a rise. 25. (a) How high a hill can a car coast up (engine disengaged) if work done by friction is negligible and its initial speed is 110 km/h? (b) If, in actuality, a 750-kg car with an initial speed of 110 km/h is observed to coast up a hill to a height 22.0 m above its starting point, how much thermal energy was generated by friction? (c) What is the average force of friction if the hill has a slope 2.5º above the horizontal? 308 Chapter 7 \| Work, Energy, and Energy Resources 7.6 Conservation of Energy 26. Using values from Table 7.1, how many DNA molecules could be broken by the energy carried by a single electron in the beam of an old-fashioned TV tube? (These electrons were not dangerous in themselves, but they did create dangerous x rays. Later model tube TVs had shielding that absorbed x rays before they escaped and exposed viewers.) 27. Using energy considerations and assuming negligible air resistance, show that a rock thrown from a bridge 20.0 m above water with an initial speed of 15.0 m/s strikes the water with a speed of 24.8 m/s independent of the direction thrown. 28. If the energy in fusion bombs were used to supply the energy needs of the world, how many of the 9-megaton variety would be needed for a year’s supply of energy (using data from Table 7.1)? This is not as far-fetched as it may sound—there are thousands of nuclear bombs, and their energy can be trapped in underground explosions and converted to electricity, as natural geothermal energy is. 29. (a) Use of hydrogen fusion to supply energy is a dream that may be realized in the next century. Fusion would be a relatively clean and almost limitless supply of energy, as can be seen from Table 7.1. To illustrate this, calculate how many years the present energy needs of the world could be supplied by one millionth of the oceans’ hydrogen fusion energy. (b) How does this time compare with historically significant events, such as the duration of stable economic systems? 7.7 Power 30. The Crab Nebula (see Figure 7.41) pulsar is the remnant of a supernova that occurred in A.D. 1054. Using data from Table 7.3, calculate the approximate factor by which the power output of this astronomical object has declined since its explosion. Figure 7.41 Crab Nebula (credit: ESO, via Wikimedia Commons) 31. Suppose a star 1000 times brighter than our Sun (that is, emitting 1000 times the power) suddenly goes supernova. Using data from Table 7.3: (a) By what factor does its power output increase? (b) How many times brighter than our entire Milky Way galaxy is the supernova? (c) Based on your answers, discuss whether it should be possible to observe supernovas in distant galaxies. Note that there are on the This content is available for free at http://cnx.org/content/col11844/1.13 order of 1011 observable galaxies, the average brightness of which is somewhat less than our own galaxy. 32. A person in good physical condition can put out 100 W of useful power for several hours at a stretch, perhaps by pedaling a mechanism that drives an electric generator. Neglecting any problems of generator efficiency and practical considerations such as resting time: (a) How many people would it take to run a 4.00-kW electric clothes dryer? (b) How many people would it take to replace a large electric power plant that generates 800 MW? 33. What is the cost of operating a 3.00-W electric clock for a year if the cost of electricity is $0.0900 per kW ⋅ h ? 34. A large household air conditioner may consume 15.0 kW of power. What is the cost of operating this air conditioner 3.00 h per day for 30.0 d if the cost of electricity is $0.110 per kW ⋅ h ? 35. (a) What is the average power consumption in watts of an appliance that uses 5.00 kW ⋅ h of energy per day? (b) How many joules of energy does this appliance consume in a year? 36. (a) What is the average useful power output of a person who does 6.00106 J of useful work in 8.00 h? (b) Working at this rate, how l
ong will it take this person to lift 2000 kg of bricks 1.50 m to a platform? (Work done to lift his body can be omitted because it is not considered useful output here.) 37. A 500-kg dragster accelerates from rest to a final speed of 110 m/s in 400 m (about a quarter of a mile) and encounters an average frictional force of 1200 N. What is its average power output in watts and horsepower if this takes 7.30 s? 38. (a) How long will it take an 850-kg car with a useful power output of 40.0 hp (1 hp = 746 W) to reach a speed of 15.0 m/ s, neglecting friction? (b) How long will this acceleration take if the car also climbs a 3.00-m-high hill in the process? 39. (a) Find the useful power output of an elevator motor that lifts a 2500-kg load a height of 35.0 m in 12.0 s, if it also increases the speed from rest to 4.00 m/s. Note that the total mass of the counterbalanced system is 10,000 kg—so that only 2500 kg is raised in height, but the full 10,000 kg is accelerated. (b) What does it cost, if electricity is $0.0900 per kW ⋅ h ? 40. (a) What is the available energy content, in joules, of a battery that operates a 2.00-W electric clock for 18 months? (b) How long can a battery that can supply 8.00104 J run a pocket calculator that consumes energy at the rate of 1.0010−3 W ? 41. (a) How long would it take a 1.50105 engines that produce 100 MW of power to reach a speed of 250 m/s and an altitude of 12.0 km if air resistance were negligible? (b) If it actually takes 900 s, what is the power? (c) Given this power, what is the average force of air resistance if the airplane takes 1200 s? (Hint: You must find the distance the plane travels in 1200 s assuming constant acceleration.) -kg airplane with 42. Calculate the power output needed for a 950-kg car to climb a 2.00º slope at a constant 30.0 m/s while encountering wind resistance and friction totaling 600 N. Explicitly show how you follow the steps in the ProblemSolving Strategies for Energy. 43. (a) Calculate the power per square meter reaching Earth’s upper atmosphere from the Sun. (Take the power output of Chapter 7 \| Work, Energy, and Energy Resources 309 the Sun to be 4.00×1026 W.) (b) Part of this is absorbed and reflected by the atmosphere, so that a maximum of 1.30 kW/m2 reaches Earth’s surface. Calculate the area in km2 of solar energy collectors needed to replace an electric power plant that generates 750 MW if the collectors convert an average of 2.00% of the maximum power into electricity. (This small conversion efficiency is due to the devices themselves, and the fact that the sun is directly overhead only briefly.) With the same assumptions, what area would be needed to meet the United States’ energy needs (1.05×1020 J)? Australia’s energy needs (5.4×1018 J)? China’s energy needs (6.3×1019 J)? (These energy consumption values are from 2006.) 7.8 Work, Energy, and Power in Humans 44. (a) How long can you rapidly climb stairs (116/min) on the 93.0 kcal of energy in a 10.0-g pat of butter? (b) How many flights is this if each flight has 16 stairs? 45. (a) What is the power output in watts and horsepower of a 70.0-kg sprinter who accelerates from rest to 10.0 m/s in 3.00 s? (b) Considering the amount of power generated, do you think a well-trained athlete could do this repetitively for long periods of time? 46. Calculate the power output in watts and horsepower of a shot-putter who takes 1.20 s to accelerate the 7.27-kg shot from rest to 14.0 m/s, while raising it 0.800 m. (Do not include the power produced to accelerate his body.) 3.00 h, and studies for 6.00 h. (Studying consumes energy at the same rate as sitting in class.) 50. What is the efficiency of a subject on a treadmill who puts out work at the rate of 100 W while consuming oxygen at the rate of 2.00 L/min? (Hint: See Table 7.5.) 51. Shoveling snow can be extremely taxing because the arms have such a low efficiency in this activity. Suppose a person shoveling a footpath metabolizes food at the rate of 800 W. (a) What is her useful power output? (b) How long will it take her to lift 3000 kg of snow 1.20 m? (This could be the amount of heavy snow on 20 m of footpath.) (c) How much waste heat transfer in kilojoules will she generate in the process? 52. Very large forces are produced in joints when a person jumps from some height to the ground. (a) Calculate the magnitude of the force produced if an 80.0-kg person jumps from a 0.600–m-high ledge and lands stiffly, compressing joint material 1.50 cm as a result. (Be certain to include the weight of the person.) (b) In practice the knees bend almost involuntarily to help extend the distance over which you stop. Calculate the magnitude of the force produced if the stopping distance is 0.300 m. (c) Compare both forces with the weight of the person. 53. Jogging on hard surfaces with insufficiently padded shoes produces large forces in the feet and legs. (a) Calculate the magnitude of the force needed to stop the downward motion of a jogger’s leg, if his leg has a mass of 13.0 kg, a speed of 6.00 m/s, and stops in a distance of 1.50 cm. (Be certain to include the weight of the 75.0-kg jogger’s body.) (b) Compare this force with the weight of the jogger. 54. (a) Calculate the energy in kJ used by a 55.0-kg woman who does 50 deep knee bends in which her center of mass is lowered and raised 0.400 m. (She does work in both directions.) You may assume her efficiency is 20%. (b) What is the average power consumption rate in watts if she does this in 3.00 min? 55. Kanellos Kanellopoulos flew 119 km from Crete to Santorini, Greece, on April 23, 1988, in the Daedalus 88, an aircraft powered by a bicycle-type drive mechanism (see Figure 7.43). His useful power output for the 234-min trip was about 350 W. Using the efficiency for cycling from Table 7.2, calculate the food energy in kilojoules he metabolized during the flight. Figure 7.42 Shot putter at the Dornoch Highland Gathering in 2007. (credit: John Haslam, Flickr) 47. (a) What is the efficiency of an out-of-condition professor who does 2.10105 J of useful work while metabolizing 500 kcal of food energy? (b) How many food calories would a well-conditioned athlete metabolize in doing the same work with an efficiency of 20%? 48. Energy that is not utilized for work or heat transfer is converted to the chemical energy of body fat containing about 39 kJ/g. How many grams of fat will you gain if you eat 10,000 kJ (about 2500 kcal) one day and do nothing but sit relaxed for 16.0 h and sleep for the other 8.00 h? Use data from Table 7.5 for the energy consumption rates of these activities. Figure 7.43 The Daedalus 88 in flight. (credit: NASA photo by Beasley) 49. Using data from Table 7.5, calculate the daily energy needs of a person who sleeps for 7.00 h, walks for 2.00 h, attends classes for 4.00 h, cycles for 2.00 h, sits relaxed for 56. The swimmer shown in Figure 7.44 exerts an average horizontal backward force of 80.0 N with his arm during each 1.80 m long stroke. (a) What is his work output in each 310 Chapter 7 \| Work, Energy, and Energy Resources stroke? (b) Calculate the power output of his arms if he does 120 strokes per minute. long time? Discuss why exercise is necessary but may not be sufficient to cause a person to lose weight. Figure 7.44 57. Mountain climbers carry bottled oxygen when at very high altitudes. (a) Assuming that a mountain climber uses oxygen at twice the rate for climbing 116 stairs per minute (because of low air temperature and winds), calculate how many liters of oxygen a climber would need for 10.0 h of climbing. (These are liters at sea level.) Note that only 40% of the inhaled oxygen is utilized; the rest is exhaled. (b) How much useful work does the climber do if he and his equipment have a mass of 90.0 kg and he gains 1000 m of altitude? (c) What is his efficiency for the 10.0-h climb? 58. The awe-inspiring Great Pyramid of Cheops was built more than 4500 years ago. Its square base, originally 230 m on a side, covered 13.1 acres, and it was 146 m high, with a mass of about 7×109 kg . (The pyramid’s dimensions are slightly different today due to quarrying and some sagging.) Historians estimate that 20,000 workers spent 20 years to construct it, working 12-hour days, 330 days per year. (a) Calculate the gravitational potential energy stored in the pyramid, given its center of mass is at one-fourth its height. (b) Only a fraction of the workers lifted blocks; most were involved in support services such as building ramps (see Figure 7.45), bringing food and water, and hauling blocks to the site. Calculate the efficiency of the workers who did the lifting, assuming there were 1000 of them and they consumed food energy at the rate of 300 kcal/h. What does your answer imply about how much of their work went into block-lifting, versus how much work went into friction and lifting and lowering their own bodies? (c) Calculate the mass of food that had to be supplied each day, assuming that the average worker required 3600 kcal per day and that their diet was 5% protein, 60% carbohydrate, and 35% fat. (These proportions neglect the mass of bulk and nondigestible materials consumed.) 7.9 World Energy Use 60. Integrated Concepts (a) Calculate the force the woman in Figure 7.46 exerts to do a push-up at constant speed, taking all data to be known to three digits. (b) How much work does she do if her center of mass rises 0.240 m? (c) What is her useful power output if she does 25 push-ups in 1 min? (Should work done lowering her body be included? See the discussion of useful work in Work, Energy, and Power in Humans. Figure 7.46 Forces involved in doing push-ups. The woman’s weight acts as a force exerted downward on her center of gravity (CG). 61. Integrated Concepts A 75.0-kg cross-country skier is climbing a 3.0º slope at a constant speed of 2.00 m/s and encounters air resistance of 25.0 N. Find his power output for work done against the gravitational force and air resistance. (b) What average
force does he exert backward on the snow to accomplish this? (c) If he continues to exert this force and to experience the same air resistance when he reaches a level area, how long will it take him to reach a velocity of 10.0 m/s? 62. Integrated Concepts The 70.0-kg swimmer in Figure 7.44 starts a race with an initial velocity of 1.25 m/s and exerts an average force of 80.0 N backward with his arms during each 1.80 m long stroke. (a) What is his initial acceleration if water resistance is 45.0 N? (b) What is the subsequent average resistance force from the water during the 5.00 s it takes him to reach his top velocity of 2.50 m/s? (c) Discuss whether water resistance seems to increase linearly with velocity. 63. Integrated Concepts A toy gun uses a spring with a force constant of 300 N/m to propel a 10.0-g steel ball. If the spring is compressed 7.00 cm and friction is negligible: (a) How much force is needed to compress the spring? (b) To what maximum height can the ball be shot? (c) At what angles above the horizontal may a child aim to hit a target 3.00 m away at the same height as the gun? (d) What is the gun’s maximum range on level ground? 64. Integrated Concepts (a) What force must be supplied by an elevator cable to produce an acceleration of 0.800 m/s2 against a 200-N frictional force, if the mass of the loaded elevator is 1500 kg? (b) How much work is done by the cable in lifting the elevator 20.0 m? (c) What is the final speed of the elevator if it starts from rest? (d) How much work went into thermal energy? Figure 7.45 Ancient pyramids were probably constructed using ramps as simple machines. (credit: Franck Monnier, Wikimedia Commons) 59. (a) How long can you play tennis on the 800 kJ (about 200 kcal) of energy in a candy bar? (b) Does this seem like a 65. Unreasonable Results A car advertisement claims that its 900-kg car accelerated from rest to 30.0 m/s and drove 100 km, gaining 3.00 km in altitude, on 1.0 gal of gasoline. The average force of friction This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 7 \| Work, Energy, and Energy Resources 311 including air resistance was 700 N. Assume all values are known to three significant figures. (a) Calculate the car’s efficiency. (b) What is unreasonable about the result? (c) Which premise is unreasonable, or which premises are inconsistent? 66. Unreasonable Results Body fat is metabolized, supplying 9.30 kcal/g, when dietary intake is less than needed to fuel metabolism. The manufacturers of an exercise bicycle claim that you can lose 0.500 kg of fat per day by vigorously exercising for 2.00 h per day on their machine. (a) How many kcal are supplied by the metabolization of 0.500 kg of fat? (b) Calculate the kcal/min that you would have to utilize to metabolize fat at the rate of 0.500 kg in 2.00 h. (c) What is unreasonable about the results? (d) Which premise is unreasonable, or which premises are inconsistent? 67. Construct Your Own Problem Consider a person climbing and descending stairs. Construct a problem in which you calculate the long-term rate at which stairs can be climbed considering the mass of the person, his ability to generate power with his legs, and the height of a single stair step. Also consider why the same person can descend stairs at a faster rate for a nearly unlimited time in spite of the fact that very similar forces are exerted going down as going up. (This points to a fundamentally different process for descending versus climbing stairs.) 68. Construct Your Own Problem Consider humans generating electricity by pedaling a device similar to a stationary bicycle. Construct a problem in which you determine the number of people it would take to replace a large electrical generation facility. Among the things to consider are the power output that is reasonable using the legs, rest time, and the need for electricity 24 hours per day. Discuss the practical implications of your results. 69. Integrated Concepts A 105-kg basketball player crouches down 0.400 m while waiting to jump. After exerting a force on the floor through this 0.400 m, his feet leave the floor and his center of gravity rises 0.950 m above its normal standing erect position. (a) Using energy considerations, calculate his velocity when he leaves the floor. (b) What average force did he exert on the floor? (Do not neglect the force to support his weight as well as that to accelerate him.) (c) What was his power output during the acceleration phase? Test Prep for AP® Courses c. 450 N d. 600 N 7.1 Work: The Scientific Definition 1. Given Table 7.7 about how much force does the rocket engine exert on the 3.0-kg payload? Table 7.7 Distance traveled with rocket engine firing (m) Payload final velocity (m/s) 500 490 1020 505 a. 150 N b. 300 N 310 300 450 312 2. You have a cart track, a cart, several masses, and a position-sensing pulley. Design an experiment to examine how the force exerted on the cart does work as it moves through a distance. 3. Look at Figure 7.10(c). You compress a spring by x, and then release it. Next you compress the spring by 2x. How much more work did you do the second time than the first? a. Half as much b. The same c. Twice as much d. Four times as much 4. You have a cart track, two carts, several masses, a position-sensing pulley, and a piece of carpet (a rough surface) that will fit over the track. Design an experiment to examine how the force exerted on the cart does work as the cart moves through a distance. 312 Chapter 7 \| Work, Energy, and Energy Resources 5. A crane is lifting construction materials from the ground to an elevation of 60 m. Over the first 10 m, the motor linearly increases the force it exerts from 0 to 10 kN. It exerts that constant force for the next 40 m, and then winds down to 0 N again over the last 10 m, as shown in the figure. What is the total work done on the construction materials? Figure 7.47 a. 500 kJ b. 600 kJ c. 300 kJ d. 18 MJ 7.2 Kinetic Energy and the Work-Energy Theorem 6. A toy car is going around a loop-the-loop. Gravity ____ the kinetic energy on the upward side of the loop, ____ the kinetic energy at the top, and ____ the kinetic energy on the downward side of the loop. a. increases, decreases, has no effect on b. decreases, has no effect on, increases c. increases, has no effect on, decreases d. decreases, increases, has no effect on 7. A roller coaster is set up with a track in the form of a perfect cosine. Describe and graph what happens to the kinetic energy of a cart as it goes through the first full period of the track. 8. If wind is blowing horizontally toward a car with an angle of 30 degrees from the direction of travel, the kinetic energy will ____. If the wind is blowing at a car at 135 degrees from the direction of travel, the kinetic energy will ____. a. increase, increase increase, decrease b. c. decrease, increase d. decrease, decrease 9. In what direction relative to the direction of travel can a force act on a car (traveling on level ground), and not change the kinetic energy? Can you give examples of such forces? 10. A 2000-kg airplane is coming in for a landing, with a velocity 5 degrees below the horizontal and a drag force of 40 kN acting directly rearward. Kinetic energy will ____ due to the net force of ____. a. increase, 20 kN b. decrease, 40 kN c. increase, 45 kN d. decrease, 45 kN 11. You are participating in the Iditarod, and your sled dogs are pulling you across a frozen lake with a force of 1200 N while a 300 N wind is blowing at you at 135 degrees from your direction of travel. What is the net force, and will your kinetic energy increase or decrease? 12. A model drag car is being accelerated along its track from rest by a motor with a force of 75 N, but there is a drag force of 30 N due to the track. What is the kinetic energy after 2 m of travel? a. 90 J This content is available for free at http://cnx.org/content/col11844/1.13 b. 150 J c. 210 J d. 60 J 13. You are launching a 2-kg potato out of a potato cannon. The cannon is 1.5 m long and is aimed 30 degrees above the horizontal. It exerts a 50 N force on the potato. What is the kinetic energy of the potato as it leaves the muzzle of the potato cannon? 14. When the force acting on an object is parallel to the direction of the motion of the center of mass, the mechanical energy ____. When the force acting on an object is antiparallel to the direction of the center of mass, the mechanical energy ____. increases, increases a. b. increases, decreases c. decreases, increases d. decreases, decreases 15. Describe a system in which the main forces acting are parallel or antiparallel to the center of mass, and justify your answer. 16. A child is pulling two red wagons, with the second one tied to the first by a (non-stretching) rope. Each wagon has a mass of 10 kg. If the child exerts a force of 30 N for 5.0 m, how much has the kinetic energy of the two-wagon system changed? a. 300 J b. 150 J c. 75 J d. 60 J 17. A child has two red wagons, with the rear one tied to the front by a (non-stretching) rope. If the child pushes on the rear wagon, what happens to the kinetic energy of each of the wagons, and the two-wagon system? 18. Draw a graph of the force parallel to displacement exerted on a stunt motorcycle going through a loop-the-loop versus the distance traveled around the loop. Explain the net change in energy. 7.3 Gravitational Potential Energy 19. A 1.0 kg baseball is flying at 10 m/s. How much kinetic energy does it have? Potential energy? a. 10 J, 20 J b. 50 J, 20 J c. unknown, 50 J d. 50 J, unknown 20. A 2.0-kg potato has been launched out of a potato cannon at 9.0 m/s. What is the kinetic energy? If you then learn that it is 4.0 m above the ground, what is the total mechanical energy relative to the ground? a. 78 J, 3 J b. 160 J, 81 J c. 81 J, 160 J d. 81 J, 3 J 21. You have a 120-g yo-yo that you are swinging at 0.9 m/s. How much energy does it have? How high can it get above the lowest
point of the swing without your doing any additional work, on Earth? How high could it get on the Moon, where gravity is 1/6 Earth’s? 7.4 Conservative Forces and Potential Energy 22. Two 4.0 kg masses are connected to each other by a spring with a force constant of 25 N/m and a rest length of 1.0 m. If the spring has been compressed to 0.80 m in length and Chapter 7 \| Work, Energy, and Energy Resources 313 the masses are traveling toward each other at 0.50 m/s (each), what is the total energy in the system? a. 1.0 J b. 1.5 J c. 9.0 J d. 8.0 J 23. A spring with a force constant of 5000 N/m and a rest length of 3.0 m is used in a catapult. When compressed to 1.0 m, it is used to launch a 50 kg rock. However, there is an error in the release mechanism, so the rock gets launched almost straight up. How high does it go, and how fast is it going when it hits the ground? 24. What information do you need to calculate the kinetic energy and potential energy of a spring? Potential energy due to gravity? How many objects do you need information about for each of these cases? 25. You are loading a toy dart gun, which has two settings, the more powerful with the spring compressed twice as far as the lower setting. If it takes 5.0 J of work to compress the dart gun to the lower setting, how much work does it take for the higher setting? a. 20 J b. 10 J c. 2.5 J d. 40 J 26. Describe a system you use daily with internal potential energy. 27. Old-fashioned pendulum clocks are powered by masses that need to be wound back to the top of the clock about once a week to counteract energy lost due to friction and to the chimes. One particular clock has three masses: 4.0 kg, 4.0 kg, and 6.0 kg. They can drop 1.3 meters. How much energy does the clock use in a week? a. 51 J b. 76 J c. 127 J d. 178 J 28. A water tower stores not only water, but (at least part of) the energy to move the water. How much? Make reasonable estimates for how much water is in the tower, and other quantities you need. 29. Old-fashioned pocket watches needed to be wound daily so they wouldn’t run down and lose time, due to the friction in the internal components. This required a large number of turns of the winding key, but not much force per turn, and it was possible to overwind and break the watch. How was the energy stored? a. A small mass raised a long distance b. A large mass raised a short distance c. A weak spring deformed a long way d. A strong spring deformed a short way 30. Some of the very first clocks invented in China were powered by water. Describe how you think this was done. 7.5 Nonconservative Forces 31. You are in a room in a basement with a smooth concrete floor (friction force equals 40 N) and a nice rug (friction force equals 55 N) that is 3 m by 4 m. However, you have to push a very heavy box from one corner of the rug to the opposite corner of the rug. Will you do more work against friction going around the floor or across the rug, and how much extra? a. Across the rug is 275 J extra b. Around the floor is 5 J extra c. Across the rug is 5 J extra d. Around the floor is 280 J extra 32. In the Appalachians, along the interstate, there are ramps of loose gravel for semis that have had their brakes fail to drive into to stop. Design an experiment to measure how effective this would be. 7.6 Conservation of Energy 33. You do 30 J of work to load a toy dart gun. However, the dart is 10 cm long and feels a frictional force of 10 N while going through the dart gun’s barrel. What is the kinetic energy of the fired dart? a. 30 J b. 29 J c. 28 J d. 27 J 34. When an object is lifted by a crane, it begins and ends its motion at rest. The same is true of an object pushed across a rough surface. Explain why this happens. What are the differences between these systems? 35. A child has two red wagons, with the rear one tied to the front by a stretchy rope (a spring). If the child pulls on the front wagon, the ____ increases. a. kinetic energy of the wagons b. potential energy stored in the spring c. both A and B d. not enough information 36. A child has two red wagons, with the rear one tied to the front by a stretchy rope (a spring). If the child pulls on the front wagon, the energy stored in the system increases. How do the relative amounts of potential and kinetic energy in this system change over time? 37. Which of the following are closed systems? a. Earth b. a car c. a frictionless pendulum d. a mass on a spring in a vacuum 38. Describe a real-world example of a closed system. 39. A 5.0-kg rock falls off of a 10 m cliff. If air resistance exerts a force of 10 N, what is the kinetic energy when the rock hits the ground? a. 400 J b. 12.6 m/s c. 100 J d. 500 J 40. Hydroelectricity is generated by storing water behind a dam, and then letting some of it run through generators in the dam to turn them. If the system is the water, what is the environment that is doing work on it? If a dam has water 100 m deep behind it, how much energy was generated if 10,000 kg of water exited the dam at 2.0 m/s? 41. Before railroads were invented, goods often traveled along canals, with mules pulling barges from the bank. If a mule is exerting a 1200 N force for 10 km, and the rope connecting the mule to the barge is at a 20 degree angle from the direction of travel, how much work did the mule do on the barge? a. 12 MJ b. 11 MJ c. 4.1 MJ d. 6 MJ 42. Describe an instance today in which you did work, by the scientific definition. Then calculate how much work you did in that instance, showing your work. 314 Chapter 7 \| Work, Energy, and Energy Resources This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 8 \| Linear Momentum and Collisions 315 8 LINEAR MOMENTUM AND COLLISIONS Figure 8.1 Each rugby player has great momentum, which will affect the outcome of their collisions with each other and the ground. (credit: ozzzie, Flickr) Chapter Outline 8.1. Linear Momentum and Force 8.2. Impulse 8.3. Conservation of Momentum 8.4. Elastic Collisions in One Dimension 8.5. Inelastic Collisions in One Dimension 8.6. Collisions of Point Masses in Two Dimensions 8.7. Introduction to Rocket Propulsion Connection for AP® courses In this chapter, you will learn about the concept of momentum and the relationship between momentum and force (both vector quantities) applied over a time interval. Have you ever considered why a glass dropped on a tile floor will often break, but a glass dropped on carpet will often remain intact? Both involve changes in momentum, but the actual collision with the floor is different in each case, just as an automobile collision without the benefit of an airbag can have a significantly different outcome than one with an airbag. You will learn that the interaction of objects (like a glass and the floor or two automobiles) results in forces, which in turn result in changes in the momentum of each object. At the same time, you will see how the law of momentum conservation can be applied to a system to help determine the outcome of a collision. The content in this chapter supports: Big Idea 3 The interactions of an object with other objects can be described by forces. Enduring Understanding 3.D A force exerted on an object can change the momentum of the object. Essential Knowledge 3.D.2 The change in momentum of an object occurs over a time interval. Big Idea 4: Interactions between systems can result in changes in those systems. Enduring Understanding 4.B Interactions with other objects or systems can change the total linear momentum of a system. Essential Knowledge 4.B.1 The change in linear momentum for a constant-mass system is the product of the mass of the system and the change in velocity of the center of mass. Big Idea 5 Changes that occur as a result of interactions are constrained by conservation laws. Enduring Understanding 5.A Certain quantities are conserved, in the sense that the changes of those quantities in a given system are always equal to the transfer of that quantity to or from the system by all possible interactions with other systems. 316 Chapter 8 \| Linear Momentum and Collisions Essential Knowledge 5.A.2 For all systems under all circumstances, energy, charge, linear momentum, and angular momentum are conserved. Essential Knowledge 5.D.1 In a collision between objects, linear momentum is conserved. In an elastic collision, kinetic energy is the same before and after. Essential Knowledge 5.D.2 In a collision between objects, linear momentum is conserved. In an inelastic collision, kinetic energy is not the same before and after the collision. 8.1 Linear Momentum and Force Learning Objectives By the end of this section, you will be able to: • Define linear momentum. • Explain the relationship between linear momentum and force. • State Newton’s second law of motion in terms of linear momentum. • Calculate linear momentum given mass and velocity. The information presented in this section supports the following AP® learning objectives and science practices: • 3.D.1.1 The student is able to justify the selection of data needed to determine the relationship between the direction of the force acting on an object and the change in momentum caused by that force. (S.P. 4.1) Linear Momentum The scientific definition of linear momentum is consistent with most people’s intuitive understanding of momentum: a large, fastmoving object has greater momentum than a smaller, slower object. Linear momentum is defined as the product of a system’s mass multiplied by its velocity. In symbols, linear momentum is expressed as p = v. (8.1) Momentum is directly proportional to the object’s mass and also its velocity. Thus the greater an object’s mass or the greater its velocity, the greater its momentum. Momentum p is a vector having the same direction as the velocity v . The SI unit for momentum is kg · m/s . Linear Momentum Linear momentum is defined as the product of a system’s mass multiplied by its velocity: p = v. (8.2) Example 8.1 Calcul
ating Momentum: A Football Player and a Football (a) Calculate the momentum of a 110-kg football player running at 8.00 m/s. (b) Compare the player’s momentum with the momentum of a hard-thrown 0.410-kg football that has a speed of 25.0 m/s. Strategy No information is given regarding direction, and so we can calculate only the magnitude of the momentum, . (As usual, a symbol that is in italics is a magnitude, whereas one that is italicized, boldfaced, and has an arrow is a vector.) In both parts of this example, the magnitude of momentum can be calculated directly from the definition of momentum given in the equation, which becomes = (8.3) when only magnitudes are considered. Solution for (a) To determine the momentum of the player, substitute the known values for the player’s mass and speed into the equation. player = 110 kg (8.00 m/s) = 880 kg · m/s Solution for (b) To determine the momentum of the ball, substitute the known values for the ball’s mass and speed into the equation. ball = 0.410 kg (25.0 m/s) = 10.3 kg · m/s (8.4) (8.5) The ratio of the player’s momentum to that of the ball is This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 8 \| Linear Momentum and Collisions player ball = 880 10.3 = 85.9. 317 (8.6) Discussion Although the ball has greater velocity, the player has a much greater mass. Thus the momentum of the player is much greater than the momentum of the football, as you might guess. As a result, the player’s motion is only slightly affected if he catches the ball. We shall quantify what happens in such collisions in terms of momentum in later sections. Momentum and Newton’s Second Law The importance of momentum, unlike the importance of energy, was recognized early in the development of classical physics. Momentum was deemed so important that it was called the “quantity of motion.” Newton actually stated his second law of motion in terms of momentum: The net external force equals the change in momentum of a system divided by the time over which it changes. Using symbols, this law is Fnet = Δp Δ where Fnet is the net external force, Δp is the change in momentum, and Δ is the change in time. Newton’s Second Law of Motion in Terms of Momentum The net external force equals the change in momentum of a system divided by the time over which it changes. Fnet = Δp Δ (8.7) (8.8) Making Connections: Force and Momentum Force and momentum are intimately related. Force acting over time can change momentum, and Newton’s second law of motion, can be stated in its most broadly applicable form in terms of momentum. Momentum continues to be a key concept in the study of atomic and subatomic particles in quantum mechanics. This statement of Newton’s second law of motion includes the more familiar Fnet =a as a special case. We can derive this form as follows. First, note that the change in momentum Δp is given by If the mass of the system is constant, then Δp = Δ v . Δ(v) = Δv. So that for constant mass, Newton’s second law of motion becomes Because Δv Δ = a , we get the familiar equation when the mass of the system is constant. Fnet = Δp Δ = Δv Δ . Fnet =a (8.9) (8.10) (8.11) (8.12) Newton’s second law of motion stated in terms of momentum is more generally applicable because it can be applied to systems where the mass is changing, such as rockets, as well as to systems of constant mass. We will consider systems with varying mass in some detail; however, the relationship between momentum and force remains useful when mass is constant, such as in the following example. Example 8.2 Calculating Force: Venus Williams’ Racquet During the 2007 French Open, Venus Williams hit the fastest recorded serve in a premier women’s match, reaching a speed of 58 m/s (209 km/h). What is the average force exerted on the 0.057-kg tennis ball by Venus Williams’ racquet, assuming that the ball’s speed just after impact is 58 m/s, that the initial horizontal component of the velocity before impact is negligible, and that the ball remained in contact with the racquet for 5.0 ms (milliseconds)? Strategy 318 Chapter 8 \| Linear Momentum and Collisions This problem involves only one dimension because the ball starts from having no horizontal velocity component before impact. Newton’s second law stated in terms of momentum is then written as As noted above, when mass is constant, the change in momentum is given by Δ = Δ = (f − i). Fnet = Δp Δ . In this example, the velocity just after impact and the change in time are given; thus, once Δ is calculated, net = can be used to find the force. Solution To determine the change in momentum, substitute the values for the initial and final velocities into the equation above. Δ = (f – i) 0.057 kg = = 3.306 kg · m/s ≈ 3.3 kg · m/s (58 m/s – 0 m/s) Now the magnitude of the net external force can determined by using net = Δ Δ : net = Δ Δ = 3.306 kg ⋅ m/s 5.0×10−3 s = 661 N ≈ 660 N, (8.13) (8.14) Δ Δ (8.15) (8.16) where we have retained only two significant figures in the final step. Discussion This quantity was the average force exerted by Venus Williams’ racquet on the tennis ball during its brief impact (note that the ball also experienced the 0.56-N force of gravity, but that force was not due to the racquet). This problem could also be solved by first finding the acceleration and then using net = , but one additional step would be required compared with the strategy used in this example. Making Connections: Illustrative Example Figure 8.2 A puck has an elastic, glancing collision with the edge of an air hockey table. In Figure 8.2, a puck is shown colliding with the edge of an air hockey table at a glancing angle. During the collision, the edge of the table exerts a force F on the puck, and the velocity of the puck changes as a result of the collision. The change in momentum is found by the equation: This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 8 \| Linear Momentum and Collisions Δp = Δv = v' - v = (v' + ( - v)) 319 (8.17) As shown, the direction of the change in velocity is the same as the direction of the change in momentum, which in turn is in the same direction as the force exerted by the edge of the table. Note that there is only a horizontal change in velocity. There is no difference in the vertical components of the initial and final velocity vectors; therefore, there is no vertical component to the change in velocity vector or the change in momentum vector. This is consistent with the fact that the force exerted by the edge of the table is purely in the horizontal direction. 8.2 Impulse Learning Objectives By the end of this section, you will be able to: • Define impulse. • Describe effects of impulses in everyday life. • Determine the average effective force using graphical representation. • Calculate average force and impulse given mass, velocity, and time. The information presented in this section supports the following AP® learning objectives and science practices: • 3.D.2.1 The student is able to justify the selection of routines for the calculation of the relationships between changes in momentum of an object, average force, impulse, and time of interaction. (S.P. 2.1) • 3.D.2.2 The student is able to predict the change in momentum of an object from the average force exerted on the object and the interval of time during which the force is exerted. (S.P. 6.4) • 3.D.2.3 The student is able to analyze data to characterize the change in momentum of an object from the average force exerted on the object and the interval of time during which the force is exerted. (S.P. 5.1) • 3.D.2.4 The student is able to design a plan for collecting data to investigate the relationship between changes in momentum and the average force exerted on an object over time. (S.P. 4.1) • 4.B.2.1 The student is able to apply mathematical routines to calculate the change in momentum of a system by analyzing the average force exerted over a certain time on the system. (S.P. 2.2) • 4.B.2.2 The student is able to perform analysis on data presented as a force-time graph and predict the change in momentum of a system. (S.P. 5.1) The effect of a force on an object depends on how long it acts, as well as how great the force is. In Example 8.1, a very large force acting for a short time had a great effect on the momentum of the tennis ball. A small force could cause the same change in momentum, but it would have to act for a much longer time. For example, if the ball were thrown upward, the gravitational force (which is much smaller than the tennis racquet’s force) would eventually reverse the momentum of the ball. Quantitatively, the effect we are talking about is the change in momentum Δp . By rearranging the equation Fnet = Δp Δ to be we can see how the change in momentum equals the average net external force multiplied by the time this force acts. The quantity Fnet Δ is given the name impulse. Impulse is the same as the change in momentum. Δp = FnetΔ, (8.18) Impulse: Change in Momentum Change in momentum equals the average net external force multiplied by the time this force acts. Δp = FnetΔ (8.19) The quantity Fnet Δ is given the name impulse. There are many ways in which an understanding of impulse can save lives, or at least limbs. The dashboard padding in a car, and certainly the airbags, allow the net force on the occupants in the car to act over a much longer time when there is a sudden stop. The momentum change is the same for an occupant, whether an air bag is deployed or not, but the force (to bring the occupant to a stop) will be much less if it acts over a larger time. Cars today have many plastic components. One advantage of plastics is their lighter weight, which results in better gas mileage. Another advantage is that a car will crumple in a collision, especially in the event of a head-on collision. A longer collision time means the force on the car will be less. Deaths during car races decreased dramatically when the rigid frames of r
acing cars were replaced with parts that could crumple or collapse in the event of an accident. Bones in a body will fracture if the force on them is too large. If you jump onto the floor from a table, the force on your legs can be immense if you land stiff-legged on a hard surface. Rolling on the ground after jumping from the table, or landing with a parachute, extends the time over which the force (on you from the ground) acts. 320 Chapter 8 \| Linear Momentum and Collisions Making Connections: Illustrations of Force Exerted Figure 8.3 This is a graph showing the force exerted by a fixed barrier on a block versus time. A 1.2-kg block slides across a horizontal, frictionless surface with a constant speed of 3.0 m/s before striking a fixed barrier and coming to a stop. In Figure 8.3, the force exerted by the barrier is assumed to be a constant 15 N during the 0.24-s collision. The impulse can be calculated using the area under the curve. Δ = Δ = (15 N)(0.24 s) = 3.6 kg•m/s Note that the initial momentum of the block is: = = (1.2 kg)( − 3.0 m/s) = − 3.6 kg•m/s (8.20) (8.21) We are assuming that the initial velocity is −3.0 m/s. We have established that the force exerted by the barrier is in the positive direction, so the initial velocity of the block must be in the negative direction. Since the final momentum of the block is zero, the impulse is equal to the change in momentum of the block. Suppose that, instead of striking a fixed barrier, the block is instead stopped by a spring.Consider the force exerted by the spring over the time interval from the beginning of the collision until the block comes to rest. Figure 8.4 This is a graph showing the force exerted by a spring on a block versus time. In this case, the impulse can be calculated again using the area under the curve (the area of a triangle): = 1 2 (base)(height) = 1 2 (0.24 s)(30 N) = 3.6 kg•m/s (8.22) Again, this is equal to the difference between the initial and final momentum of the block, so the impulse is equal to the change in momentum. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 8 \| Linear Momentum and Collisions 321 Example 8.3 Calculating Magnitudes of Impulses: Two Billiard Balls Striking a Rigid Wall Two identical billiard balls strike a rigid wall with the same speed, and are reflected without any change of speed. The first ball strikes perpendicular to the wall. The second ball strikes the wall at an angle of 30º from the perpendicular, and bounces off at an angle of 30º from perpendicular to the wall. (a) Determine the direction of the force on the wall due to each ball. (b) Calculate the ratio of the magnitudes of impulses on the two balls by the wall. Strategy for (a) In order to determine the force on the wall, consider the force on the ball due to the wall using Newton’s second law and then apply Newton’s third law to determine the direction. Assume the -axis to be normal to the wall and to be positive in the initial direction of motion. Choose the -axis to be along the wall in the plane of the second ball’s motion. The momentum direction and the velocity direction are the same. Solution for (a) The first ball bounces directly into the wall and exerts a force on it in the direction. Therefore the wall exerts a force on the ball in the direction. The second ball continues with the same momentum component in the direction, but reverses its -component of momentum, as seen by sketching a diagram of the angles involved and keeping in mind the proportionality between velocity and momentum. These changes mean the change in momentum for both balls is in the direction, so the force of the wall on each ball is along the direction. Strategy for (b) Calculate the change in momentum for each ball, which is equal to the impulse imparted to the ball. Solution for (b) Let be the speed of each ball before and after collision with the wall, and the mass of each ball. Choose the -axis and -axis as previously described, and consider the change in momentum of the first ball which strikes perpendicular to the wall. xi = ; yi = 0 xf = −; yf = 0 Impulse is the change in momentum vector. Therefore the -component of impulse is equal to −2 and the component of impulse is equal to zero. Now consider the change in momentum of the second ball. xi = cos 30º; yi = sin 30º xf = – cos 30º; yf = − sin 30º It should be noted here that while x changes sign after the collision, y does not. Therefore the -component of impulse is equal to −2 cos 30º and the -component of impulse is equal to zero. The ratio of the magnitudes of the impulse imparted to the balls is 2 2 cos 30º = 2 3 = 1.155. (8.23) (8.24) (8.25) (8.26) (8.27) Discussion The direction of impulse and force is the same as in the case of (a); it is normal to the wall and along the negative -direction. Making use of Newton’s third law, the force on the wall due to each ball is normal to the wall along the positive -direction. Our definition of impulse includes an assumption that the force is constant over the time interval Δ . Forces are usually not constant. Forces vary considerably even during the brief time intervals considered. It is, however, possible to find an average effective force eff that produces the same result as the corresponding time-varying force. Figure 8.5 shows a graph of what an actual force looks like as a function of time for a ball bouncing off the floor. The area under the curve has units of momentum and is equal to the impulse or change in momentum between times 1 and 2 . That area is equal to the area inside the 322 Chapter 8 \| Linear Momentum and Collisions rectangle bounded by eff , 1 , and 2 . Thus the impulses and their effects are the same for both the actual and effective forces. Figure 8.5 A graph of force versus time with time along the -axis and force along the -axis for an actual force and an equivalent effective force. The areas under the two curves are equal. Making Connections: Baseball In most real-life collisions, the forces acting on an object are not constant. For example, when a bat strikes a baseball, the force is very small at the beginning of the collision since only a small portion of the ball is initially in contact with the bat. As the collision continues, the ball deforms so that a greater fraction of the ball is in contact with the bat, resulting in a greater force. As the ball begins to leave the bat, the force drops to zero, much like the force curve in Figure 8.5. Although the changing force is difficult to precisely calculate at each instant, the average force can be estimated very well in most cases. Suppose that a 150-g baseball experiences an average force of 480 N in a direction opposite the initial 32 m/s speed of the baseball over a time interval of 0.017 s. What is the final velocity of the baseball after the collision? Δ = Δ = (480)(0.017) = 8.16 kg•m/s − = 8.16 kg • m/s (0.150 kg) − (0.150 kg)( − 32 m/s) = 8.16 kg • m/s = 22 m/s (8.28) (8.29) (8.30) (8.31) Note in the above example that the initial velocity of the baseball prior to the collision is negative, consistent with the assumption we initially made that the force exerted by the bat is positive and in the direction opposite the initial velocity of the baseball. In this case, even though the force acting on the baseball varies with time, the average force is a good approximation of the effective force acting on the ball for the purposes of calculating the impulse and the change in momentum. Making Connections: Take-Home Investigation—Hand Movement and Impulse Try catching a ball while “giving” with the ball, pulling your hands toward your body. Then, try catching a ball while keeping your hands still. Hit water in a tub with your full palm. After the water has settled, hit the water again by diving your hand with your fingers first into the water. (Your full palm represents a swimmer doing a belly flop and your diving hand represents a swimmer doing a dive.) Explain what happens in each case and why. Which orientations would you advise people to avoid and why? Making Connections: Constant Force and Constant Acceleration The assumption of a constant force in the definition of impulse is analogous to the assumption of a constant acceleration in kinematics. In both cases, nature is adequately described without the use of calculus. Applying the Science Practices: Verifying the Relationship between Force and Change in Linear Momentum Design an experiment in order to experimentally verify the relationship between the impulse of a force and change in linear momentum. For simplicity, it would be best to ensure that frictional forces are very small or zero in your experiment so that the effect of friction can be neglected. As you design your experiment, consider the following: • Would it be easier to analyze a one-dimensional collision or a two-dimensional collision? • How will you measure the force? This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 8 \| Linear Momentum and Collisions 323 • Should you have two objects in motion or one object bouncing off a rigid surface? • How will you measure the duration of the collision? • How will you measure the initial and final velocities of the object(s)? • Would it be easier to analyze an elastic or inelastic collision? • Should you verify the relationship mathematically or graphically? 8.3 Conservation of Momentum Learning Objectives By the end of this section, you will be able to: • Describe the law of conservation of linear momentum. • Derive an expression for the conservation of momentum. • Explain conservation of momentum with examples. • Explain the law of conservation of momentum as it relates to atomic and subatomic particles. The information presented in this section supports the following AP® learning objectives and science practices: • 5.A.2.1 The student is able to define open and closed systems for everyday situations and apply conservation concepts for energy, charge, and lin
ear momentum to those situations. (S.P. 6.4, 7.2) • 5.D.1.4 The student is able to design an experimental test of an application of the principle of the conservation of linear momentum, predict an outcome of the experiment using the principle, analyze data generated by that experiment whose uncertainties are expressed numerically, and evaluate the match between the prediction and the outcome. (S.P. 4.2, 5.1, 5.3, 6.4) • 5.D.2.1 The student is able to qualitatively predict, in terms of linear momentum and kinetic energy, how the outcome of a collision between two objects changes depending on whether the collision is elastic or inelastic. (S.P. 6.4, 7.2) • 5.D.2.2 The student is able to plan data collection strategies to test the law of conservation of momentum in a two- object collision that is elastic or inelastic and analyze the resulting data graphically. (S.P.4.1, 4.2, 5.1) • 5.D.3.1 The student is able to predict the velocity of the center of mass of a system when there is no interaction outside of the system but there is an interaction within the system (i.e., the student simply recognizes that interactions within a system do not affect the center of mass motion of the system and is able to determine that there is no external force). (S.P. 6.4) Momentum is an important quantity because it is conserved. Yet it was not conserved in the examples in Impulse and Linear Momentum and Force, where large changes in momentum were produced by forces acting on the system of interest. Under what circumstances is momentum conserved? The answer to this question entails considering a sufficiently large system. It is always possible to find a larger system in which total momentum is constant, even if momentum changes for components of the system. If a football player runs into the goalpost in the end zone, there will be a force on him that causes him to bounce backward. However, the Earth also recoils —conserving momentum—because of the force applied to it through the goalpost. Because Earth is many orders of magnitude more massive than the player, its recoil is immeasurably small and can be neglected in any practical sense, but it is real nevertheless. Consider what happens if the masses of two colliding objects are more similar than the masses of a football player and Earth—for example, one car bumping into another, as shown in Figure 8.6. Both cars are coasting in the same direction when the lead car (labeled 2) is bumped by the trailing car (labeled 1). The only unbalanced force on each car is the force of the collision. (Assume that the effects due to friction are negligible.) Car 1 slows down as a result of the collision, losing some momentum, while car 2 speeds up and gains some momentum. We shall now show that the total momentum of the two-car system remains constant. 324 Chapter 8 \| Linear Momentum and Collisions Figure 8.6 A car of mass 1 moving with a velocity of 1 bumps into another car of mass 2 and velocity 2 that it is following. As a result, the first car slows down to a velocity of v′1 and the second speeds up to a velocity of v′2 . The momentum of each car is changed, but the total momentum tot of the two cars is the same before and after the collision (if you assume friction is negligible). Using the definition of impulse, the change in momentum of car 1 is given by Δ1 = 1Δ, (8.32) where 1 is the force on car 1 due to car 2, and Δ is the time the force acts (the duration of the collision). Intuitively, it seems obvious that the collision time is the same for both cars, but it is only true for objects traveling at ordinary speeds. This assumption must be modified for objects travelling near the speed of light, without affecting the result that momentum is conserved. Similarly, the change in momentum of car 2 is Δ2 = 2Δ (8.33) where 2 is the force on car 2 due to car 1, and we assume the duration of the collision Δ is the same for both cars. We know from Newton’s third law that 2 = – 1 , and so Thus, the changes in momentum are equal and opposite, and Δ1 + Δ2 = 0. Δ2 = −1Δ = −Δ1. Because the changes in momentum add to zero, the total momentum of the two-car system is constant. That is, 1 + 2 = constant, 1 + 2 = ′1 + ′2, where ′1 and ′2 are the momenta of cars 1 and 2 after the collision. (We often use primes to denote the final state.) This result—that momentum is conserved—has validity far beyond the preceding one-dimensional case. It can be similarly shown that total momentum is conserved for any isolated system, with any number of objects in it. In equation form, the conservation of momentum principle for an isolated system is written or ptot = constant, ptot = p′tot, (8.34) (8.35) (8.36) (8.37) (8.38) (8.39) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 8 \| Linear Momentum and Collisions 325 where ptot is the total momentum (the sum of the momenta of the individual objects in the system) and p′tot is the total momentum some time later. (The total momentum can be shown to be the momentum of the center of mass of the system.) An isolated system is defined to be one for which the net external force is zero Fnet = 0 . Conservation of Momentum Principle ptot = constant ptot = p′tot (isolated system) (8.40) Isolated System An isolated system is defined to be one for which the net external force is zero Fnet = 0 . Making Connections: Cart Collisions Consider two air carts with equal mass (m) on a linear track. The first cart moves with a speed v towards the second cart, which is initially at rest. We will take the initial direction of motion of the first cart as the positive direction. The momentum of the system will be conserved in the collision. If the collision is elastic, then the first cart will stop after the collision. Conservation of momentum therefore tells us that the second cart will have a final velocity v after the collision in the same direction as the initial velocity of the first cart. The kinetic energy of the system will be conserved since the masses are equal and the final velocity of cart 2 is equal to the initial velocity of cart 1. What would a graph of total momentum vs. time look like in this case? What would a graph of total kinetic energy vs. time look like in this case? Consider the center of mass of this system as the frame of reference. As cart 1 approaches cart 2, the center of mass remains exactly halfway between the two carts. The center of mass moves toward the stationary cart 2 at a speed 2 . After the collision, the center of mass continues moving in the same direction, away from (now stationary) cart 1 at a speed 2 . How would a graph of center-of-mass velocity vs. time compare to a graph of momentum vs. time? Suppose instead that the two carts move with equal speeds v in opposite directions towards the center of mass. Again, they have an elastic collision, so after the collision, they exchange velocities (each cart moving in the opposite direction of its initial motion with the same speed). As the two carts approach, the center of mass is exactly between the two carts, at the point where they will collide. In this case, how would a graph of center-of-mass velocity vs. time compare to a graph of the momentum of the system vs. time? Let us return to the example where the first cart is moving with a speed v toward the second cart, initially at rest. Suppose the second cart has some putty on one end so that, when the collision occurs, the two carts stick together in an inelastic collision. In this case, conservation of momentum tells us that the final velocity of the two-cart system will be half the initial velocity of the first cart, in the same direction as the first cart’s initial motion. Kinetic energy will not be conserved in this case, however. Compared to the moving cart before the collision, the overall moving mass after the collision is doubled, but the velocity is halved. The initial kinetic energy of the system is: = 1 22(1st cart)+0(2nd cart)=1 22 The final kinetic energy of the two carts (2m) moving together (at speed v/2) is: = 1 2 (2) 2 2 =1 42 (8.41) (8.42) What would a graph of total momentum vs. time look like in this case? What would a graph of total kinetic energy vs. time look like in this case? Consider the center of mass of this system. As cart 1 approaches cart 2, the center of mass remains exactly halfway between the two carts. The center of mass moves toward the stationary cart 2 at a speed 2 . After the collision, the two carts move together at a speed 2 momentum vs. time? . How would a graph of center-of-mass velocity vs. time compare to a graph of Suppose instead that the two carts move with equal speeds v in opposite directions towards the center of mass. They have putty on the end of each cart so that they stick together after the collision. As the two carts approach, the center of mass is exactly between the two carts, at the point where they will collide. In this case, how would a graph of center-of-mass velocity vs. time compare to a graph of the momentum of the system vs. time? 326 Chapter 8 \| Linear Momentum and Collisions Perhaps an easier way to see that momentum is conserved for an isolated system is to consider Newton’s second law in terms of momentum, Fnet = Δptot Δ . For an isolated system, Fnet = 0 ; thus, Δptot = 0 , and ptot is constant. We have noted that the three length dimensions in nature— , , and —are independent, and it is interesting to note that momentum can be conserved in different ways along each dimension. For example, during projectile motion and where air resistance is negligible, momentum is conserved in the horizontal direction because horizontal forces are zero and momentum is unchanged. But along the vertical direction, the net vertical force is not zero and the momentum of the projectile is not conserved. (See Figure 8.7.) However, if the momentum of the projectile-Earth system is considered in the vertical direction, we find that the total momentum is conserved. Figure 8.7
The horizontal component of a projectile’s momentum is conserved if air resistance is negligible, even in this case where a space probe separates. The forces causing the separation are internal to the system, so that the net external horizontal force – net is still zero. The vertical component of the momentum is not conserved, because the net vertical force – net is not zero. In the vertical direction, the space probe-Earth system needs to be considered and we find that the total momentum is conserved. The center of mass of the space probe takes the same path it would if the separation did not occur. The conservation of momentum principle can be applied to systems as different as a comet striking Earth and a gas containing huge numbers of atoms and molecules. Conservation of momentum is violated only when the net external force is not zero. But another larger system can always be considered in which momentum is conserved by simply including the source of the external force. For example, in the collision of two cars considered above, the two-car system conserves momentum while each one-car system does not. Making Connections: Take-Home Investigation—Drop of Tennis Ball and a Basketball Hold a tennis ball side by side and in contact with a basketball. Drop the balls together. (Be careful!) What happens? Explain your observations. Now hold the tennis ball above and in contact with the basketball. What happened? Explain your observations. What do you think will happen if the basketball ball is held above and in contact with the tennis ball? Making Connections: Take-Home Investigation—Two Tennis Balls in a Ballistic Trajectory Tie two tennis balls together with a string about a foot long. Hold one ball and let the other hang down and throw it in a ballistic trajectory. Explain your observations. Now mark the center of the string with bright ink or attach a brightly colored sticker to it and throw again. What happened? Explain your observations. Some aquatic animals such as jellyfish move around based on the principles of conservation of momentum. A jellyfish fills its umbrella section with water and then pushes the water out resulting in motion in the opposite direction to that of the jet of water. Squids propel themselves in a similar manner but, in contrast with jellyfish, are able to control the direction in which they move by aiming their nozzle forward or backward. Typical squids can move at speeds of 8 to 12 km/h. The ballistocardiograph (BCG) was a diagnostic tool used in the second half of the 20th century to study the strength of the heart. About once a second, your heart beats, forcing blood into the aorta. A force in the opposite direction is exerted on the rest of your body (recall Newton’s third law). A ballistocardiograph is a device that can measure this reaction force. This measurement is done by using a sensor (resting on the person) or by using a moving table suspended from the ceiling. This technique can gather information on the strength of the heart beat and the volume of blood passing from the heart. However, the electrocardiogram (ECG or EKG) and the echocardiogram (cardiac ECHO or ECHO; a technique that uses ultrasound to see an image of the heart) are more widely used in the practice of cardiology. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 8 \| Linear Momentum and Collisions 327 Applying Science Practices: Verifying the Conservation of Linear Momentum Design an experiment to verify the conservation of linear momentum in a one-dimensional collision, both elastic and inelastic. For simplicity, try to ensure that friction is minimized so that it has a negligible effect on your experiment. As you consider your experiment, consider the following questions: • Predict how the final momentum of the system will compare to the initial momentum of the system that you will measure. Justify your prediction. • How will you measure the momentum of each object? • Should you have two objects in motion or one object bouncing off a rigid surface? • Should you verify the relationship mathematically or graphically? • How will you estimate the uncertainty of your measurements? How will you express this uncertainty in your data? When you have completed each experiment, compare the outcome to your prediction about the initial and final momentum of the system and evaluate your results. Making Connections: Conservation of Momentum and Collision Conservation of momentum is quite useful in describing collisions. Momentum is crucial to our understanding of atomic and subatomic particles because much of what we know about these particles comes from collision experiments. Subatomic Collisions and Momentum The conservation of momentum principle not only applies to the macroscopic objects, it is also essential to our explorations of atomic and subatomic particles. Giant machines hurl subatomic particles at one another, and researchers evaluate the results by assuming conservation of momentum (among other things). On the small scale, we find that particles and their properties are invisible to the naked eye but can be measured with our instruments, and models of these subatomic particles can be constructed to describe the results. Momentum is found to be a property of all subatomic particles including massless particles such as photons that compose light. Momentum being a property of particles hints that momentum may have an identity beyond the description of an object’s mass multiplied by the object’s velocity. Indeed, momentum relates to wave properties and plays a fundamental role in what measurements are taken and how we take these measurements. Furthermore, we find that the conservation of momentum principle is valid when considering systems of particles. We use this principle to analyze the masses and other properties of previously undetected particles, such as the nucleus of an atom and the existence of quarks that make up particles of nuclei. Figure 8.8 below illustrates how a particle scattering backward from another implies that its target is massive and dense. Experiments seeking evidence that quarks make up protons (one type of particle that makes up nuclei) scattered high-energy electrons off of protons (nuclei of hydrogen atoms). Electrons occasionally scattered straight backward in a manner that implied a very small and very dense particle makes up the proton—this observation is considered nearly direct evidence of quarks. The analysis was based partly on the same conservation of momentum principle that works so well on the large scale. Figure 8.8 A subatomic particle scatters straight backward from a target particle. In experiments seeking evidence for quarks, electrons were observed to occasionally scatter straight backward from a proton. 328 Chapter 8 \| Linear Momentum and Collisions 8.4 Elastic Collisions in One Dimension Learning Objectives By the end of this section, you will be able to: • Describe an elastic collision of two objects in one dimension. • Define internal kinetic energy. • Derive an expression for conservation of internal kinetic energy in a one-dimensional collision. • Determine the final velocities in an elastic collision given masses and initial velocities. The information presented in this section supports the following AP® learning objectives and science practices: • 4.B.1.1 The student is able to calculate the change in linear momentum of a two-object system with constant mass in linear motion from a representation of the system (data, graphs, etc.). (S.P. 1.4, 2.2) • 4.B.1.2 The student is able to analyze data to find the change in linear momentum for a constant-mass system using the product of the mass and the change in velocity of the center of mass. (S.P. 5.1) • 5.A.2.1 The student is able to define open and closed systems for everyday situations and apply conservation concepts for energy, charge, and linear momentum to those situations. (S.P. 6.4, 7.2) • 5.D.1.1 The student is able to make qualitative predictions about natural phenomena based on conservation of linear momentum and restoration of kinetic energy in elastic collisions. (S.P. 6.4, 7.2) • 5.D.1.2 The student is able to apply the principles of conservation of momentum and restoration of kinetic energy to reconcile a situation that appears to be isolated and elastic, but in which data indicate that linear momentum and kinetic energy are not the same after the interaction, by refining a scientific question to identify interactions that have not been considered. Students will be expected to solve qualitatively and/or quantitatively for one-dimensional situations and only qualitatively in two-dimensional situations. (S.P. 2.2, 3.2, 5.1, 5.3) • 5.D.1.5 The student is able to classify a given collision situation as elastic or inelastic, justify the selection of conservation of linear momentum and restoration of kinetic energy as the appropriate principles for analyzing an elastic collision, solve for missing variables, and calculate their values. (S.P. 2.1, 2.2) • 5.D.1.6 The student is able to make predictions of the dynamical properties of a system undergoing a collision by application of the principle of linear momentum conservation and the principle of the conservation of energy in situations in which an elastic collision may also be assumed. (S.P. 6.4) • 5.D.1.7 The student is able to classify a given collision situation as elastic or inelastic, justify the selection of conservation of linear momentum and restoration of kinetic energy as the appropriate principles for analyzing an elastic collision, solve for missing variables, and calculate their values. (S.P. 2.1, 2.2) • 5.D.2.1 The student is able to qualitatively predict, in terms of linear momentum and kinetic energy, how the outcome of a collision between two objects changes depending on whether the collision is elastic or inelastic. (S.P. 6.4, 7.2) • 5.D.2.2 The student is able to plan data collection strategies to test t
he law of conservation of momentum in a two- object collision that is elastic or inelastic and analyze the resulting data graphically. (S.P.4.1, 4.2, 5.1) • 5.D.3.2 The student is able to make predictions about the velocity of the center of mass for interactions within a defined one-dimensional system. (S.P. 6.4) Let us consider various types of two-object collisions. These collisions are the easiest to analyze, and they illustrate many of the physical principles involved in collisions. The conservation of momentum principle is very useful here, and it can be used whenever the net external force on a system is zero. We start with the elastic collision of two objects moving along the same line—a one-dimensional problem. An elastic collision is one that also conserves internal kinetic energy. Internal kinetic energy is the sum of the kinetic energies of the objects in the system. Figure 8.9 illustrates an elastic collision in which internal kinetic energy and momentum are conserved. Truly elastic collisions can only be achieved with subatomic particles, such as electrons striking nuclei. Macroscopic collisions can be very nearly, but not quite, elastic—some kinetic energy is always converted into other forms of energy such as heat transfer due to friction and sound. One macroscopic collision that is nearly elastic is that of two steel blocks on ice. Another nearly elastic collision is that between two carts with spring bumpers on an air track. Icy surfaces and air tracks are nearly frictionless, more readily allowing nearly elastic collisions on them. Elastic Collision An elastic collision is one that conserves internal kinetic energy. Internal Kinetic Energy Internal kinetic energy is the sum of the kinetic energies of the objects in the system. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 8 \| Linear Momentum and Collisions 329 Figure 8.9 An elastic one-dimensional two-object collision. Momentum and internal kinetic energy are conserved. Now, to solve problems involving one-dimensional elastic collisions between two objects we can use the equations for conservation of momentum and conservation of internal kinetic energy. First, the equation for conservation of momentum for two objects in a one-dimensional collision is or 1 + 2 = ′1+ ′2 net = 0 1 1 + 22 = 1′1 + 2′2 net = 0 , (8.43) (8.44) where the primes (') indicate values after the collision. By definition, an elastic collision conserves internal kinetic energy, and so the sum of kinetic energies before the collision equals the sum after the collision. Thus, 2 = 1 2 (two-object elastic collision) 2 + 1 2 + 1 (8.45) 22 2 21 ′1 22 ′2 1 21 1 expresses the equation for conservation of internal kinetic energy in a one-dimensional collision. Making Connections: Collisions Suppose data are collected on a collision between two masses sliding across a frictionless surface. Mass A (1.0 kg) moves with a velocity of +12 m/s, and mass B (2.0 kg) moves with a velocity of −12 m/s. The two masses collide and stick together after the collision. The table below shows the measured velocities of each mass at times before and after the collision: Table 8.1 Time (s) Velocity A (m/s) Velocity B (m/s) 0 1.0 s 2.0 s 3.0 s +12 +12 −4.0 −4.0 −12 −12 −4.0 −4.0 The total mass of the system is 3.0 kg. The velocity of the center of mass of this system can be determined from the conservation of momentum. Consider the system before the collision: ( + ) = + (3.0) = (1)(12) + (2)( − 12) = − 4.0 m/s (8.46) (8.47) (8.48) 330 Chapter 8 \| Linear Momentum and Collisions After the collision, the center-of-mass velocity is the same: ( + ) = ( + ) (3.0) = (3)( − 4.0) = − 4.0 m/s The total momentum of the system before the collision is: + = (1)(12) + (2)( − 12) = − 12 kg • m/s The total momentum of the system after the collision is: ( + ) = (3)( − 4) = − 12 kg • m/s (8.49) (8.50) (8.51) (8.52) (8.53) Thus, the change in momentum of the system is zero when measured this way. We get a similar result when we calculate the momentum using the center-of-mass velocity. Since the center-of-mass velocity is the same both before and after the collision, we calculate the same momentum for the system using this method both before and after the collision. Example 8.4 Calculating Velocities Following an Elastic Collision Calculate the velocities of two objects following an elastic collision, given that 1 = 0.500 kg, 2 = 3.50 kg, 1 = 4.00 m/s, and 2 = 0. (8.54) Strategy and Concept First, visualize what the initial conditions mean—a small object strikes a larger object that is initially at rest. This situation is slightly simpler than the situation shown in Figure 8.9 where both objects are initially moving. We are asked to find two unknowns (the final velocities ′1 and ′2 ). To find two unknowns, we must use two independent equations. Because this collision is elastic, we can use the above two equations. Both can be simplified by the fact that object 2 is initially at rest, and thus 2 = 0 . Once we simplify these equations, we combine them algebraically to solve for the unknowns. Solution For this problem, note that 2 = 0 and use conservation of momentum. Thus, or 1 = ′1 + ′2 1 1 = 1′1 + 2′2. Using conservation of internal kinetic energy and that 2 = 0 , Solving the first equation (momentum equation) for ′2 , we obtain 1 21 1 2 = 1 21 ′1 2 + 1 22 ′2 2. ′2 = 1 2 1 − ′1 . (8.55) (8.56) (8.57) (8.58) Substituting this expression into the second equation (internal kinetic energy equation) eliminates the variable ′2 , leaving only ′1 as an unknown (the algebra is left as an exercise for the reader). There are two solutions to any quadratic equation; in this example, they are and ′1 = 4.00 m/s ′1 = −3.00 m/s. (8.59) (8.60) As noted when quadratic equations were encountered in earlier chapters, both solutions may or may not be meaningful. In this case, the first solution is the same as the initial condition. The first solution thus represents the situation before the collision and is discarded. The second solution (′1 = −3.00 m/s) is negative, meaning that the first object bounces backward. When this negative value of ′1 is used to find the velocity of the second object after the collision, we get This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 8 \| Linear Momentum and Collisions ′2 = 1 2 1 − ′1 = 0.500 kg 3.50 kg 4.00 − (−3.00) m/s or Discussion ′2 = 1.00 m/s. 331 (8.61) (8.62) The result of this example is intuitively reasonable. A small object strikes a larger one at rest and bounces backward. The larger one is knocked forward, but with a low speed. (This is like a compact car bouncing backward off a full-size SUV that is initially at rest.) As a check, try calculating the internal kinetic energy before and after the collision. You will see that the internal kinetic energy is unchanged at 4.00 J. Also check the total momentum before and after the collision; you will find it, too, is unchanged. The equations for conservation of momentum and internal kinetic energy as written above can be used to describe any onedimensional elastic collision of two objects. These equations can be extended to more objects if needed. Making Connections: Take-Home Investigation—Ice Cubes and Elastic Collision Find a few ice cubes which are about the same size and a smooth kitchen tabletop or a table with a glass top. Place the ice cubes on the surface several centimeters away from each other. Flick one ice cube toward a stationary ice cube and observe the path and velocities of the ice cubes after the collision. Try to avoid edge-on collisions and collisions with rotating ice cubes. Have you created approximately elastic collisions? Explain the speeds and directions of the ice cubes using momentum. PhET Explorations: Collision Lab Investigate collisions on an air hockey table. Set up your own experiments: vary the number of discs, masses and initial conditions. Is momentum conserved? Is kinetic energy conserved? Vary the elasticity and see what happens. Figure 8.10 Collision Lab (http://cnx.org/content/m55171/1.3/collision-lab_en.jar) 8.5 Inelastic Collisions in One Dimension Learning Objectives By the end of this section, you will be able to: • Define inelastic collision. • Explain perfectly inelastic collisions. • Apply an understanding of collisions to sports. • Determine recoil velocity and loss in kinetic energy given mass and initial velocity. The information presented in this section supports the following AP® learning objectives and science practices: • 4.B.1.1 The student is able to calculate the change in linear momentum of a two-object system with constant mass in linear motion from a representation of the system (data, graphs, etc.). (S.P. 1.4, 2.2) • 5.A.2.1 The student is able to define open and closed systems for everyday situations and apply conservation concepts for energy, charge, and linear momentum to those situations. (S.P. 6.4, 7.2) • 5.D.1.3 The student is able to apply mathematical routines appropriately to problems involving elastic collisions in one dimension and justify the selection of those mathematical routines based on conservation of momentum and restoration of kinetic energy. (S.P. 2.1, 2.2) • 5.D.1.5 The student is able to classify a given collision situation as elastic or inelastic, justify the selection of conservation of linear momentum and restoration of kinetic energy as the appropriate principles for analyzing an elastic collision, solve for missing variables, and calculate their values. (S.P. 2.1, 2.2) • 5.D.2.1 The student is able to qualitatively predict, in terms of linear momentum and kinetic energy, how the outcome of a collision between two objects changes depending on whether the collision is elastic or inelastic. (S.P. 6.4, 7.2) • 5.D.2.2 The student is able to plan data collection strategies to test the law of conservation of momentum in a two- object collision that is elastic or inelastic and analyze the resulting data
graphically. (S.P.4.1, 4.2, 5.1) • 5.D.2.3 The student is able to apply the conservation of linear momentum to a closed system of objects involved in an inelastic collision to predict the change in kinetic energy. (S.P. 6.4, 7.2) 332 Chapter 8 \| Linear Momentum and Collisions • 5.D.2.4 The student is able to analyze data that verify conservation of momentum in collisions with and without an external friction force. (S.P. 4.1, 4.2, 4.4, 5.1, 5.3) • 5.D.2.5 The student is able to classify a given collision situation as elastic or inelastic, justify the selection of conservation of linear momentum as the appropriate solution method for an inelastic collision, recognize that there is a common final velocity for the colliding objects in the totally inelastic case, solve for missing variables, and calculate their values. (S.P. 2.1 2.2) • 5.D.2.6 The student is able to apply the conservation of linear momentum to an isolated system of objects involved in an inelastic collision to predict the change in kinetic energy. (S.P. 6.4, 7.2) We have seen that in an elastic collision, internal kinetic energy is conserved. An inelastic collision is one in which the internal kinetic energy changes (it is not conserved). This lack of conservation means that the forces between colliding objects may remove or add internal kinetic energy. Work done by internal forces may change the forms of energy within a system. For inelastic collisions, such as when colliding objects stick together, this internal work may transform some internal kinetic energy into heat transfer. Or it may convert stored energy into internal kinetic energy, such as when exploding bolts separate a satellite from its launch vehicle. Inelastic Collision An inelastic collision is one in which the internal kinetic energy changes (it is not conserved). Figure 8.11 shows an example of an inelastic collision. Two objects that have equal masses head toward one another at equal speeds and then stick together. Their total internal kinetic energy is initially 1 22 + 1 22 = 2 . The two objects come to rest after sticking together, conserving momentum. But the internal kinetic energy is zero after the collision. A collision in which the objects stick together is sometimes called a perfectly inelastic collision because it reduces internal kinetic energy more than does any other type of inelastic collision. In fact, such a collision reduces internal kinetic energy to the minimum it can have while still conserving momentum. Perfectly Inelastic Collision A collision in which the objects stick together is sometimes called “perfectly inelastic.” Figure 8.11 An inelastic one-dimensional two-object collision. Momentum is conserved, but internal kinetic energy is not conserved. (a) Two objects of equal mass initially head directly toward one another at the same speed. (b) The objects stick together (a perfectly inelastic collision), and so their final velocity is zero. The internal kinetic energy of the system changes in any inelastic collision and is reduced to zero in this example. Example 8.5 Calculating Velocity and Change in Kinetic Energy: Inelastic Collision of a Puck and a Goalie (a) Find the recoil velocity of a 70.0-kg ice hockey goalie, originally at rest, who catches a 0.150-kg hockey puck slapped at him at a velocity of 35.0 m/s. (b) How much kinetic energy is lost during the collision? Assume friction between the ice and the puck-goalie system is negligible. (See Figure 8.12 ) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 8 \| Linear Momentum and Collisions 333 Figure 8.12 An ice hockey goalie catches a hockey puck and recoils backward. The initial kinetic energy of the puck is almost entirely converted to thermal energy and sound in this inelastic collision. Strategy Momentum is conserved because the net external force on the puck-goalie system is zero. We can thus use conservation of momentum to find the final velocity of the puck and goalie system. Note that the initial velocity of the goalie is zero and that the final velocity of the puck and goalie are the same. Once the final velocity is found, the kinetic energies can be calculated before and after the collision and compared as requested. Solution for (a) Momentum is conserved because the net external force on the puck-goalie system is zero. Conservation of momentum is or 1 + 2 = ′1 + ′2 1 1 + 22 = 1′1 + 2′2. (8.63) (8.64) Because the goalie is initially at rest, we know 2 = 0 . Because the goalie catches the puck, the final velocities are equal, or ′1 = ′2 = ′ . Thus, the conservation of momentum equation simplifies to Solving for ′ yields 1 1 = (1 + 2)′. ′ = 1 1 + 2 1. Entering known values in this equation, we get ′ = 0.150 kg 70.0 kg + 0.150 kg (35.0 m/s) = 7.48×10−2 m/s. Discussion for (a) (8.65) (8.66) (8.67) This recoil velocity is small and in the same direction as the puck’s original velocity, as we might expect. Solution for (b) Before the collision, the internal kinetic energy KEint of the system is that of the hockey puck, because the goalie is initially at rest. Therefore, KEint is initially KEint = 1 22 = 1 2 = 91.9 J. 0.150 kg (35.0 m/s)2 After the collision, the internal kinetic energy is KE′int = 1 2 ( + )2 = 1 2 = 0.196 J. The change in internal kinetic energy is thus 70.15 kg 7.48×10−2 m/s (8.68) 2 (8.69) 334 Chapter 8 \| Linear Momentum and Collisions KE′int − KEint = 0.196 J − 91.9 J = − 91.7 J (8.70) where the minus sign indicates that the energy was lost. Discussion for (b) Nearly all of the initial internal kinetic energy is lost in this perfectly inelastic collision. KEint is mostly converted to thermal energy and sound. During some collisions, the objects do not stick together and less of the internal kinetic energy is removed—such as happens in most automobile accidents. Alternatively, stored energy may be converted into internal kinetic energy during a collision. Figure 8.13 shows a one-dimensional example in which two carts on an air track collide, releasing potential energy from a compressed spring. Example 8.6 deals with data from such a collision. Figure 8.13 An air track is nearly frictionless, so that momentum is conserved. Motion is one-dimensional. In this collision, examined in Example 8.6, the potential energy of a compressed spring is released during the collision and is converted to internal kinetic energy. Collisions are particularly important in sports and the sporting and leisure industry utilizes elastic and inelastic collisions. Let us look briefly at tennis. Recall that in a collision, it is momentum and not force that is important. So, a heavier tennis racquet will have the advantage over a lighter one. This conclusion also holds true for other sports—a lightweight bat (such as a softball bat) cannot hit a hardball very far. The location of the impact of the tennis ball on the racquet is also important, as is the part of the stroke during which the impact occurs. A smooth motion results in the maximizing of the velocity of the ball after impact and reduces sports injuries such as tennis elbow. A tennis player tries to hit the ball on the “sweet spot” on the racquet, where the vibration and impact are minimized and the ball is able to be given more velocity. Sports science and technologies also use physics concepts such as momentum and rotational motion and vibrations. Take-Home Experiment—Bouncing of Tennis Ball 1. Find a racquet (a tennis, badminton, or other racquet will do). Place the racquet on the floor and stand on the handle. Drop a tennis ball on the strings from a measured height. Measure how high the ball bounces. Now ask a friend to hold the racquet firmly by the handle and drop a tennis ball from the same measured height above the racquet. Measure how high the ball bounces and observe what happens to your friend’s hand during the collision. Explain your observations and measurements. 2. The coefficient of restitution () is a measure of the elasticity of a collision between a ball and an object, and is defined as the ratio of the speeds after and before the collision. A perfectly elastic collision has a of 1. For a ball bouncing off the floor (or a racquet on the floor), can be shown to be = ( / )1 / 2 which the ball bounces and is the height from which the ball is dropped. Determine for the cases in Part 1 and where is the height to This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 8 \| Linear Momentum and Collisions 335 for the case of a tennis ball bouncing off a concrete or wooden floor ( = 0.85 for new tennis balls used on a tennis court). Example 8.6 Calculating Final Velocity and Energy Release: Two Carts Collide In the collision pictured in Figure 8.13, two carts collide inelastically. Cart 1 (denoted 1 carries a spring which is initially compressed. During the collision, the spring releases its potential energy and converts it to internal kinetic energy. The mass of cart 1 and the spring is 0.350 kg, and the cart and the spring together have an initial velocity of 2.00 m/s . Cart 2 (denoted 2 in Figure 8.13) has a mass of 0.500 kg and an initial velocity of −0.500 m/s . After the collision, cart 1 is observed to recoil with a velocity of −4.00 m/s . (a) What is the final velocity of cart 2? (b) How much energy was released by the spring (assuming all of it was converted into internal kinetic energy)? Strategy We can use conservation of momentum to find the final velocity of cart 2, because net = 0 (the track is frictionless and the force of the spring is internal). Once this velocity is determined, we can compare the internal kinetic energy before and after the collision to see how much energy was released by the spring. Solution for (a) As before, the equation for conservation of momentum in a two-object system is The only unknown in this equation is ′2 . Solving for ′2 and substituting known values into the previous equation yields 1 1 + 22 = 1′1 + 2′2 . (8.71) ′2 = = 1 1 + 22 − 1
′1 2 (2.00 m/s) + 0.350 kg 0.500 kg (−0.500 m/s) 0.500 kg − 0.350 kg (−4.00 m/s) 0.500 kg = 3.70 m/s. Solution for (b) The internal kinetic energy before the collision is 2 + 1 KEint = 1 = 1 2 21 1 0.350 kg 2 22 2 (2.00 m/s)2 + 1 2 0.500 kg ( – 0.500 m/s)2 After the collision, the internal kinetic energy is = 0.763 J. 2 + 1 KE′int = 1 = 1 2 = 6.22 J. 21 ′1 0.350 kg 2 22 ′2 (-4.00 m/s)2 + 1 2 0.500 kg (3.70 m/s)2 The change in internal kinetic energy is thus KE′int − KEint = 6.22 J − 0.763 J = 5.46 J. (8.72) (8.73) (8.74) (8.75) Discussion The final velocity of cart 2 is large and positive, meaning that it is moving to the right after the collision. The internal kinetic energy in this collision increases by 5.46 J. That energy was released by the spring. 8.6 Collisions of Point Masses in Two Dimensions By the end of this section, you will be able to: • Discuss two-dimensional collisions as an extension of one-dimensional analysis. Learning Objectives 336 Chapter 8 \| Linear Momentum and Collisions • Define point masses. • Derive an expression for conservation of momentum along the x-axis and y-axis. • Describe elastic collisions of two objects with equal mass. • Determine the magnitude and direction of the final velocity given initial velocity and scattering angle. The information presented in this section supports the following AP® learning objectives and science practices: • 5.D.1.2 The student is able to apply the principles of conservation of momentum and restoration of kinetic energy to reconcile a situation that appears to be isolated and elastic, but in which data indicate that linear momentum and kinetic energy are not the same after the interaction, by refining a scientific question to identify interactions that have not been considered. Students will be expected to solve qualitatively and/or quantitatively for one-dimensional situations and only qualitatively in two-dimensional situations. • 5.D.3.3 The student is able to make predictions about the velocity of the center of mass for interactions within a defined two-dimensional system. In the previous two sections, we considered only one-dimensional collisions; during such collisions, the incoming and outgoing velocities are all along the same line. But what about collisions, such as those between billiard balls, in which objects scatter to the side? These are two-dimensional collisions, and we shall see that their study is an extension of the one-dimensional analysis already presented. The approach taken (similar to the approach in discussing two-dimensional kinematics and dynamics) is to choose a convenient coordinate system and resolve the motion into components along perpendicular axes. Resolving the motion yields a pair of one-dimensional problems to be solved simultaneously. One complication arising in two-dimensional collisions is that the objects might rotate before or after their collision. For example, if two ice skaters hook arms as they pass by one another, they will spin in circles. We will not consider such rotation until later, and so for now we arrange things so that no rotation is possible. To avoid rotation, we consider only the scattering of point masses—that is, structureless particles that cannot rotate or spin. We start by assuming that Fnet = 0 , so that momentum p is conserved. The simplest collision is one in which one of the particles is initially at rest. (See Figure 8.14.) The best choice for a coordinate system is one with an axis parallel to the velocity of the incoming particle, as shown in Figure 8.14. Because momentum is conserved, the components of momentum along the - and -axes ( and ) will also be conserved, but with the chosen coordinate system, is initially zero and is the momentum of the incoming particle. Both facts simplify the analysis. (Even with the simplifying assumptions of point masses, one particle initially at rest, and a convenient coordinate system, we still gain new insights into nature from the analysis of twodimensional collisions.) Figure 8.14 A two-dimensional collision with the coordinate system chosen so that 2 is initially at rest and 1 is parallel to the -axis. This coordinate system is sometimes called the laboratory coordinate system, because many scattering experiments have a target that is stationary in the laboratory, while particles are scattered from it to determine the particles that make-up the target and how they are bound together. The particles may not be observed directly, but their initial and final velocities are. Along the -axis, the equation for conservation of momentum is Where the subscripts denote the particles and axes and the primes denote the situation after the collision. In terms of masses and velocities, this equation is 1 + 2 = ′1 + ′2. (8.76) 1 1 + 22 = 11 + 22. But because particle 2 is initially at rest, this equation becomes 1 1 = 1′1 + 2′2. (8.77) (8.78) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 8 \| Linear Momentum and Collisions 337 The components of the velocities along the -axis have the form cos . Because particle 1 initially moves along the -axis, we find 1 = 1 . Conservation of momentum along the -axis gives the following equation: 1 1 = 1′1 cos 1 + 2′2 cos 2, where 1 and 2 are as shown in Figure 8.14. Conservation of Momentum along the -axis 1 1 = 1′1 cos 1 + 2′2 cos 2 Along the -axis, the equation for conservation of momentum is or 1 + 2 = ′1 + ′2 1 1 + 22 = 1′1 + 2′2. (8.79) (8.80) (8.81) (8.82) But 1 is zero, because particle 1 initially moves along the -axis. Because particle 2 is initially at rest, 2 is also zero. The equation for conservation of momentum along the -axis becomes 0 = 1′1 + 2′2. The components of the velocities along the -axis have the form sin . Thus, conservation of momentum along the -axis gives the following equation: 0 = 1′1 sin 1 + 2′2 sin 2. Conservation of Momentum along the -axis 0 = 1′1 sin 1 + 2′2 sin 2 (8.83) (8.84) (8.85) The equations of conservation of momentum along the -axis and -axis are very useful in analyzing two-dimensional collisions of particles, where one is originally stationary (a common laboratory situation). But two equations can only be used to find two unknowns, and so other data may be necessary when collision experiments are used to explore nature at the subatomic level. Making Connections: Real World Connections We have seen, in one-dimensional collisions when momentum is conserved, that the center-of-mass velocity of the system remains unchanged as a result of the collision. If you calculate the momentum and center-of-mass velocity before the collision, you will get the same answer as if you calculate both quantities after the collision. This logic also works for twodimensional collisions. For example, consider two cars of equal mass. Car A is driving east (+x-direction) with a speed of 40 m/s. Car B is driving north (+y-direction) with a speed of 80 m/s. What is the velocity of the center-of-mass of this system before and after an inelastic collision, in which the cars move together as one mass after the collision? Since both cars have equal mass, the center-of-mass velocity components are just the average of the components of the individual velocities before the collision. The x-component of the center of mass velocity is 20 m/s, and the y-component is 40 m/s. Using momentum conservation for the collision in both the x-component and y-component yields similar answers: (40) + (0) = (2)final() final() = 20 m/s (0) + (80) = (2)final() final() = 40 m/s (8.86) (8.87) (8.88) (8.89) 338 Chapter 8 \| Linear Momentum and Collisions Since the two masses move together after the collision, the velocity of this combined object is equal to the center-of-mass velocity. Thus, the center-of-mass velocity before and after the collision is identical, even in two-dimensional collisions, when momentum is conserved. Example 8.7 Determining the Final Velocity of an Unseen Object from the Scattering of Another Object Suppose the following experiment is performed. A 0.250-kg object (1) is slid on a frictionless surface into a dark room, where it strikes an initially stationary object with mass of 0.400 kg (2) . The 0.250-kg object emerges from the room at an angle of 45.0º with its incoming direction. The speed of the 0.250-kg object is originally 2.00 m/s and is 1.50 m/s after the collision. Calculate the magnitude and direction of the velocity (′2 and 2) of the 0.400-kg object after the collision. Strategy Momentum is conserved because the surface is frictionless. The coordinate system shown in Figure 8.15 is one in which 2 is originally at rest and the initial velocity is parallel to the -axis, so that conservation of momentum along the - and -axes is applicable. Everything is known in these equations except ′2 and 2 , which are precisely the quantities we wish to find. We can find two unknowns because we have two independent equations: the equations describing the conservation of momentum in the - and -directions. Solution Solving 1 1 = 1′1 cos 1 + 2′2 cos 2 for 2′ cos 2 and 0 = 1′1 sin 1 + 2′2 sin 2 for ′2 sin 2 and taking the ratio yields an equation (in which θ2 is the only unknown quantity. Applying the identity tan = sin cos , we obtain: tan 2 = ′1 sin 1 ′1 cos 1 − 1 . Entering known values into the previous equation gives tan 2 = (1.50 m/s)(0.7071) (1.50 m/s)(0.7071) − 2.00 m/s = −1.129. Thus, 2 = tan−1(−1.129) = 311.5º ≈ 312º. (8.90) (8.91) (8.92) Angles are defined as positive in the counter clockwise direction, so this angle indicates that 2 is scattered to the right in Figure 8.15, as expected (this angle is in the fourth quadrant). Either equation for the - or -axis can now be used to solve for ′2 , but the latter equation is easiest because it has fewer terms. Entering known values into this equation gives ′2 = − 1 2 ′1 sin 1 sin 2 ′2 = − 0.250 kg 0.400 kg (1.50 m/s) 0.7071 −0.7485 . ′2 = 0.886 m/s. Thus, Discussion (8.93) (8.94) (8.95) It is instructive to calculate
the internal kinetic energy of this two-object system before and after the collision. (This calculation is left as an end-of-chapter problem.) If you do this calculation, you will find that the internal kinetic energy is less after the collision, and so the collision is inelastic. This type of result makes a physicist want to explore the system further. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 8 \| Linear Momentum and Collisions 339 Figure 8.15 A collision taking place in a dark room is explored in Example 8.7. The incoming object 1 is scattered by an initially stationary object. Only the stationary object’s mass 2 is known. By measuring the angle and speed at which 1 emerges from the room, it is possible to calculate the magnitude and direction of the initially stationary object’s velocity after the collision. Elastic Collisions of Two Objects with Equal Mass Some interesting situations arise when the two colliding objects have equal mass and the collision is elastic. This situation is nearly the case with colliding billiard balls, and precisely the case with some subatomic particle collisions. We can thus get a mental image of a collision of subatomic particles by thinking about billiards (or pool). (Refer to Figure 8.14 for masses and angles.) First, an elastic collision conserves internal kinetic energy. Again, let us assume object 2 (2) is initially at rest. Then, the internal kinetic energy before and after the collision of two objects that have equal masses is 1 21 2 = 1 2′1 2 + 1 2′2 2. (8.96) Because the masses are equal, 1 = 2 = . Algebraic manipulation (left to the reader) of conservation of momentum in the - and -directions can show that 2′1 (Remember that 2 is negative here.) The two preceding equations can both be true only if 1 − 2 2′2 2 + ′1′2 cos 1 21 2 = 1 2 + 1 . ′1 ′2 cos 1 − 2 = 0. (8.97) (8.98) • There are three ways that this term can be zero. They are ′1 = 0 : head-on collision; incoming ball stops ′2 = 0 : no collision; incoming ball continues unaffected cos(1 − 2) = 0 : angle of separation (1 − 2) is 90º after the collision • • All three of these ways are familiar occurrences in billiards and pool, although most of us try to avoid the second. If you play enough pool, you will notice that the angle between the balls is very close to 90º after the collision, although it will vary from this value if a great deal of spin is placed on the ball. (Large spin carries in extra energy and a quantity called angular momentum, which must also be conserved.) The assumption that the scattering of billiard balls is elastic is reasonable based on the correctness of the three results it produces. This assumption also implies that, to a good approximation, momentum is conserved for the two-ball system in billiards and pool. The problems below explore these and other characteristics of two-dimensional collisions. 340 Chapter 8 \| Linear Momentum and Collisions Connections to Nuclear and Particle Physics Two-dimensional collision experiments have revealed much of what we know about subatomic particles, as we shall see in Medical Applications of Nuclear Physics and Particle Physics. Ernest Rutherford, for example, discovered the nature of the atomic nucleus from such experiments. 8.7 Introduction to Rocket Propulsion Learning Objectives By the end of this section, you will be able to: • State Newton’s third law of motion. • Explain the principle involved in propulsion of rockets and jet engines. • Derive an expression for the acceleration of the rocket. • Discuss the factors that affect the rocket’s acceleration. • Describe the function of a space shuttle. Rockets range in size from fireworks so small that ordinary people use them to immense Saturn Vs that once propelled massive payloads toward the Moon. The propulsion of all rockets, jet engines, deflating balloons, and even squids and octopuses is explained by the same physical principle—Newton’s third law of motion. Matter is forcefully ejected from a system, producing an equal and opposite reaction on what remains. Another common example is the recoil of a gun. The gun exerts a force on a bullet to accelerate it and consequently experiences an equal and opposite force, causing the gun’s recoil or kick. Making Connections: Take-Home Experiment—Propulsion of a Balloon Hold a balloon and fill it with air. Then, let the balloon go. In which direction does the air come out of the balloon and in which direction does the balloon get propelled? If you fill the balloon with water and then let the balloon go, does the balloon’s direction change? Explain your answer. Figure 8.16 shows a rocket accelerating straight up. In part (a), the rocket has a mass and a velocity relative to Earth, and hence a momentum . In part (b), a time Δ has elapsed in which the rocket has ejected a mass Δ of hot gas at a velocity e relative to the rocket. The remainder of the mass ( − Δ) now has a greater velocity ( + Δ) . The momentum of the entire system (rocket plus expelled gas) has actually decreased because the force of gravity has acted for a time Δ , producing a negative impulse Δ = −Δ . (Remember that impulse is the net external force on a system multiplied by the time it acts, and it equals the change in momentum of the system.) So, the center of mass of the system is in free fall but, by rapidly expelling mass, part of the system can accelerate upward. It is a commonly held misconception that the rocket exhaust pushes on the ground. If we consider thrust; that is, the force exerted on the rocket by the exhaust gases, then a rocket’s thrust is greater in outer space than in the atmosphere or on the launch pad. In fact, gases are easier to expel into a vacuum. By calculating the change in momentum for the entire system over Δ , and equating this change to the impulse, the following expression can be shown to be a good approximation for the acceleration of the rocket. “The rocket” is that part of the system remaining after the gas is ejected, and is the acceleration due to gravity. = e Δ Δ − Acceleration of a Rocket Acceleration of a rocket is = e Δ Δ − (8.99) (8.100) where is the acceleration of the rocket, e is the escape velocity, is the mass of the rocket, Δ is the mass of the ejected gas, and Δ is the time in which the gas is ejected. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 8 \| Linear Momentum and Collisions 341 Figure 8.16 (a) This rocket has a mass and an upward velocity . The net external force on the system is − , if air resistance is neglected. (b) A time Δ later the system has two main parts, the ejected gas and the remainder of the rocket. The reaction force on the rocket is what overcomes the gravitational force and accelerates it upward. A rocket’s acceleration depends on three major factors, consistent with the equation for acceleration of a rocket . First, the greater the exhaust velocity of the gases relative to the rocket, e , the greater the acceleration is. The practical limit for e is about 2.5×103 m/s for conventional (non-nuclear) hot-gas propulsion systems. The second factor is the rate at which mass is ejected from the rocket. This is the factor Δ / Δ in the equation. The quantity (Δ / Δ)e , with units of newtons, is called "thrust.” The faster the rocket burns its fuel, the greater its thrust, and the greater its acceleration. The third factor is the mass of the rocket. The smaller the mass is (all other factors being the same), the greater the acceleration. The rocket mass decreases dramatically during flight because most of the rocket is fuel to begin with, so that acceleration increases continuously, reaching a maximum just before the fuel is exhausted. Factors Affecting a Rocket’s Acceleration • The greater the exhaust velocity e of the gases relative to the rocket, the greater the acceleration. • The faster the rocket burns its fuel, the greater its acceleration. • The smaller the rocket’s mass (all other factors being the same), the greater the acceleration. Example 8.8 Calculating Acceleration: Initial Acceleration of a Moon Launch A Saturn V’s mass at liftoff was 2.80×106 kg , its fuel-burn rate was 1.40×104 kg/s , and the exhaust velocity was 2.40×103 m/s . Calculate its initial acceleration. Strategy This problem is a straightforward application of the expression for acceleration because is the unknown and all of the terms on the right side of the equation are given. Solution Substituting the given values into the equation for acceleration yields 342 Chapter 8 \| Linear Momentum and Collisions = e − Δ Δ = 2.40×103 m/s 2.80×106 kg 1.40×104 kg/s − 9.80 m/s2 (8.101) Discussion = 2.20 m/s2 . This value is fairly small, even for an initial acceleration. The acceleration does increase steadily as the rocket burns fuel, because decreases while e and Δ Δ show that the thrust of the engines was 3.36×107 N . remain constant. Knowing this acceleration and the mass of the rocket, you can To achieve the high speeds needed to hop continents, obtain orbit, or escape Earth’s gravity altogether, the mass of the rocket other than fuel must be as small as possible. It can be shown that, in the absence of air resistance and neglecting gravity, the final velocity of a one-stage rocket initially at rest is = e ln 0 r (8.102) where ln 0 / r is the natural logarithm of the ratio of the initial mass of the rocket (0) to what is left (r) after all of the fuel is exhausted. (Note that is actually the change in velocity, so the equation can be used for any segment of the flight. If we start from rest, the change in velocity equals the final velocity.) For example, let us calculate the mass ratio needed to escape Earth’s gravity starting from rest, given that the escape velocity from Earth is about 11.2×103 m/s , and assuming an exhaust velocity e = 2.5×103 m/s . Solving for 0 / r gives Thus, the mass of the rocket is ln 0 r = e = 11.2×103 m/s 2.5×103 m/s = 4.48 0 r = 4.48 = 88. r = 0 88
. (8.103) (8.104) (8.105) This result means that only 1 / 88 of the mass is left when the fuel is burnt, and 87 / 88 of the initial mass was fuel. Expressed as percentages, 98.9% of the rocket is fuel, while payload, engines, fuel tanks, and other components make up only 1.10%. Taking air resistance and gravitational force into account, the mass r remaining can only be about 0 / 180 . It is difficult to build a rocket in which the fuel has a mass 180 times everything else. The solution is multistage rockets. Each stage only needs to achieve part of the final velocity and is discarded after it burns its fuel. The result is that each successive stage can have smaller engines and more payload relative to its fuel. Once out of the atmosphere, the ratio of payload to fuel becomes more favorable, too. The space shuttle was an attempt at an economical vehicle with some reusable parts, such as the solid fuel boosters and the craft itself. (See Figure 8.17) The shuttle’s need to be operated by humans, however, made it at least as costly for launching satellites as expendable, unmanned rockets. Ideally, the shuttle would only have been used when human activities were required for the success of a mission, such as the repair of the Hubble space telescope. Rockets with satellites can also be launched from airplanes. Using airplanes has the double advantage that the initial velocity is significantly above zero and a rocket can avoid most of the atmosphere’s resistance. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 8 \| Linear Momentum and Collisions 343 Figure 8.17 The space shuttle had a number of reusable parts. Solid fuel boosters on either side were recovered and refueled after each flight, and the entire orbiter returned to Earth for use in subsequent flights. The large liquid fuel tank was expended. The space shuttle was a complex assemblage of technologies, employing both solid and liquid fuel and pioneering ceramic tiles as reentry heat shields. As a result, it permitted multiple launches as opposed to single-use rockets. (credit: NASA) PhET Explorations: Lunar Lander Can you avoid the boulder field and land safely, just before your fuel runs out, as Neil Armstrong did in 1969? Our version of this classic video game accurately simulates the real motion of the lunar lander with the correct mass, thrust, fuel consumption rate, and lunar gravity. The real lunar lander is very hard to control. Figure 8.18 Lunar Lander (http://cnx.org/content/m55174/1.2/lunar-lander_en.jar) Glossary change in momentum: the difference between the final and initial momentum; the mass times the change in velocity conservation of momentum principle: when the net external force is zero, the total momentum of the system is conserved or constant elastic collision: a collision that also conserves internal kinetic energy impulse: the average net external force times the time it acts; equal to the change in momentum inelastic collision: a collision in which internal kinetic energy is not conserved internal kinetic energy: the sum of the kinetic energies of the objects in a system isolated system: a system in which the net external force is zero linear momentum: the product of mass and velocity perfectly inelastic collision: a collision in which the colliding objects stick together point masses: structureless particles with no rotation or spin quark: fundamental constituent of matter and an elementary particle 344 Chapter 8 \| Linear Momentum and Collisions second law of motion: physical law that states that the net external force equals the change in momentum of a system divided by the time over which it changes Section Summary 8.1 Linear Momentum and Force • Linear momentum (momentum for brevity) is defined as the product of a system’s mass multiplied by its velocity. • In symbols, linear momentum p is defined to be where is the mass of the system and v is its velocity. • The SI unit for momentum is kg · m/s . p = v, • Newton’s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes. In symbols, Newton’s second law of motion is defined to be • Fnet is the net external force, Δp is the change in momentum, and Δ is the change time. Fnet = Δp Δ , 8.2 Impulse • Impulse, or change in momentum, equals the average net external force multiplied by the time this force acts: • Forces are usually not constant over a period of time. Δp = FnetΔ. 8.3 Conservation of Momentum • The conservation of momentum principle is written or ptot = constant ptot = p′tot (isolated system), ptot is the initial total momentum and p′tot is the total momentum some time later. • An isolated system is defined to be one for which the net external force is zero Fnet = 0 . • During projectile motion and where air resistance is negligible, momentum is conserved in the horizontal direction because horizontal forces are zero. • Conservation of momentum applies only when the net external force is zero. • The conservation of momentum principle is valid when considering systems of particles. 8.4 Elastic Collisions in One Dimension • An elastic collision is one that conserves internal kinetic energy. • Conservation of kinetic energy and momentum together allow the final velocities to be calculated in terms of initial velocities and masses in one dimensional two-body collisions. 8.5 Inelastic Collisions in One Dimension • An inelastic collision is one in which the internal kinetic energy changes (it is not conserved). • A collision in which the objects stick together is sometimes called perfectly inelastic because it reduces internal kinetic energy more than does any other type of inelastic collision. • Sports science and technologies also use physics concepts such as momentum and rotational motion and vibrations. 8.6 Collisions of Point Masses in Two Dimensions • The approach to two-dimensional collisions is to choose a convenient coordinate system and break the motion into components along perpendicular axes. Choose a coordinate system with the -axis parallel to the velocity of the incoming particle. • Two-dimensional collisions of point masses where mass 2 is initially at rest conserve momentum along the initial direction of mass 1 (the -axis), stated by 1 1 = 1′1 cos 1 + 2′2 cos 2 and along the direction perpendicular to the initial direction (the -axis) stated by 0 = 1′1 +2′2 . • The internal kinetic before and after the collision of two objects that have equal masses is 2 = 1 2 + 1 2 + ′1′2 cos 1 − 2 . 2′1 2′2 1 21 This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 8 \| Linear Momentum and Collisions 345 • Point masses are structureless particles that cannot spin. 8.7 Introduction to Rocket Propulsion • Newton’s third law of motion states that to every action, there is an equal and opposite reaction. • Acceleration of a rocket is = e − . Δ Δ • A rocket’s acceleration depends on three main factors. They are 1. The greater the exhaust velocity of the gases, the greater the acceleration. 2. The faster the rocket burns its fuel, the greater its acceleration. 3. The smaller the rocket's mass, the greater the acceleration. Conceptual Questions 8.1 Linear Momentum and Force 1. An object that has a small mass and an object that has a large mass have the same momentum. Which object has the largest kinetic energy? 2. An object that has a small mass and an object that has a large mass have the same kinetic energy. Which mass has the largest momentum? 3. Professional Application Football coaches advise players to block, hit, and tackle with their feet on the ground rather than by leaping through the air. Using the concepts of momentum, work, and energy, explain how a football player can be more effective with his feet on the ground. 4. How can a small force impart the same momentum to an object as a large force? 8.2 Impulse 5. Professional Application Explain in terms of impulse how padding reduces forces in a collision. State this in terms of a real example, such as the advantages of a carpeted vs. tile floor for a day care center. 6. While jumping on a trampoline, sometimes you land on your back and other times on your feet. In which case can you reach a greater height and why? 7. Professional Application Tennis racquets have “sweet spots.” If the ball hits a sweet spot then the player's arm is not jarred as much as it would be otherwise. Explain why this is the case. 8.3 Conservation of Momentum 8. Professional Application If you dive into water, you reach greater depths than if you do a belly flop. Explain this difference in depth using the concept of conservation of energy. Explain this difference in depth using what you have learned in this chapter. 9. Under what circumstances is momentum conserved? 10. Can momentum be conserved for a system if there are external forces acting on the system? If so, under what conditions? If not, why not? 11. Momentum for a system can be conserved in one direction while not being conserved in another. What is the angle between the directions? Give an example. 12. Professional Application Explain in terms of momentum and Newton’s laws how a car’s air resistance is due in part to the fact that it pushes air in its direction of motion. 13. Can objects in a system have momentum while the momentum of the system is zero? Explain your answer. 14. Must the total energy of a system be conserved whenever its momentum is conserved? Explain why or why not. 8.4 Elastic Collisions in One Dimension 15. What is an elastic collision? 8.5 Inelastic Collisions in One Dimension 16. What is an inelastic collision? What is a perfectly inelastic collision? 17. Mixed-pair ice skaters performing in a show are standing motionless at arms length just before starting a routine. They reach out, clasp hands, and pull themselves together by only using their arms. Assuming there is no friction between the blades of thei
r skates and the ice, what is their velocity after their bodies meet? 346 Chapter 8 \| Linear Momentum and Collisions 18. A small pickup truck that has a camper shell slowly coasts toward a red light with negligible friction. Two dogs in the back of the truck are moving and making various inelastic collisions with each other and the walls. What is the effect of the dogs on the motion of the center of mass of the system (truck plus entire load)? What is their effect on the motion of the truck? 8.6 Collisions of Point Masses in Two Dimensions 19. Figure 8.19 shows a cube at rest and a small object heading toward it. (a) Describe the directions (angle 1 ) at which the small object can emerge after colliding elastically with the cube. How does 1 depend on , the so-called impact parameter? Ignore any effects that might be due to rotation after the collision, and assume that the cube is much more massive than the small object. (b) Answer the same questions if the small object instead collides with a massive sphere. Figure 8.19 A small object approaches a collision with a much more massive cube, after which its velocity has the direction 1 . The angles at which the small object can be scattered are determined by the shape of the object it strikes and the impact parameter . 8.7 Introduction to Rocket Propulsion 20. Professional Application Suppose a fireworks shell explodes, breaking into three large pieces for which air resistance is negligible. How is the motion of the center of mass affected by the explosion? How would it be affected if the pieces experienced significantly more air resistance than the intact shell? 21. Professional Application During a visit to the International Space Station, an astronaut was positioned motionless in the center of the station, out of reach of any solid object on which he could exert a force. Suggest a method by which he could move himself away from this position, and explain the physics involved. 22. Professional Application It is possible for the velocity of a rocket to be greater than the exhaust velocity of the gases it ejects. When that is the case, the gas velocity and gas momentum are in the same direction as that of the rocket. How is the rocket still able to obtain thrust by ejecting the gases? This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 8 \| Linear Momentum and Collisions 347 Problems & Exercises 8.1 Linear Momentum and Force 1. (a) Calculate the momentum of a 2000-kg elephant charging a hunter at a speed of 7.50 m/s . (b) Compare the elephant’s momentum with the momentum of a 0.0400-kg tranquilizer dart fired at a speed of 600 m/s . (c) What is the momentum of the 90.0-kg hunter running at 7.40 m/s after missing the elephant? 2. (a) What is the mass of a large ship that has a momentum of 1.60×109 kg · m/s , when the ship is moving at a speed of 48.0 km/h? (b) Compare the ship’s momentum to the momentum of a 1100-kg artillery shell fired at a speed of 1200 m/s . 3. (a) At what speed would a 2.00×104-kg airplane have to fly to have a momentum of 1.60×109 kg · m/s (the same as the ship’s momentum in the problem above)? (b) What is the plane’s momentum when it is taking off at a speed of 60.0 m/s ? (c) If the ship is an aircraft carrier that launches these airplanes with a catapult, discuss the implications of your answer to (b) as it relates to recoil effects of the catapult on the ship. 4. (a) What is the momentum of a garbage truck that is 1.20×104 kg and is moving at 10.0 m/s ? (b) At what speed would an 8.00-kg trash can have the same momentum as the truck? 5. A runaway train car that has a mass of 15,000 kg travels at a speed of 5.4 m/s down a track. Compute the time required for a force of 1500 N to bring the car to rest. 6. The mass of Earth is 5.972×1024 kg and its orbital radius is an average of 1.496×1011 m . Calculate its linear momentum. 8.2 Impulse 7. A bullet is accelerated down the barrel of a gun by hot gases produced in the combustion of gun powder. What is the average force exerted on a 0.0300-kg bullet to accelerate it to a speed of 600 m/s in a time of 2.00 ms (milliseconds)? 8. Professional Application A car moving at 10 m/s crashes into a tree and stops in 0.26 s. Calculate the force the seat belt exerts on a passenger in the car to bring him to a halt. The mass of the passenger is 70 kg. 9. A person slaps her leg with her hand, bringing her hand to rest in 2.50 milliseconds from an initial speed of 4.00 m/s. (a) What is the average force exerted on the leg, taking the effective mass of the hand and forearm to be 1.50 kg? (b) Would the force be any different if the woman clapped her hands together at the same speed and brought them to rest in the same time? Explain why or why not. 10. Professional Application A professional boxer hits his opponent with a 1000-N horizontal blow that lasts for 0.150 s. (a) Calculate the impulse imparted by this blow. (b) What is the opponent’s final velocity, if his mass is 105 kg and he is motionless in midair when struck near his center of mass? (c) Calculate the recoil velocity of the opponent’s 10.0-kg head if hit in this manner, assuming the head does not initially transfer significant momentum to the boxer’s body. (d) Discuss the implications of your answers for parts (b) and (c). 11. Professional Application Suppose a child drives a bumper car head on into the side rail, which exerts a force of 4000 N on the car for 0.200 s. (a) What impulse is imparted by this force? (b) Find the final velocity of the bumper car if its initial velocity was 2.80 m/s and the car plus driver have a mass of 200 kg. You may neglect friction between the car and floor. 12. Professional Application One hazard of space travel is debris left by previous missions. There are several thousand objects orbiting Earth that are large enough to be detected by radar, but there are far greater numbers of very small objects, such as flakes of paint. Calculate the force exerted by a 0.100-mg chip of paint that strikes a spacecraft window at a relative speed of 4.00×103 m/s , given the collision lasts 6.00×10 – 8 s . 13. Professional Application A 75.0-kg person is riding in a car moving at 20.0 m/s when the car runs into a bridge abutment. (a) Calculate the average force on the person if he is stopped by a padded dashboard that compresses an average of 1.00 cm. (b) Calculate the average force on the person if he is stopped by an air bag that compresses an average of 15.0 cm. 14. Professional Application Military rifles have a mechanism for reducing the recoil forces of the gun on the person firing it. An internal part recoils over a relatively large distance and is stopped by damping mechanisms in the gun. The larger distance reduces the average force needed to stop the internal part. (a) Calculate the recoil velocity of a 1.00-kg plunger that directly interacts with a 0.0200-kg bullet fired at 600 m/s from the gun. (b) If this part is stopped over a distance of 20.0 cm, what average force is exerted upon it by the gun? (c) Compare this to the force exerted on the gun if the bullet is accelerated to its velocity in 10.0 ms (milliseconds). 15. A cruise ship with a mass of 1.00×107 kg strikes a pier at a speed of 0.750 m/s. It comes to rest 6.00 m later, damaging the ship, the pier, and the tugboat captain’s finances. Calculate the average force exerted on the pier using the concept of impulse. (Hint: First calculate the time it took to bring the ship to rest.) 16. Calculate the final speed of a 110-kg rugby player who is initially running at 8.00 m/s but collides head-on with a padded goalpost and experiences a backward force of 1.76×104 N for 5.50×10–2 s . 17. Water from a fire hose is directed horizontally against a wall at a rate of 50.0 kg/s and a speed of 42.0 m/s. Calculate the magnitude of the force exerted on the wall, assuming the water’s horizontal momentum is reduced to zero. 18. A 0.450-kg hammer is moving horizontally at 7.00 m/s when it strikes a nail and comes to rest after driving the nail 1.00 cm into a board. (a) Calculate the duration of the impact. (b) What was the average force exerted on the nail? 19. Starting with the definitions of momentum and kinetic energy, derive an equation for the kinetic energy of a particle expressed as a function of its momentum. 348 Chapter 8 \| Linear Momentum and Collisions 20. A ball with an initial velocity of 10 m/s moves at an angle 60º above the -direction. The ball hits a vertical wall and bounces off so that it is moving 60º above the − -direction with the same speed. What is the impulse delivered by the wall? 21. When serving a tennis ball, a player hits the ball when its velocity is zero (at the highest point of a vertical toss). The racquet exerts a force of 540 N on the ball for 5.00 ms, giving it a final velocity of 45.0 m/s. Using these data, find the mass of the ball. 22. A punter drops a ball from rest vertically 1 meter down onto his foot. The ball leaves the foot with a speed of 18 m/s at an angle 55º above the horizontal. What is the impulse delivered by the foot (magnitude and direction)? 8.3 Conservation of Momentum 23. Professional Application Train cars are coupled together by being bumped into one another. Suppose two loaded train cars are moving toward one another, the first having a mass of 150,000 kg and a velocity of 0.300 m/s, and the second having a mass of 110,000 kg and a velocity of −0.120 m/s . (The minus indicates direction of motion.) What is their final velocity? 24. Suppose a clay model of a koala bear has a mass of 0.200 kg and slides on ice at a speed of 0.750 m/s. It runs into another clay model, which is initially motionless and has a mass of 0.350 kg. Both being soft clay, they naturally stick together. What is their final velocity? 25. Professional Application Consider the following question: A car moving at 10 m/s crashes into a tree and stops in 0.26 s. Calculate the force the seatbelt exerts on a passenger in the car t
o bring him to a halt. The mass of the passenger is 70 kg. Would the answer to this question be different if the car with the 70-kg passenger had collided with a car that has a mass equal to and is traveling in the opposite direction and at the same speed? Explain your answer. 26. What is the velocity of a 900-kg car initially moving at 30.0 m/s, just after it hits a 150-kg deer initially running at 12.0 m/s in the same direction? Assume the deer remains on the car. 27. A 1.80-kg falcon catches a 0.650-kg dove from behind in midair. What is their velocity after impact if the falcon’s velocity is initially 28.0 m/s and the dove’s velocity is 7.00 m/s in the same direction? which it came. What would their final velocities be in this case? 8.5 Inelastic Collisions in One Dimension 31. A 0.240-kg billiard ball that is moving at 3.00 m/s strikes the bumper of a pool table and bounces straight back at 2.40 m/s (80% of its original speed). The collision lasts 0.0150 s. (a) Calculate the average force exerted on the ball by the bumper. (b) How much kinetic energy in joules is lost during the collision? (c) What percent of the original energy is left? 32. During an ice show, a 60.0-kg skater leaps into the air and is caught by an initially stationary 75.0-kg skater. (a) What is their final velocity assuming negligible friction and that the 60.0-kg skater’s original horizontal velocity is 4.00 m/s? (b) How much kinetic energy is lost? 33. Professional Application Using mass and speed data from Example 8.1 and assuming that the football player catches the ball with his feet off the ground with both of them moving horizontally, calculate: (a) the final velocity if the ball and player are going in the same direction and (b) the loss of kinetic energy in this case. (c) Repeat parts (a) and (b) for the situation in which the ball and the player are going in opposite directions. Might the loss of kinetic energy be related to how much it hurts to catch the pass? 34. A battleship that is 6.00×107 kg and is originally at rest fires a 1100-kg artillery shell horizontally with a velocity of 575 m/s. (a) If the shell is fired straight aft (toward the rear of the ship), there will be negligible friction opposing the ship’s recoil. Calculate its recoil velocity. (b) Calculate the increase in internal kinetic energy (that is, for the ship and the shell). This energy is less than the energy released by the gun powder—significant heat transfer occurs. 35. Professional Application Two manned satellites approaching one another, at a relative speed of 0.250 m/s, intending to dock. The first has a mass of 4.00×103 kg , and the second a mass of 7.50×103 kg . (a) Calculate the final velocity (after docking) by using the frame of reference in which the first satellite was originally at rest. (b) What is the loss of kinetic energy in this inelastic collision? (c) Repeat both parts by using the frame of reference in which the second satellite was originally at rest. Explain why the change in velocity is different in the two frames, whereas the change in kinetic energy is the same in both. 8.4 Elastic Collisions in One Dimension 36. Professional Application 28. Two identical objects (such as billiard balls) have a onedimensional collision in which one is initially motionless. After the collision, the moving object is stationary and the other moves with the same speed as the other originally had. Show that both momentum and kinetic energy are conserved. 29. Professional Application Two manned satellites approach one another at a relative speed of 0.250 m/s, intending to dock. The first has a mass of 4.00×103 kg , and the second a mass of 7.50×103 kg . If the two satellites collide elastically rather than dock, what is their final relative velocity? A 30,000-kg freight car is coasting at 0.850 m/s with negligible friction under a hopper that dumps 110,000 kg of scrap metal into it. (a) What is the final velocity of the loaded freight car? (b) How much kinetic energy is lost? 37. Professional Application Space probes may be separated from their launchers by exploding bolts. (They bolt away from one another.) Suppose a 4800-kg satellite uses this method to separate from the 1500-kg remains of its launcher, and that 5000 J of kinetic energy is supplied to the two parts. What are their subsequent velocities using the frame of reference in which they were at rest before separation? 30. A 70.0-kg ice hockey goalie, originally at rest, catches a 0.150-kg hockey puck slapped at him at a velocity of 35.0 m/ s. Suppose the goalie and the ice puck have an elastic collision and the puck is reflected back in the direction from 38. A 0.0250-kg bullet is accelerated from rest to a speed of 550 m/s in a 3.00-kg rifle. The pain of the rifle’s kick is much worse if you hold the gun loosely a few centimeters from your shoulder rather than holding it tightly against your shoulder. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 8 \| Linear Momentum and Collisions 349 (a) Calculate the recoil velocity of the rifle if it is held loosely away from the shoulder. (b) How much kinetic energy does the rifle gain? (c) What is the recoil velocity if the rifle is held tightly against the shoulder, making the effective mass 28.0 kg? (d) How much kinetic energy is transferred to the rifleshoulder combination? The pain is related to the amount of kinetic energy, which is significantly less in this latter situation. (e) Calculate the momentum of a 110-kg football player running at 8.00 m/s. Compare the player’s momentum with the momentum of a hard-thrown 0.410-kg football that has a speed of 25.0 m/s. Discuss its relationship to this problem. 39. Professional Application One of the waste products of a nuclear reactor is plutonium-239 . This nucleus is radioactive and 239 Pu decays by splitting into a helium-4 nucleus and a uranium-235 nucleus , the latter of which is also 4 He + 235 U radioactive and will itself decay some time later. The energy emitted in the plutonium decay is 8.40×10 – 13 J and is entirely converted to kinetic energy of the helium and uranium nuclei. The mass of the helium nucleus is 6.68×10 – 27 kg , while that of the uranium is 3.92×10 – 25 kg (note that the ratio of the masses is 4 to 235). (a) Calculate the velocities of the two nuclei, assuming the plutonium nucleus is originally at rest. (b) How much kinetic energy does each nucleus carry away? Note that the data given here are accurate to three digits only. 40. Professional Application The Moon’s craters are remnants of meteorite collisions. Suppose a fairly large asteroid that has a mass of 5.00×1012 kg (about a kilometer across) strikes the Moon at a speed of 15.0 km/s. (a) At what speed does the Moon recoil after the perfectly inelastic collision (the mass of the Moon is 7.36×1022 kg ) ? (b) How much kinetic energy is lost in the collision? Such an event may have been observed by medieval English monks who reported observing a red glow and subsequent haze about the Moon. (c) In October 2009, NASA crashed a rocket into the Moon, and analyzed the plume produced by the impact. (Significant amounts of water were detected.) Answer part (a) and (b) for this real-life experiment. The mass of the rocket was 2000 kg and its speed upon impact was 9000 km/h. How does the plume produced alter these results? 41. Professional Application Two football players collide head-on in midair while trying to catch a thrown football. The first player is 95.0 kg and has an initial velocity of 6.00 m/s, while the second player is 115 kg and has an initial velocity of –3.50 m/s. What is their velocity just after impact if they cling together? 42. What is the speed of a garbage truck that is 1.20×104 kg and is initially moving at 25.0 m/s just after it hits and adheres to a trash can that is 80.0 kg and is initially at rest? 43. During a circus act, an elderly performer thrills the crowd by catching a cannon ball shot at him. The cannon ball has a mass of 10.0 kg and the horizontal component of its velocity is 8.00 m/s when the 65.0-kg performer catches it. If the performer is on nearly frictionless roller skates, what is his recoil velocity? 44. (a) During an ice skating performance, an initially motionless 80.0-kg clown throws a fake barbell away. The clown’s ice skates allow her to recoil frictionlessly. If the clown recoils with a velocity of 0.500 m/s and the barbell is thrown with a velocity of 10.0 m/s, what is the mass of the barbell? (b) How much kinetic energy is gained by this maneuver? (c) Where does the kinetic energy come from? 8.6 Collisions of Point Masses in Two Dimensions 45. Two identical pucks collide on an air hockey table. One puck was originally at rest. (a) If the incoming puck has a speed of 6.00 m/s and scatters to an angle of 30.0º ,what is the velocity (magnitude and direction) of the second puck? (You may use the result that 1 − 2 = 90º for elastic collisions of objects that have identical masses.) (b) Confirm that the collision is elastic. 46. Confirm that the results of the example Example 8.7 do conserve momentum in both the - and -directions. 47. A 3000-kg cannon is mounted so that it can recoil only in the horizontal direction. (a) Calculate its recoil velocity when it fires a 15.0-kg shell at 480 m/s at an angle of 20.0º above the horizontal. (b) What is the kinetic energy of the cannon? This energy is dissipated as heat transfer in shock absorbers that stop its recoil. (c) What happens to the vertical component of momentum that is imparted to the cannon when it is fired? 48. Professional Application A 5.50-kg bowling ball moving at 9.00 m/s collides with a 0.850-kg bowling pin, which is scattered at an angle of 85.0º to the initial direction of the bowling ball and with a speed of 15.0 m/s. (a) Calculate the final velocity (magnitude and direction) of the bowling ball. (b) Is the collision elastic? (c) Linear kinetic energy is greater after the collision. Disc
uss how spin on the ball might be converted to linear kinetic energy in the collision. 49. Professional Application Ernest Rutherford (the first New Zealander to be awarded the Nobel Prize in Chemistry) demonstrated that nuclei were very small and dense by scattering helium-4 nuclei from 4 He 197 Au gold-197 nuclei . The energy of the incoming helium nucleus was 8.00×10−13 J , and the masses of the helium and gold nuclei were 6.68×10−27 kg and 3.29×10−25 kg , respectively (note that their mass ratio is 4 to 197). (a) If a helium nucleus scatters to an angle of 120º during an elastic collision with a gold nucleus, calculate the helium nucleus’s final speed and the final velocity (magnitude and direction) of the gold nucleus. (b) What is the final kinetic energy of the helium nucleus? 50. Professional Application Two cars collide at an icy intersection and stick together afterward. The first car has a mass of 1200 kg and is approaching at 8.00 m/s due south. The second car has a mass of 850 kg and is approaching at 17.0 m/s due west. (a) Calculate the final velocity (magnitude and direction) of the cars. (b) How much kinetic energy is lost in the collision? (This energy goes into deformation of the cars.) Note that 350 Chapter 8 \| Linear Momentum and Collisions gravity. The mass of the rocket just as it runs out of fuel is 75,000-kg, and its exhaust velocity is 2.40×103 m/s . Assume that the acceleration of gravity is the same as on Earth’s surface . (b) Why might it be necessary 9.80 m/s2 to limit the acceleration of a rocket? 59. Given the following data for a fire extinguisher-toy wagon rocket experiment, calculate the average exhaust velocity of the gases expelled from the extinguisher. Starting from rest, the final velocity is 10.0 m/s. The total mass is initially 75.0 kg and is 70.0 kg after the extinguisher is fired. 60. How much of a single-stage rocket that is 100,000 kg can be anything but fuel if the rocket is to have a final speed of 8.00 km/s , given that it expels gases at an exhaust velocity of 2.20×103 m/s? 61. Professional Application (a) A 5.00-kg squid initially at rest ejects 0.250-kg of fluid with a velocity of 10.0 m/s. What is the recoil velocity of the squid if the ejection is done in 0.100 s and there is a 5.00-N frictional force opposing the squid’s movement. (b) How much energy is lost to work done against friction? 62. Unreasonable Results Squids have been reported to jump from the ocean and travel 30.0 m (measured horizontally) before re-entering the water. (a) Calculate the initial speed of the squid if it leaves the water at an angle of 20.0º , assuming negligible lift from the air and negligible air resistance. (b) The squid propels itself by squirting water. What fraction of its mass would it have to eject in order to achieve the speed found in the previous part? The water is ejected at 12.0 m/s ; gravitational force and friction are neglected. (c) What is unreasonable about the results? (d) Which premise is unreasonable, or which premises are inconsistent? 63. Construct Your Own Problem Consider an astronaut in deep space cut free from her space ship and needing to get back to it. The astronaut has a few packages that she can throw away to move herself toward the ship. Construct a problem in which you calculate the time it takes her to get back by throwing all the packages at one time compared to throwing them one at a time. Among the things to be considered are the masses involved, the force she can exert on the packages through some distance, and the distance to the ship. 64. Construct Your Own Problem Consider an artillery projectile striking armor plating. Construct a problem in which you find the force exerted by the projectile on the plate. Among the things to be considered are the mass and speed of the projectile and the distance over which its speed is reduced. Your instructor may also wish for you to consider the relative merits of depleted uranium versus lead projectiles based on the greater density of uranium. because both cars have an initial velocity, you cannot use the equations for conservation of momentum along the -axis and -axis; instead, you must look for other simplifying aspects. 51. Starting with equations 1 1 = 1′1 cos 1 + 2′2 cos 2 and 0 = 1′1 sin 1 + 2′2 sin 2 for conservation of momentum in the - and -directions and assuming that one object is originally stationary, prove that for an elastic collision of two objects of equal masses, 2+ 1 1 21 as discussed in the text. 2+′1′2 cos 1 − 2 2′2 2′1 2 = 1 52. Integrated Concepts A 90.0-kg ice hockey player hits a 0.150-kg puck, giving the puck a velocity of 45.0 m/s. If both are initially at rest and if the ice is frictionless, how far does the player recoil in the time it takes the puck to reach the goal 15.0 m away? 8.7 Introduction to Rocket Propulsion 53. Professional Application Antiballistic missiles (ABMs) are designed to have very large accelerations so that they may intercept fast-moving incoming missiles in the short time available. What is the takeoff acceleration of a 10,000-kg ABM that expels 196 kg of gas per second at an exhaust velocity of 2.50×103 m/s? 54. Professional Application What is the acceleration of a 5000-kg rocket taking off from the Moon, where the acceleration due to gravity is only 1.6 m/s2 , if the rocket expels 8.00 kg of gas per second at an exhaust velocity of 2.20×103 m/s? 55. Professional Application Calculate the increase in velocity of a 4000-kg space probe that expels 3500 kg of its mass at an exhaust velocity of 2.00×103 m/s . You may assume the gravitational force is negligible at the probe’s location. 56. Professional Application Ion-propulsion rockets have been proposed for use in space. They employ atomic ionization techniques and nuclear energy sources to produce extremely high exhaust velocities, perhaps as great as 8.00×106 m/s . These techniques allow a much more favorable payload-to-fuel ratio. To illustrate this fact: (a) Calculate the increase in velocity of a 20,000-kg space probe that expels only 40.0-kg of its mass at the given exhaust velocity. (b) These engines are usually designed to produce a very small thrust for a very long time—the type of engine that might be useful on a trip to the outer planets, for example. Calculate the acceleration of such an engine if it expels 4.50×10−6 kg/s at the given velocity, assuming the acceleration due to gravity is negligible. 57. Derive the equation for the vertical acceleration of a rocket. 58. Professional Application (a) Calculate the maximum rate at which a rocket can expel gases if its acceleration cannot exceed seven times that of This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 8 \| Linear Momentum and Collisions 351 Test Prep for AP® Courses 8.1 Linear Momentum and Force 1. A boy standing on a frictionless ice rink is initially at rest. He throws a snowball in the +x-direction, and it travels on a ballistic trajectory, hitting the ground some distance away. Which of the following is true about the boy while he is in the act of throwing the snowball? a. He feels an upward force to compensate for the downward trajectory of the snowball. b. He feels a backward force exerted by the snowball he is throwing. c. He feels no net force. d. He feels a forward force, the same force that propels the snowball. 2. A 150-g baseball is initially moving 80 mi/h in the –xdirection. After colliding with a baseball bat for 20 ms, the baseball moves 80 mi/h in the +x-direction. What is the magnitude and direction of the average force exerted by the bat on the baseball? 8.2 Impulse 3. A 1.0-kg ball of putty is released from rest and falls vertically 1.5 m until it strikes a hard floor, where it comes to rest in a 0.045-s time interval. What is the magnitude and direction of the average force exerted on the ball by the floor during the collision? a. 33 N, up b. 120 N, up c. 120 N, down d. 240 N, down 4. A 75-g ball is dropped from rest from a height of 2.2 m. It bounces off the floor and rebounds to a maximum height of 1.7 m. If the ball is in contact with the floor for 0.024 s, what is the magnitude and direction of the average force exerted on the ball by the floor during the collision? 5. A 2.4-kg ceramic bowl falls to the floor. During the 0.018-s impact, the bowl experiences an average force of 750 N from the floor. The bowl is at rest after the impact. From what initial height did the bowl fall? a. 1.6 m b. 2.8 m c. 3.2 m d. 5.6 m 6. Whether or not an object (such as a plate, glass, or bone) breaks upon impact depends on the average force exerted on that object by the surface. When a 1.2-kg glass figure hits the floor, it will break if it experiences an average force of 330 N. When it hits a tile floor, the glass comes to a stop in 0.015 s. From what minimum height must the glass fall to experience sufficient force to break? How would your answer change if the figure were falling to a padded or carpeted surface? Explain. 7. A 2.5-kg block slides across a frictionless table toward a horizontal spring.As the block bounces off the spring, a probe measures the velocity of the block (initially negative, moving away from the probe) over time as follows: Table 8.2 Velocity (m/s) Time (s) −12.0 −10.0 −6.0 0 6.0 10.0 12.0 0 0.10 0.20 0.30 0.40 0.50 0.60 What is the average force exerted on the block by the spring over the entire 0.60-s time interval of the collision? a. 50 N b. 60 N c. 100 N d. 120 N 8. During an automobile crash test, the average force exerted by a solid wall on a 2500-kg car that hits the wall is measured to be 740,000 N over a 0.22-s time interval. What was the initial speed of the car prior to the collision, assuming the car is at rest at the end of the time interval? 9. A test car is driving toward a solid crash-test barrier with a speed of 45 mi/h. Two seconds prior to impact, the car begins to brake, but it is still moving when it hits the wall. After the collision with the w
all, the car crumples somewhat and comes to a complete stop. In order to estimate the average force exerted by the wall on the car, what information would you need to collect? a. The (negative) acceleration of the car before it hits the wall and the distance the car travels while braking. b. The (negative) acceleration of the car before it hits the wall and the velocity of the car just before impact. c. The velocity of the car just before impact and the duration of the collision with the wall. d. The duration of the collision with the wall and the distance the car travels while braking. 10. Design an experiment to verify the relationship between the average force exerted on an object and the change in momentum of that object. As part of your explanation, list the equipment you would use and describe your experimental setup. What would you measure and how? How exactly would you verify the relationship? Explain. 11. A 22-g puck hits the wall of an air hockey table perpendicular to the wall with an initial speed of 14 m/s.The puck is in contact with the wall for 0.0055 s, and it rebounds from the wall with a speed of 14 m/s in the opposite direction.What is the magnitude of the average force exerted by the wall on the puck? a. 0.308 N b. 0.616 N c. 56 N d. 112 N 12. A 22-g puck hits the wall of an air hockey table perpendicular to the wall with an initial speed of 7 m/s. The puck is in contact with the wall for 0.011 s, and the wall exerts an average force of 28 N on the puck during that time. Calculate the magnitude and direction of the change in momentum of the puck. 13. 352 Chapter 8 \| Linear Momentum and Collisions Which of the following will be true about the total momentum of the two cars? It will be greater before the collision. It will be equal before and after the collision. It will be greater after the collision. a. b. c. d. The answer depends on whether the collision is elastic or inelastic. 18. A group of students has two carts, A and B, with wheels that turn with negligible friction. The carts can travel along a straight horizontal track. Cart A has known mass mA. The students are asked to use a one-dimensional collision between the carts to determine the mass of cart B. Before the collision, cart A travels to the right and cart B is initially at rest. After the collision, the carts stick together. a. Describe an experimental procedure to determine the velocities of the carts before and after a collision, including all the additional equipment you would need. You may include a labeled diagram of your setup to help in your description. Indicate what measurements you would take and how you would take them. Include enough detail so that another student could carry out your procedure. b. There will be sources of error in the measurements taken in the experiment, both before and after the collision. For your experimental procedure, will the uncertainty in the calculated value of the mass of cart B be affected more by the error in the measurements taken before the collision or by those taken after the collision, or will it be equally affected by both sets of measurements? Justify your answer. A group of students took measurements for one collision. A graph of the students’ data is shown below. Figure 8.22 The image shows a graph with position in meters on the vertical axis and time in seconds on the horizontal axis. c. Given mA = 0.50 kg, use the graph to calculate the mass of cart B. Explicitly indicate the principles used in your calculations. d. The students are now asked to Consider the kinetic energy changes in an inelastic collision, specifically whether the initial values of one of the physical quantities affect the fraction of mechanical energy dissipated in the collision. How could you modify the experiment to investigate this question? Be sure to explicitly describe the calculations you would make, specifying all equations you would use (but do not actually do any algebra or arithmetic). 19. Cart A is moving with an initial velocity +v (in the positive direction) toward cart B, initially at rest. Both carts have equal mass and are on a frictionless surface. Which of the following Figure 8.20 This is a graph showing the force exerted by a rigid wall versus time. The graph in Figure 8.20 represents the force exerted on a particle during a collision. What is the magnitude of the change in momentum of the particle as a result of the collision? a. 1.2 kg • m/s b. 2.4 kg • m/s c. 3.6 kg • m/s d. 4.8 kg • m/s 14. Figure 8.21 This is a graph showing the force exerted by a rigid wall versus time. The graph in Figure 8.21 represents the force exerted on a particle during a collision. What is the magnitude of the change in momentum of the particle as a result of the collision? 8.3 Conservation of Momentum 15. Which of the following is an example of an open system? a. Two air cars colliding on a track elastically. b. Two air cars colliding on a track and sticking together. c. A bullet being fired into a hanging wooden block and becoming embedded in the block, with the system then acting as a ballistic pendulum. d. A bullet being fired into a hillside and becoming buried in the earth. 16. A 40-kg girl runs across a mat with a speed of 5.0 m/s and jumps onto a 120-kg hanging platform initially at rest, causing the girl and platform to swing back and forth like a pendulum together after her jump. What is the combined velocity of the girl and platform after the jump? What is the combined momentum of the girl and platform both before and after the collision? A 50-kg boy runs across a mat with a speed of 6.0 m/s and collides with a soft barrier on the wall, rebounding off the wall and falling to the ground. The boy is at rest after the collision. What is the momentum of the boy before and after the collision? Is momentum conserved in this collision? Explain. Which of these is an example of an open system and which is an example of a closed system? Explain your answer. 17. A student sets up an experiment to measure the momentum of a system of two air cars, A and B, of equal mass, moving on a linear, frictionless track. Before the collision, car A has a certain speed, and car B is at rest. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 8 \| Linear Momentum and Collisions 353 before, − 2 statements correctly characterizes the velocity of the center of mass of the system before and after the collision? a. + 2 b. + 2 c. + 2 before, 0 after after after before, + 2 d. 0 before, 0 after 20. Cart A is moving with a velocity of +10 m/s toward cart B, which is moving with a velocity of +4 m/s. Both carts have equal mass and are moving on a frictionless surface. The two carts have an inelastic collision and stick together after the collision. Calculate the velocity of the center of mass of the system before and after the collision. If there were friction present in this problem, how would this external force affect the center-of-mass velocity both before and after the collision? 8.4 Elastic Collisions in One Dimension 21. Two cars (A and B) of mass 1.5 kg collide. Car A is initially moving at 12 m/s, and car B is initially moving in the same direction with a speed of 6 m/s. The two cars are moving along a straight line before and after the collision. What will be the change in momentum of this system after the collision? a. −27 kg • m/s b. zero c. +27 kg • m/s d. It depends on whether the collision is elastic or inelastic. 22. Two cars (A and B) of mass 1.5 kg collide. Car A is initially moving at 24 m/s, and car B is initially moving in the opposite direction with a speed of 12 m/s. The two cars are moving along a straight line before and after the collision. (a) If the two cars have an elastic collision, calculate the change in momentum of the two-car system. (b) If the two cars have a completely inelastic collision, calculate the change in momentum of the two-car system. 23. Puck A (200 g) slides across a frictionless surface to collide with puck B (800 g), initially at rest. The velocity of each puck is measured during the experiment as follows: Table 8.3 Time Velocity A Velocity B 0 1.0 s 2.0 s 3.0 s +8.0 m/s +8.0 m/s 0 0 −2.0 m/s +2.5 m/s −2.0 m/s +2.5 m/s What is the change in momentum of the center of mass of the system as a result of the collision? a. +1.6 kg•m/s b. +0.8 kg•m/s c. 0 d. −1.6 kg•m/s 24. For the table above, calculate the center-of-mass velocity of the system both before and after the collision, then calculate the center-of-mass momentum of the system both before and after the collision. From this, determine the change in the momentum of the system as a result of the collision. 25. Two cars (A and B) of equal mass have an elastic collision. Prior to the collision, car A is moving at 15 m/s in the +x-direction, and car B is moving at 10 m/s in the –x-direction. Assuming that both cars continue moving along the x-axis after the collision, what will be the velocity of car A after the collision? a. same as the original 15 m/s speed, opposite direction b. equal to car B’s velocity prior to the collision c. equal to the average of the two velocities, in its original direction d. equal to the average of the two velocities, in the opposite direction 26. Two cars (A and B) of equal mass have an elastic collision. Prior to the collision, car A is moving at 20 m/s in the +x-direction, and car B is moving at 10 m/s in the –x-direction. Assuming that both cars continue moving along the x-axis after the collision, what will be the velocities of each car after the collision? 27. A rubber ball is dropped from rest at a fixed height. It bounces off a hard floor and rebounds upward, but it only reaches 90% of its original fixed height. What is the best way to explain the loss of kinetic energy of the ball during the collision? a. Energy was required to deform the ball’s shape during the collision with the floor. b. Energy was lost due to work done by the ball pushing on the floor durin
g the collision. c. Energy was lost due to friction between the ball and the floor. d. Energy was lost due to the work done by gravity during the motion. 28. A tennis ball strikes a wall with an initial speed of 15 m/s. The ball bounces off the wall but rebounds with slightly less speed (14 m/s) after the collision. Explain (a) what else changed its momentum in response to the ball’s change in momentum so that overall momentum is conserved, and (b) how some of the ball’s kinetic energy was lost. 29. Two objects, A and B, have equal mass. Prior to the collision, mass A is moving 10 m/s in the +x-direction, and mass B is moving 4 m/s in the +x-direction. Which of the following results represents an inelastic collision between A and B? a. After the collision, mass A is at rest, and mass B moves 14 m/s in the +x-direction. b. After the collision, mass A moves 4 m/s in the –x- direction, and mass B moves 18 m/s in the +x-direction. c. After the collision, the two masses stick together and move 7 m/s in the +x-direction. d. After the collision, mass A moves 4 m/s in the +x- direction, and mass B moves 10 m/s in the +x-direction. 30. Mass A is three times more massive than mass B. Mass A is initially moving 12 m/s in the +x-direction. Mass B is initially moving 12 m/s in the –x-direction. Assuming that the collision is elastic, calculate the final velocity of both masses after the collision. Show that your results are consistent with conservation of momentum and conservation of kinetic energy. 31. Two objects (A and B) of equal mass collide elastically. Mass A is initially moving 5.0 m/s in the +x-direction prior to the collision. Mass B is initially moving 3.0 m/s in the –xdirection prior to the collision. After the collision, mass A will be moving with a velocity of 3.0 m/s in the –x-direction. What will be the velocity of mass B after the collision? a. 3.0 m/s in the +x-direction b. 5.0 m/s in the +x-direction c. 3.0 m/s in the –x-direction 354 Chapter 8 \| Linear Momentum and Collisions d. 5.0 m/s in the –x-direction 32. Two objects (A and B) of equal mass collide elastically. Mass A is initially moving 4.0 m/s in the +x-direction prior to the collision. Mass B is initially moving 8.0 m/s in the –xdirection prior to the collision. After the collision, mass A will be moving with a velocity of 8.0 m/s in the –x-direction. (a) Use the principle of conservation of momentum to predict the velocity of mass B after the collision. (b) Use the fact that kinetic energy is conserved in elastic collisions to predict the velocity of mass B after the collision. 33. Two objects of equal mass collide. Object A is initially moving in the +x-direction with a speed of 12 m/s, and object B is initially at rest. After the collision, object A is at rest, and object B is moving away with some unknown velocity. There are no external forces acting on the system of two masses. What statement can we make about this collision? a. Both momentum and kinetic energy are conserved. b. Momentum is conserved, but kinetic energy is not conserved. a. There will be no change in the center-of-mass velocity. b. The center-of-mass velocity will decrease by 2 m/s. c. The center-of-mass velocity will decrease by 6 m/s. d. The center-of-mass velocity will decrease by 8 m/s. 40. Mass A (1.0 kg) slides across a frictionless surface with a velocity of 4 m/s in the positive direction. Mass B (1.0 kg) slides across the same surface in the opposite direction with a velocity of −8 m/s. The two objects collide and stick together after the collision. Predict how the center-of-mass velocity will change as a result of the collision, and explain your prediction. Calculate the center-of-mass velocity of the system both before and after the collision and explain why it remains the same or why it has changed. 8.5 Inelastic Collisions in One Dimension 41. Mass A (2.0 kg) has an initial velocity of 4 m/s in the +xdirection. Mass B (2.0 kg) has an initial velocity of 5 m/s in the –x-direction. If the two masses have an elastic collision, what will be the final velocities of the masses after the collision? c. Neither momentum nor kinetic energy is conserved. d. More information is needed in order to determine which a. Both will move 0.5 m/s in the –x-direction. b. Mass A will stop; mass B will move 9 m/s in the +x- is conserved. 34. Two objects of equal mass collide. Object A is initially moving with a velocity of 15 m/s in the +x-direction, and object B is initially at rest. After the collision, object A is at rest. There are no external forces acting on the system of two masses. (a) Use momentum conservation to deduce the velocity of object B after the collision. (b) Is this collision elastic? Justify your answer. 35. Which of the following statements is true about an inelastic collision? a. Momentum is conserved, and kinetic energy is conserved. b. Momentum is conserved, and kinetic energy is not conserved. c. Momentum is not conserved, and kinetic energy is conserved. d. Momentum is not conserved, and kinetic energy is not conserved. 36. Explain how the momentum and kinetic energy of a system of two colliding objects changes as a result of (a) an elastic collision and (b) an inelastic collision. 37. Figure 8.9 shows the positions of two colliding objects measured before, during, and after a collision. Mass A is 1.0 kg. Mass B is 3.0 kg. Which of the following statements is true? a. This is an elastic collision, with a total momentum of 0 kg • m/s. b. This is an elastic collision, with a total momentum of 1.67 kg • m/s. c. This is an inelastic collision, with a total momentum of 0 kg • m/s. d. This is an inelastic collision, with a total momentum of 1.67 kg • m/s. 38. For the above graph, determine the initial and final momentum for both objects, assuming mass A is 1.0 kg and mass B is 3.0 kg. Also, determine the initial and final kinetic energies for both objects. Based on your results, explain whether momentum is conserved in this collision, and state whether the collision is elastic or inelastic. 39. Mass A (1.0 kg) slides across a frictionless surface with a velocity of 8 m/s in the positive direction. Mass B (3.0 kg) is initially at rest. The two objects collide and stick together. What will be the change in the center-of-mass velocity of the system as a result of the collision? This content is available for free at http://cnx.org/content/col11844/1.13 direction. c. Mass B will stop; mass A will move 9 m/s in the –x- direction. d. Mass A will move 5 m/s in the –x-direction; mass B will move 4 m/s in the +x-direction. 42. Mass A has an initial velocity of 22 m/s in the +x-direction. Mass B is three times more massive than mass A and has an initial velocity of 22 m/s in the –x-direction. If the two masses have an elastic collision, what will be the final velocities of the masses after the collision? 43. Mass A (2.0 kg) is moving with an initial velocity of 15 m/s in the +x-direction, and it collides with mass B (5.0 kg), initially at rest. After the collision, the two objects stick together and move as one. What is the change in kinetic energy of the system as a result of the collision? a. no change b. decrease by 225 J c. decrease by 161 J d. decrease by 64 J 44. Mass A (2.0 kg) is moving with an initial velocity of 15 m/s in the +x-direction, and it collides with mass B (4.0 kg), initially moving 7.0 m/s in the +x-direction. After the collision, the two objects stick together and move as one. What is the change in kinetic energy of the system as a result of the collision? 45. Mass A slides across a rough table with an initial velocity of 12 m/s in the +x-direction. By the time mass A collides with mass B (a stationary object with equal mass), mass A has slowed to 10 m/s. After the collision, the two objects stick together and move as one. Immediately after the collision, the velocity of the system is measured to be 5 m/s in the +xdirection, and the system eventually slides to a stop. Which of the following statements is true about this motion? a. Momentum is conserved during the collision, but it is not conserved during the motion before and after the collision. b. Momentum is not conserved at any time during this analysis. c. Momentum is conserved at all times during this analysis. d. Momentum is not conserved during the collision, but it is conserved during the motion before and after the collision. 46. Mass A is initially moving with a velocity of 12 m/s in the +x-direction. Mass B is twice as massive as mass A and is Chapter 8 \| Linear Momentum and Collisions 355 initially at rest. After the two objects collide, the two masses move together as one with a velocity of 4 m/s in the +xdirection. Is momentum conserved in this collision? 47. Mass A is initially moving with a velocity of 24 m/s in the +x-direction. Mass B is twice as massive as mass A and is initially at rest. The two objects experience a totally inelastic collision. What is the final speed of both objects after the collision? a. A is not moving; B is moving 24 m/s in the +x-direction. b. Neither A nor B is moving. c. A is moving 24 m/s in the –x-direction. B is not moving. d. Both A and B are moving together 8 m/s in the +x- direction. 48. Mass A is initially moving with some unknown velocity in the +x-direction. Mass B is twice as massive as mass A and initially at rest. The two objects collide, and after the collision, they move together with a speed of 6 m/s in the +x-direction. (a) Is this collision elastic or inelastic? Explain. (b) Determine the initial velocity of mass A. 49. Mass A is initially moving with a velocity of 2 m/s in the +x-direction. Mass B is initially moving with a velocity of 6 m/s in the –x-direction. The two objects have equal masses. After they collide, mass A moves with a speed of 4 m/s in the –xdirection. What is the final velocity of mass B after the collision? a. 6 m/s in the +x-direction b. 4 m/s in the +x-direction c. zero d. 4 m/s in the –x-direction 50. Mass A is init

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