Physics & Practice Of

VOLUME 1

MAZUR

PRINCIPLES PRINCIPLES & PRACTICE OF PHYSICS & PRACTICE OF

Putting Principles First Based on his storied research and teaching, Eric Mazur’s Principles & Practice of Physics builds an understanding of physics that is both thorough and accessible. Unique organization and pedagogy allow students to develop a true conceptual understanding of physics alongside the quantitative skills needed in the course. • New learning architecture: The book is structured to help students learn physics in an organized way that encourages comprehension and reduces distraction. • Physics on a contemporary foundation: Traditional texts delay the introduction of ideas PHYSICS that we now see as unifying and foundational. This text builds physics on those unifying foundations, helping students to develop an understanding that is stronger, deeper, and fundamentally simpler. • Research-based instruction: This text uses a range of research-based instructional techniques to teach physics in the most effective possible manner. The result is a groundbreaking book that puts principles first, thereby making it more accessible to students and easier for instructors to teach. MasteringPhysics® works with the text to create a learning program that enables students to learn both in and out of the classroom.

About the Cover Two jets of water collide and mix to form a single swirling wave. The image conveys the elegance and symmetry of physics and how the separate Principles and Practice volumes of this text meld together to teach students the beauty of physics.

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Eric Mazur Harvard University

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MAZU0930_FM_Principles_V1_R4.indd 1 12/11/13 4:48 AM Executive Editor: Becky Ruden Publisher: Jim Smith Senior Development Editor: Margot Otway Project Managers: Martha Steele and Beth Collins Vice-President of Marketing: Christy Lesko Marketing Manager: Will Moore Photo Researcher: Eric Schrader Manufacturing Buyer: Jeff Sargent Managing Development Editor: Cathy Murphy Development Editor: Irene Nunes Program Manager: Katie Conley Image Lead, Senior Project Manager: Maya Melenchuk Copyeditor: Carol Reitz Associate Content Producer: Megan Power Full-Service Production and Composition: Cenveo® Publisher Services Illustrators: Rolin Graphics Senior Market Development Manager: Michelle Cadden Text Designer: Hespenheide Design Cover Designer: Tandem Creative, Inc. Cover Photo Credit: Franklin Kappa

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MAZU0930_FM_Principles_V1_R4.indd 2 12/11/13 4:48 AM Brief Contents Volume 1 of Principles of Physics includes Chapters 1–21. Volume 2 of Principles of Physics includes Chapters 22–34.

Chapter 1 Foundations 1 Chapter 2 Motion in One Dimension 28 Chapter 3 Acceleration 53 Chapter 4 Momentum 75 Chapter 5 Energy 101 Chapter 6 Principle of Relativity 121 Chapter 7 Interactions 148 Chapter 8 Force 176 Chapter 9 Work 202 Chapter 10 Motion in a Plane 226 Chapter 11 Motion in a Circle 254 Chapter 12 Torque 281 Chapter 13 Gravity 308 Chapter 14 Special Relativity 337 Chapter 15 Periodic Motion 374 Chapter 16 Waves in One Dimension 400 Chapter 17 Waves in Two and Three Dimensions 432 Chapter 18 Fluids 463 Chapter 19 Entropy 501 Chapter 20 Energy Transferred Thermally 530 Chapter 21 Degradation of Energy 562 Chapter 22 Electric Interactions Chapter 23 The Electric Field Chapter 24 Gauss’s Law Chapter 25 Work and Energy in Electrostatics Chapter 26 Charge Separation and Storage Chapter 27 Magnetic Interactions Chapter 28 Magnetic Fields of Charged Particles in Motion Chapter 29 Changing Magnetic Fields Chapter 30 Changing Electric Fields Chapter 31 Electric Circuits Chapter 32 Electronics Chapter 33 Ray Optics Chapter 34 Wave and Particle Optics

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MAZU0930_FM_Principles_V1_R4.indd 3 12/11/13 4:48 AM About the Author

ric Mazur is the Balkanski Professor of Physics and Applied Physics at Harvard University and Area Dean of Applied Physics. Dr. Mazur is a renowned scientist and researcher in optical physics and in education Eresearch, and a sought-after author and speaker. Dr. Mazur joined the faculty at Harvard shortly after obtaining his Ph.D. at the University of Leiden in the Netherlands. In 2012 he was awarded an Honorary Doctorate from the École Polytechnique and the University of Montreal. He is a Member of the Royal Academy of Sciences of the Netherlands and holds honorary professorships at the Institute of Semiconductor Physics of the Chinese Academy of Sciences in Beijing, the Institute of Laser Engineering at the Beijing University of Technology, and the Beijing Normal University. Dr. Mazur has held appointments as Visiting Professor or Distinguished Lecturer at Carnegie Mellon University, the Ohio State University, the Pennsylvania State University, Princeton University, Vanderbilt University, Hong Kong University, the University of Leuven in Belgium, and National Taiwan University in Taiwan, among others. In addition to his work in optical physics, Dr. Mazur is interested in education, science policy, outreach, and the public perception of science. In 1990 he began developing peer instruction, a method for teaching large lecture classes interactively. This teaching method has developed a large following, both nationally and internationally, and has been adopted across many science disciplines. Dr. Mazur is author or co-author of over 250 scientific publications and holds two dozen patents. He has also written on education and is the author of Peer Instruction: A User’s Manual (Pearson, 1997), a book that explains how to teach large lecture classes interactively. In 2006 he helped produce the award-winning DVD Interactive Teaching. He is the co-founder of Learning Catalytics, a platform for promoting interactive problem solving in the classroom, which is available in MasteringPhysics®.

MAZU0930_FM_Principles_V1_R4.indd 4 12/11/13 4:48 AM To the Student

Let me tell you a bit about myself. others to see the beauty of the universe—is a wonderful I always knew exactly what I wanted to do. It just never combination. worked out that way. When I started teaching, I did what all teachers did at the When I was seven years old, my grandfather gave me time: lecture. It took almost a decade to discover that my a book about astronomy. Growing up in the Netherlands award-winning lecturing did for my students exactly what I became fascinated by the structure of the solar system, the courses I took in college had done for me: It turned the the Milky Way, the universe. I remember struggling with subject that I was teaching into a collection of facts that my the concept of infinite space and asking endless questions students memorized by rote. Instead of transmitting the without getting satisfactory answers. I developed an early beauty of my field, I was essentially regurgitating facts to passion for space and space exploration. I knew I was going my students. to be an astronomer. In high school I was good at physics, When I discovered that my students were not master- but when I entered university and had to choose a major, ing even the most basic principles, I decided to completely I chose astronomy. change my approach to teaching. Instead of lecturing, I It took only a few months for my romance with the heav- asked students to read my lecture notes at home, and then, ens to unravel. Instead of teaching me about the mysteries in class, I taught by questioning—by asking my students to and structure of the universe, astronomy had been reduced reflect on concepts, discuss in pairs, and experience their to a mind-numbing web of facts, from declinations and own “aha!” moments. right ascensions to semi-major axes and eccentricities. Dis- Over the course of more than twenty years, the lecture illusioned about astronomy, I switched majors to physics. notes have evolved into this book. Consider this book to be Physics initially turned out to be no better than astronomy, my best possible “lecturing” to you. But instead of listening and I struggled to remain engaged. I managed to make it to me without having the opportunity to reflect and think, through my courses, often by rote memorization, but the this book will permit you to pause and think; to hopefully beauty of science eluded me. experience many “aha!” moments on your own. It wasn’t until doing research in graduate school that I re- I hope this book will help you develop the thinking skills discovered the beauty of science. I knew one thing for sure, that will make you successful in your career. And remem- though: I was never going to be an academic. I was going ber: your future may be—and likely will be—very different to do something useful in my life. Just before obtaining my from what you imagine. doctorate, I lined up my dream job working on the develop- I welcome any feedback you have. Feel free to send me ment of the compact disc, but I decided to spend one year email or tweets. doing postdoctoral research first. I wrote this book for you. It was a long year. After my postdoc, I accepted a junior Eric Mazur faculty position and started teaching. That’s when I discov- @eric_mazur ered that the combination of doing research—uncovering [email protected] the mysteries of the universe—and teaching—helping Cambridge, MA

MAZU0930_FM_Principles_V1_R4.indd 5 12/11/13 4:48 AM To the Instructor

They say that the person who teaches is the one who Setting a new standard learns the most in the classroom. Indeed, teaching led me The tenacity of the standard approach in textbooks can be to many unexpected insights. So, also, with the writing attributed to a combination of inertia and familiarity. Teach- of this book, which has been a formidably exciting intel- ing large introductory courses is a major chore, and once a lectual journey. course is developed, changing it is not easy. Furthermore, the standard texts worked for us, so it’s natural to feel that Why write a new physics text? they should work for our students, too. The fallacy in the latter line of reasoning is now well- In May 1993 I was driving to Troy, NY, to speak at a meeting known thanks to education research. Very few of our stu- held in honor of Robert Resnick’s retirement. In the car with dents are like us at all. Most take physics because they are me was a dear friend and colleague, Albert Altman, professor required to do so; many will take no physics beyond the at the University of Massachusetts, Lowell. He asked me if I introductory course. Physics education research makes it was familiar with the approach to physics taken by Ernst Mach clear that the standard approach fails these students. in his popular lectures. I wasn’t. Mach treats conservation of Because of pressure on physics departments to deliver momentum before discussing the laws of motion, and his for- better education to non-majors, changes are occurring in mulation of mechanics had a profound influence on Einstein. the way physics is taught. These changes, in turn, create a The idea of using conservation principles derived from need for a textbook that embodies a new educational phi- experimental observations as the basis for a text—rather losophy in both format and presentation. than Newton’s laws and the concept of force—appealed to me immediately. After all, most physicists never use the concept of force because it relates only to mechanics. It has Organization of this book no role in quantum physics, for example. The conservation As I considered the best way to convey the conceptual principles, however, hold throughout all of physics. In that framework of mechanics, it became clear that the standard sense they are much more fundamental than Newton’s laws. curriculum truly deserved to be rethought. For example, stan- Furthermore, conservation principles involve only algebra, dard texts are forced to redefine certain concepts more than whereas Newton’s second law is a differential equation. once—a strategy that we know befuddles students. (Examples It occurred to me, however, that Mach’s approach could be are work, the standard definition of which is incompatible taken further. Wouldn’t it be nice to start with conservation of with the first law of thermodynamics, and energy, which is both momentum and energy, and only later bring in the con- redefined when modern physics is discussed.) cept of force? After all, physics education research has shown Another point that has always bothered me is the arbi- that the concept of force is fraught with pitfalls. What’s more, trary division between “modern” and “classical” physics. after tediously deriving many results using kinematics and In most texts, the first thirty-odd chapters present physics dynamics, most physics textbooks show that you can derive essentially as it was known at the end of the 19th century; the same results from conservation principles in just one or “modern physics” gets tacked on at the end. There’s no need two lines. Why not do it the easy way first? for this separation. Our goal should be to explain physics in It took me many years to reorganize introductory phys- the way that works best for students, using our full contem- ics around the conservation principles, but the resulting ap- porary understanding. All physics is modern! proach is one that is much more unified and modern—the That is why my table of contents departs from the “standard conservation principles are the theme that runs throughout organization” in the following specific ways. this entire book. Additional motives for writing this text came from my own Emphasis on conservation laws. As mentioned earlier, this teaching. Most textbooks focus on the acquisition of infor- book introduces the conservation laws early and treats them mation and on the development of procedural knowledge. the way they should be: as the backbone of physics. The ad- This focus comes at the expense of conceptual understand- vantages of this shift are many. First, it avoids many of the ing or the ability to transfer knowledge to a new context. As standard pitfalls related to the concept of force, and it leads explained below, I have structured this text to redress that naturally to the two-body character of forces and the laws balance. I also have drawn deeply on the results of physics of motion. Second, the conservation laws enable students education research, including that of my own research group. to solve a wide variety of problems without any calculus. I have written this text to be accessible and easy for stu- Indeed, for complex systems, the conservation laws are often dents to understand. My hope is that it can take on the the natural (or only) way to solve problems. Third, the book burden of basic teaching, freeing class time for synthesis, deduces the conservation laws from observations, helping discussion, and problem solving. to make clear their connection with the world around us.

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Table 1 Scheduling matrix

Chapters that can be omitted Topic Chapters Can be inserted after chapter… without affecting continuity Mechanics 1–14 6, 13–14 Waves 15–17 12 16–17 Fluids 18 9 Thermal Physics 19–21 10 21 Electricity & Magnetism 22–30 12 (but 17 is needed for 29–30) 29–30 Circuits 31–32 26 (but 30 is needed for 32) 32 Optics 33–34 17 34

I and several other instructors have tested this approach parts by themselves can support an in-depth two-semester extensively in our classes and found markedly improved or three-quarter course that presents a complete picture of performance on problems involving momentum and energy, physics embodying the fundamental ideas of modern phys- with large gains on assessment instruments like the Force ics. Additional parts can be added for a longer or faster-paced Concept Inventory. course. The five shorter parts are more or less self-contained, although they do build on previous material, so their place- Early emphasis on the concept of system. Fundamental to ment is flexible. Within each part or chapter, more advanced most physical models is the separation of a system from its or difficult material is placed at the end. environment. This separation is so basic that physicists tend to carry it out unconsciously, and traditional texts largely Pedagogy gloss over it. This text introduces the concept in the context of conservation principles and uses it consistently. This text draws on many models and techniques derived from my own teaching and from physics education research. Postponement of vectors. Most introductory physics con- The following are major themes that I have incorporated cerns phenomena that take place along one dimension. Prob- throughout. lems that involve more than one dimension can be broken down into one-dimensional problems using vectorial nota- Separation of conceptual and mathematical frameworks. tion. So a solid understanding of physics in one dimension is Each chapter is divided into two parts: Concepts and Quan- of fundamental importance. However, by introducing vectors titative Tools. The first part, Concepts, develops the full in more than one dimension from the start, standard texts conceptual framework of the topic and addresses many of distract the student from the basic concepts of kinematics. the common questions students have. It concentrates on the In this book, I develop the complete framework of me- underlying ideas and paints the big picture, whenever possible chanics for motions and interactions in one dimension. I without equations. The second part of the chapter, Quantita- introduce the second dimension when it is needed, starting tive Tools, then develops the mathematical framework. with rotational motion. Hence, students are free to concen- Deductive approach; focus on ideas before names and trate on the actual physics. equations. To the extent possible, this text develops argu- ments deductively, starting from observations, rather than Just-in-time introduction of concepts. Wherever possible, stating principles and then “deriving” them. This approach I introduce concepts only when they are necessary. This ap- makes the material easier to assimilate for students. In the proach allows students to put ideas into immediate practice, same vein, this text introduces and explains each idea before leading to better assimilation. giving it a formal name or mathematical definition. Integration of modern physics. A survey of syllabi shows Stronger connection to experiment and experience. that less than half the calculus-based courses in the United Physics stems from observations, and this text is structured so States cover modern physics. I have therefore integrated se- that it can do the same. As much as possible, I develop the ma- lected “modern” topics throughout the text. For example, spe- terial from experimental observations (and preferably those cial relativity is covered in Chapter 14, at the end of mechanics. that students can make) rather than assertions. Most chap- Chapter 32, Electronics, includes sections on semiconductors ters use actual data in developing ideas, and new notions are and semiconductor devices. Chapter 34, Wave and Particle always introduced by going from the specific to the general— Optics, contains sections on quantization and photons. whenever possible by interpreting everyday examples. Modularity. I have written the book in a modular fashion By contrast, standard texts often introduce laws in their so it can accommodate a variety of curricula (See Table 1, most general form and then show that these laws are “Scheduling matrix”). consistent with specific (and often highly idealized) cases. The book contains two major parts, Mechanics and Elec- Consequently the world of physics and the “real” world tricity and Magnetism, plus five shorter parts. The two major remain two different things in the minds of students.

MAZU0930_FM_Principles_V1_R4.indd 7 12/11/13 4:48 AM viii to the instructor

Addressing physical complications. I also strongly oppose that equations are more important than the concepts behind presenting unnatural situations; real life complications must them, no equations are highlighted or boxed. always be confronted head-on. For example, the use of un- Both parts of the Principles chapters contain worked ex- physical words like frictionless or massless sends a message amples to help students develop problem-solving skills. to the students that physics is unrealistic or, worse, that the world of physics and the real world are unrelated entities. Structure of the Practice chapters This can easily be avoided by pointing out that friction or This volume contains material to put into practice the mass may be neglected under certain circumstances and concepts and principles developed in the corresponding pointing out why this may be done. chapters in the Principles volume. Each chapter contains Engaging the student. Education is more than just transfer the following sections: of information. Engaging the student’s mind so the infor- 1. Chapter Summary. This section provides a brief tabular mation can be assimilated is essential. To this end, the text summary of the material presented in the corresponding is written as a dialog between author and reader (often in- Principles chapter. voking the reader—you—in examples) and is punctuated by 2. Review Questions. The goal of this section is to allow stu- Checkpoints—questions that require the reader to stop and dents to quickly review the corresponding Principles chap- think. The text following a Checkpoint often refers directly ter. The questions are straightforward one-liners starting to its conclusions. Students will find complete solutions to with “what” and “how” (rather than “why” or “what if”). all the Checkpoints at the back of the book; these solutions 3. Developing a Feel. The goals of this section are to develop are written to emphasize physical reasoning and discovery. a quantitative feel for the quantities introduced in the Visualization. Visual representations are central to physics, chapter; to connect the subject of the chapter to the so I developed each chapter by designing the figures before real world; to train students in making estimates and writing the text. Many figures use multiple representations assumptions; to bolster students’ confidence in dealing to help students make connections (for example, a sketch with unfamiliar material. It can be used for self-study or may be combined with a graph and a bar diagram). Also, in for a homework or recitation assignment. This section, accordance with research, the illustration style is spare and which has no equivalent in existing books, combines a simple, putting the emphasis on the ideas and relationships number of ideas (specifically, Fermi problems and tutor- rather than on irrelevant details. The figures do not use per- ing in the style of the Princeton Learning Guide). The idea spective unless it is needed, for instance. is to start with simple estimation problems and then build up to Fermi problems (in early chapters Fermi problems are hard to compose because few concepts have been Structure of this text introduced). Because students initially find these questions Division into Principles and Practice volumes hard, the section provides many hints, which take the form I’ve divided this text into a Principles volume, which teaches of questions. A key then provides answers to these “hints.” the physics, and a Practice volume, which puts the phys- 4. Worked and Guided Problems. This section contains ics into practice and develops problem-solving skills. This complex worked examples whose primary goal is to division helps address two separate intellectually demand- teach problem solving. The Worked Problems are fully ing tasks: understanding the physics and learning to solve solved; the Guided Problems have a list of questions and problems. When these two tasks are mixed together, as suggestions to help the student think about how to solve they are in standard texts, students are easily overwhelmed. the problem. Typically, each Worked Problem is followed Consequently many students focus disproportionately on by a related Guided Problem. worked examples and procedural knowledge, at the expense 5. Questions and Problems. This is the chapter’s problem set. of the physics. The problems 1) offer a range of levels, 2) include problems relating to client disciplines (life sciences, engineer- Structure of Principles chapters ing, chemistry, astronomy, etc.), 3) use the second person as much as possible to draw in the student, and 4) do not As pointed out earlier, each Principles chapter is divided spoon-feed the students with information and unnecessary into two parts. The first part (Concepts) develops the con- diagrams. The problems are classified into three levels as ceptual framework in an accessible way, relying primarily follows: (⦁) application of single concept; numerical plug- on qualitative descriptions and illustrations. In addition to and-chug; (⦁⦁) nonobvious application of single concept including Checkpoints, each Concepts section ends with a or application of multiple concepts from current chapter; one-page Self-quiz consisting of qualitative questions. straightforward numerical or algebraic computation; (⦁⦁⦁) The second part of each chapter (Quantitative Tools) for- application of multiple concepts, possibly spanning mul- malizes the ideas developed in the first part in mathematical tiple chapters. Context-rich problems are designated CR. terms. While concise, it is relatively traditional in nature— teachers should be able to continue to use material devel- As I was developing and class-testing this book, my oped for earlier courses. To avoid creating the impression students provided extensive feedback. I have endeavored to

MAZU0930_FM_Principles_V1_R4.indd 8 12/11/13 4:48 AM to the instructor ix

incorporate all of their feedback to make the book as useful • Hints (declarative and Socratic) can provide problem- as possible for future generations of students. In addition, solving strategies or break the main problem into simpler the book was class-tested at a large number of institutions, exercises. and many of these institutions have reported significant in- • Feedback lets the student know precisely what miscon- creases in learning gains after switching to this manuscript. ception or misunderstanding is evident from their answer I am confident the book will help increase the learning gains and offers ideas to consider when attempting the problem in your class as well. It will help you, as the instructor, coach again. your students to be the best they can be. Learning Catalytics™ is a “bring your own device” student engagement, assessment, and classroom intelligence Instructor supplements system available within MasteringPhysics. With Learning The Instructor Resource DVD (ISBN 978-0-321-56175- Catalytics you can: 6/0-321-56175-9) includes an Image Library, the Procedure • Assess students in real time, using open-ended tasks to and special topic boxes from Principles, and a library of pre- probe student understanding. sentation applets from ActivPhysics, PhET simulations, and • Understand immediately where students are and adjust PhET Clicker Questions. Lecture Outlines with embedded your lecture accordingly. Clicker Questions in PowerPoint® are provided, as well as • Improve your students’ critical-thinking skills. the Instructor’s Guide and Instructor’s Solutions Manual. • Access rich analytics to understand student performance. The Instructor’s Guide (ISBN 978-0-321-94993-6/0-321- • Add your own questions to make Learning Catalytics fit 94993-5) provides chapter-by-chapter ideas for lesson plan- your course exactly. ning using Principles & Practice of Physics in class, including • Manage student interactions with intelligent grouping and strategies for addressing common student difficulties. timing. The Instructor’s Solutions Manual (ISBN 978-0-321- 95053-6/0-321-95053-4) is a comprehensive solutions The Test Bank (ISBN 978-0-130-64688-0/0-130-64688-1) manual containing complete answers and solutions to all contains more than 2000 high-quality problems, with a Developing a Feel questions, Guided Problems, and Ques- range of multiple-choice, true-false, short-answer, and tions and Problems from the Practice volume. The solutions conceptual questions correlated to Principles & Practice of to the Guided Problems use the book’s four-step problem- Physics chapters. Test files are provided in both TestGen® solving strategy (Getting Started, Devise Plan, Execute Plan, and Microsoft® Word for Mac and PC. Evaluate Result). Instructor supplements are available on the Instructor MasteringPhysics® is the leading online homework, tuto- Resource DVD, the Instructor Resource Center at www. rial, and assessment product designed to improve results by pearsonhighered.com/irc, and in the Instructor Resource area helping students quickly master concepts. Students benefit at www.masteringphysics.com. from self-paced tutorials that feature specific wrong-answer feedback, hints, and a wide variety of educationally effective Student supplements content to keep them engaged and on track. Robust diag- MasteringPhysics (www.masteringphysics.com) is de- nostics and unrivalled gradebook reporting allow instruc- signed to provide students with customized coaching and tors to pinpoint the weaknesses and misconceptions of a individualized feedback to help improve problem-solving student or class to provide timely intervention. skills. Students complete homework efficiently and effec- MasteringPhysics enables instructors to: tively with tutorials that provide targeted help. • Easily assign tutorials that provide individualized Interactive eText allows you to highlight text, add your coaching. own study notes, and review your instructor’s personalized • Mastering’s hallmark Hints and Feedback offer scaffolded notes, 24/7. The eText is available through MasteringPhysics, instruction similar to what students would experience in www.masteringphysics.com. an office hour.

MAZU0930_FM_Principles_V1_R4.indd 9 12/12/13 1:25 AM Acknowledgments

his book would not exist without the contributions Lisa Morris provided material for many of the Self-quizzes from many people. It was Tim Bozik, currently and my graduate students James Carey, Mark Winkler, and President, Higher Education at Pearson plc, who Ben Franta helped with data analysis and the appendices. I firstT approached me about writing a physics textbook. If it would also like to thank my uncle, Erich Lessing, for letting wasn’t for his persuasion and his belief in me, I don’t think I me use some of his beautiful pictures as chapter openers. would have ever undertaken the writing of a textbook. Tim’s Many people helped put together the Practice volume. With- suggestion to develop the art electronically also had a major out Daryl Pedigo’s hard work authoring and editing content, as impact on my approach to the development of the visual well as coordinating the contributions to that volume, the man- part of this book. uscript would never have taken shape. Along with Daryl, the Albert Altman pointed out Ernst Mach’s approach to de- following people provided the material for the Practice volume: veloping mechanics starting with the law of conservation of Wayne Anderson, Linda Barton, Ronald Bieniek, Michael momentum. Al encouraged me throughout the years as I Boss, Anthony Buffa, Catherine Crouch, Peter Dourmashkin, struggled to reorganize the material around the conserva- Paul Draper, Andrew Duffy, Edward Ginsberg, William tion principles. Hogan, Gerd Kortemeyer, Rafael Lopez-Mobilia, Christopher I am thankful to Irene Nunes, who served as Develop- Porter, David Rosengrant, Gay Stewart, Christopher Watts, ment Editor through several iterations of the manuscript. Lawrence Weinstein, Fred Wietfeldt, and Michael Wofsey. Irene forced me to continuously rethink what I had written I would also like to thank the editorial and production and her insights in physics kept surprising me. Her inces- staff at Pearson. Margot Otway helped realize my vision for sant questioning taught me that one doesn’t need to be a sci- the art program. Martha Steele and Beth Collins made sure ence major to obtain a deep understanding of how the world the production stayed on track. In addition, I would like to around us works and that it is possible to explain physics in thank Frank Chmely for his meticulous accuracy checking a way that makes sense for non-physics majors. of the manuscript. I am indebted to Jim Smith and Becky Catherine Crouch helped write the final chapters of elec- Ruden for supporting me through the final stages of this tricity and magnetism and the chapters on circuits and op- process and for Alison Reeves of Prentice Hall for keeping tics, permitting me to focus on the overall approach and the me on track during the early stages of the writing of this art program. Peter Dourmashkin helped me write the chap- book. Finally, I am grateful to Will Moore for his enthusi- ters on special relativity and thermodynamics. Without his asm in developing the marketing program for this book. help, I would not have been able to rethink how to introduce I am also grateful to the participants of the NSF Faculty the ideas of modern physics in a consistent way. Development Conference “Teaching Physics Conservation Many people provided feedback during the development Laws First” held in Cambridge, MA, in 1997. This confer- of the manuscript. I am particularly indebted to the late ence helped validate and cement the approach in this book. Ronald Newburgh and to Edward Ginsberg, who meticu- Finally, I am indebted to the hundreds of students in Phys- lously checked many of the chapters. I am also grateful to ics 1, Physics 11, and Applied Physics 50 who used early ver- Edwin Taylor for his critical feedback on the special relativ- sions of this text in their course and provided the feedback ity chapter and to my colleague Gary Feldman for his sug- that ended up turning my manuscript into a text that works gestions for improving that chapter. not just for instructors but, more importantly, for students.

MAZU0930_FM_Principles_V1_R4.indd 10 12/11/13 4:48 AM Acknowledgments xi

Reviewers of Principles & Practice of John Lyon, Dartmouth College Physics Trecia Markes, University of Nebraska, Kearney Peter Markowitz, Florida International University Over the years many people reviewed and class-tested the Bruce Mason, University of Oklahoma manuscript. The author and publisher are grateful for all of John McCullen, University of Arizona the feedback the reviewers provided, and we apologize if there James McGuire, Tulane University are any names on this list that have been inadvertently omitted. Timothy McKay, University of Michigan Carl Michal, University of British Columbia Edward Adelson, Ohio State University Kimball Milton, University of Oklahoma Albert Altman, University of Massachusetts, Lowell Charles Misner, University of Maryland, College Park Susan Amador Kane, Haverford College Sudipa Mitra-Kirtley, Rose-Hulman Institute of Technology James Andrews, Youngstown State University Delo Mook, Dartmouth College Arnold Arons, University of Washington Lisa Morris, Washington State University Robert Beichner, North Carolina State University Edmund Myers, Florida State University Bruce Birkett, University of California, Berkeley Alan Nathan, University of Illinois David Branning, Trinity College K.W. Nicholson, Central Alabama Community College Bernard Chasan, Boston University Fredrick Olness, Southern Methodist University Stéphane Coutu, Pennsylvania State University Dugan O’Neil, Simon Fraser University Corbin Covault, Case Western Reserve University Patrick Papin, San Diego State University Catherine Crouch, Swarthmore College George Parker, North Carolina State University Paul D’Alessandris, Monroe Community College Claude Penchina, University of Massachusetts, Amherst Paul Debevec, University of Illinois at Urbana-Champaign William Pollard, Valdosta State University N. John DiNardo, Drexel University Amy Pope, Clemson University Margaret Dobrowolska-Furdyna, Notre Dame University Joseph Priest, Miami University (deceased) Paul Draper, University of Texas, Arlington Joel Primack, University of California, Santa Cruz David Elmore, Purdue University Rex Ramsier, University of Akron Robert Endorf, University of Cincinnati Steven Rauseo, University of Pittsburgh Thomas Furtak, Colorado School of Mines Lawrence Rees, Brigham Young University Ian Gatland, Georgia Institute of Technology Carl Rotter, West Virginia University J. David Gavenda, University of Texas, Austin Leonard Scarfone, University of Vermont Edward Ginsberg, University of Massachusetts, Boston Michael Schatz, Georgia Institute of Technology Gary Gladding, University of Illinois Cindy Schwarz, Vassar College Christopher Gould, University of Southern California Hugh Scott, Illinois Institute of Technology Victoria Greene, Vanderbilt University Janet Segar, Creighton University Benjamin Grinstein, University of California, San Diego Shahid Shaheen, Florida State University Kenneth Hardy, Florida International University David Sokoloff, University of Oregon Gregory Hassold, Kettering University Gay Stewart, University of Arkansas Peter Heller, Brandeis University Roger Stockbauer, Louisiana State University Laurent Hodges, Iowa State University William Sturrus, Youngstown State University Mark Holtz, Texas Tech University Carl Tomizuka, University of Illinois Zafar Ismail, Daemen College Mani Tripathi, University of California–Davis Ramanathan Jambunathan, University of Wisconsin Oshkosh Rebecca Trousil, Skidmore College Brad Johnson, Western Washington University Christopher Watts, Auburn University Dorina Kosztin, University of Missouri Columbia Robert Weidman, Michigan Technological University Arthur Kovacs, Rochester Institute of Technology (deceased) Ranjith Wijesinghe, Ball State University Dale Long, Virginia Polytechnic Institute (deceased) Augden Windelborn, Northern Illinois University

MAZU0930_FM_Principles_V1_R4.indd 11 12/11/13 4:48 AM Detailed Contents

About the Author iv 3.6 Free-fall equations 66 To the Student v 3.7 Inclined planes 69 To the Instructor vi 3.8 Instantaneous acceleration 70 Acknowledgments x Chapter 4 Momentum 75 Chapter 1 Foundations 1 4.1 Friction 76 1.1 The scientific method 2 4.2 Inertia 76 1.2 Symmetry 4 4.3 What determines inertia? 80 1.3 Matter and the universe 6 4.4 Systems 81 1.4 Time and change 8 4.5 Inertial standard 86 1.5 Representations 9 4.6 Momentum 87 1.6 Physical quantities and units 14 4.7 Isolated systems 89 1.7 Significant digits 17 4.8 Conservation of momentum 94 1.8 Solving problems 20 1.9 Developing a feel 23 Chapter 5 Energy 101 5.1 Classification of collisions 102 Chapter 2 Motion in One Dimension 28 5.2 Kinetic energy 103 2.1 From reality to model 29 5.3 Internal energy 105 2.2 Position and displacement 30 5.4 Closed systems 108 2.3 Representing motion 32 5.5 Elastic collisions 112 2.4 Average speed and average velocity 34 5.6 Inelastic collisions 115 2.5 Scalars and vectors 39 5.7 Conservation of energy 116 2.6 Position and displacement vectors 41 5.8 Explosive separations 118 2.7 Velocity as a vector 45 2.8 Motion at constant velocity 46 Chapter 6 Principle of Relativity 121 2.9 Instantaneous velocity 48 6.1 Relativity of motion 122 6.2 Inertial reference frames 124 6.3 Principle of relativity 126 6.4 Zero-momentum reference frame 130 6.5 Galilean relativity 133 6.6 Center of mass 137 6.7 Convertible kinetic energy 142 6.8 Conservation laws and relativity 145

Chapter 7 Interactions 148 7.1 The effects of interactions 149 7.2 Potential energy 152 7.3 Energy dissipation 153 7.4 Source energy 156 Chapter 3 Acceleration 53 7.5 Interaction range 158 3.1 Changes in velocity 54 7.6 Fundamental interactions 160 3.2 Acceleration due to gravity 55 7.7 Interactions and accelerations 164 3.3 Projectile motion 57 7.8 Nondissipative interactions 165 3.4 Motion diagrams 59 7.9 Potential energy near Earth’s surface 168 3.5 Motion with constant acceleration 63 7.10 Dissipative interactions 171

xii

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Chapter 8 Force 176 10.9 Work as the product of two vectors 243 8.1 Momentum and force 177 10.10 Coefficients of friction 248 8.2 The reciprocity of forces 178 8.3 Identifying forces 180 Chapter 11 Motion in a Circle 254 8.4 Translational equilibrium 181 11.1 Circular motion at constant speed 255 8.5 Free-body diagrams 182 11.2 Forces and circular motion 259 8.6 Springs and tension 184 11.3 Rotational inertia 262 8.7 Equation of motion 188 11.4 Rotational kinematics 264 8.8 Force of gravity 191 11.5 Angular momentum 269 8.9 Hooke’s law 192 11.6 Rotational inertia of extended objects 274 8.10 Impulse 194 8.11 Systems of two interacting objects 196 Chapter 12 Torque 281 8.12 Systems of many interacting objects 198 12.1 Torque and angular momentum 282 12.2 Free rotation 285 12.3 Extended free-body diagrams 286 12.4 The vectorial nature of rotation 288 12.5 Conservation of angular momentum 293 12.6 Rolling motion 297 12.7 Torque and energy 302 12.8 The vector product 304

Chapter 13 Gravity 308 13.1 Universal gravity 309 13.2 Gravity and angular momentum 314 13.3 Weight 317 13.4 Principle of equivalence 320 13.5 Gravitational constant 325 Chapter 9 Work 202 13.6 Gravitational potential energy 326 9.1 Force displacement 203 13.7 Celestial mechanics 329 9.2 Positive and negative work 204 13.8 Gravitational force exerted by a 9.3 Energy diagrams 206 sphere 334 9.4 Choice of system 208 9.5 Work done on a single particle 213 9.6 W ork done on a many-particle system 216 9.7 Variable and distributed forces 220 9.8 Power 223

Chapter 10 Motion in a Plane 226 10.1 Straight is a relative term 227 10.2 Vectors in a plane 228 10.3 Decomposition of forces 231 10.4 Friction 234 10.5 Work and friction 235 10.6 Vector algebra 238 Chapter 14 Special Relativity 337 10.7 Projectile motion in two dimensions 240 14.1 Time measurements 338 10.8 Collisions and momentum in two 14.2 Simultaneous is a relative term 341 dimensions 242 14.3 Space-time 345

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14.4 Matter and energy 350 17.6 Beats 451 14.5 Time dilation 355 17.7 Doppler effect 454 14.6 Length contraction 360 17.8 Shock waves 459 14.7 Conservation of momentum 364 14.8 Conservation of energy 368 Chapter 15 Periodic Motion 374 15.1 Periodic motion and energy 375 15.2 Simple harmonic motion 377 15.3 Fourier’s theorem 379 15.4 Restoring forces in simple harmonic motion 381 15.5 Energy of a simple harmonic oscillator 385 15.6 Simple harmonic motion and springs 389 15.7 Restoring torques 393 15.8 Damped oscillations 396 Chapter 18 Fluids 463 18.1 Forces in a fluid 464 18.2 Buoyancy 469 18.3 Fluid flow 471 18.4 Surface effects 475 18.5 Pressure and gravity 482 18.6 Working with pressure 487 18.7 Bernoulli’s equation 491 18.8 Viscosity and surface tension 494 Chapter 19 Entropy 501 19.1 States 502 19.2 Equipartition of energy 505 19.3 Equipartition of space 507 Chapter 16 Waves in One Dimension 400 19.4 Evolution toward the most probable 16.1 Representing waves graphically 401 macrostate 509 16.2 Wave propagation 404 19.5 Dependence of entropy on 16.3 Superposition of waves 409 volume 514 16.4 Boundary effects 411 19.6 Dependence of entropy on energy 519 16.5 Wave functions 416 Properties of a monatomic ideal Standing waves 421 19.7 16.6 gas 523 Wave speed 424 16.7 19.8 Entropy of a monatomic ideal 16.8 Energy transport in waves 426 gas 526 16.9 The wave equation 429 Chapter Energy Transferred Chapter Waves in Two and Three 20 17 Thermally 530 Dimensions 432 17.1 Wavefronts 433 20.1 Thermal interactions 530 17.2 Sound 436 20.2 Temperature measurement 534 17.3 Interference 439 20.3 Heat capacity 537 17.4 Diffraction 445 20.4 PV diagrams and processes 541 17.5 Intensity 448 20.5 Change in energy and work 547

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20.6 Isochoric and isentropic ideal gas Chapter 21 Degradation of Energy 562 processes 549 21.1 Converting energy 563 20.7 Isobaric and isothermal ideal gas 21.2 Quality of energy 566 processes 551 21.3 Heat engines and heat pumps 570 Entropy change in ideal gas 20.8 Thermodynamic cycles 575 processes 555 21.4 Entropy constraints on energy Entropy change in nonideal gas 21.5 20.9 transfers 580 processes 559 21.6 Heat engine performance 583 21.7 Carnot cycle 587 21.8 Brayton cycle 589

Appendix A: Notation A-1 Appendix B: Mathematics Review A-11 Appendix C: Useful Data A-16 Appendix D: The Center of Mass of Extended Objects A-20 Appendix E: Lorentz Transformations A-21 Solutions to Checkpoints A-25 Credits C-1 Index I-1

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Matter and the universe Developing a feel Foundations 1.1 the scientific method 1.2 Symmetry 1.3 1.4 time and change 1.5 Representations 1.6 physical quantities and units 1.7 significant digits 1.8 solving problems 1.9 1 M01_MAZU0930_PRIN_Ch01_pp001-027.indd 1 M01_MAZU0930_PRIN_Ch01_pp001-027.indd 2

Concepts are going. are weritory are going to explore that so you know where we we embark on exciting this journey, let’s map out ter the step into unknown with more territory confidence. Before better problem a solver is empowering: It allows you to problems that have solved), already and been becoming problem solver (and I mean real reasoning Knowing skills. physics means becoming abetter else, course this offers you an opportunity to your sharpen psychology, engineering, biology, physics, or something of world. the Furthermore, you whether are majoring in foremost, physics provides afundamental understanding good reason for aphysics taking course is that, first and ( process for going from observations to validated theories. was development the of method scientific the One of main the changes that in that occurred century revolution scientific the that century. 16th beganinthe speculation—andcal adistinctdiscipline became during explain behavior the of universe the through philosophi- knowledge accumulated inancient inan times attempt to ture. Physics evolved from natural philosophy thatfact about curiosity world the of is part human na- ancient civilizations that to day this to survive the testify principles the use of physics. geology, and many so other disciplines you might name all verse. Indeed, biology, engineering, chemistry, astronomy, physics then, sense, is study the of there all is uni- inthe in your computer, motion inthe you of aball throw. In a provides that your daylight, structure in the of your bones, ity and magnetism. Physics around is all you: in Sunthe annihilation of matter; evaporation and melting;- electric Physics explores motion, light, and sound; creation the and solids, andgases, liquids; and objects, everyday black holes. axies, and planets. Physics with atoms deals and molecules; DNA and of molecules cells, tostars, cosmic scale the - gal of subatomic scale the to microscopic particles, the world of underlie absolutely that happens everything around us, from discovering wonderfully the simple patterns unifying that as study the fined of matter and motion. Physics is about Physics, from Greek the word for “nature,” is commonly de- 1.1 C 2 observed phenomenon.observed The hypothesis to predict is used hypothesis, is which a tentative a explanation mulate the of researcherthe observations lead These then speeds). to for dropped Styrofoam peanut travel at to floor the different during alaboratory experiment (adropped brick and a (a volcano erupting, for instance) or something happening concerning either something happening natural inthe world Figure 1.1

In its simplest form, works method scientific the as follows The many remarkable accomplishments scientific of T he scientificmethod not clear be to you why someonebecause told you to it, take and it may hances are you are course this in taking physics Chapter 1 Founda ): Aresearcher makes anumber of observations tions you it. should taking One be problems, not textbook , —a body of—a body an iterative - -

which iswhich tested then by observations. making new pothesis, is which inferred from to observations, make is used aprediction, able and verifiable by others. tigating universe: the The results obtained must repeat be - suchmake method tool scientific apowerful the for inves- experiment. The constant testing and retesting are what is capable of making predictions thatby verified can be tested explanation of a natural phenomenon, one that not amere conjecture or speculation. It is athoroughly processesbasic and relationships. is theory Ascientific thing happens and explains phenomena interms of more between observable quantities. tells observable us why between Atheory Laws are usually expressed form inthe of relationships law after test, hypothesis the is elevated to status the of either a must If modified. predictionsthe be prove accurate in test Iftime). predictions the prove inaccurate, hypothesis the and abook asheetwhen of paper aresame dropped at the behave) or related laboratory experiment (what happens similarly shaped mountain near erupting the volcano will outcomethe of some related natural (how occurrence a Figure 1.1 E the testingthe never ends. In other words, it is not possible to important, any or law theory always but rather a living and changing of body knowledge. More is science notmethod, scientific of collection a stale facts xercise 1.1Hypothesisornot? Solution is false, itis false, is is in principle testable. (d ) Even though we know statement this of exploring distant or closely observing planets, statement the (c)Although wehypothesis. humans currently have no means perimental verification and means statement this is not avalid beings on Mars which precludes cannot observed, be any ex- one hits ground the first. (b object and a lighter one at samethe instant which and observing statementverifiable. this (a)Ican verify by dropping aheavy W ( of (c type observation. Mars is inhabited by invisible that beings are able to elude any faster tothan Earth lighter fall objects ones. (b d A law tells us what Because of constantBecause the reevaluation demanded by the ) Handling toads causes warts. or atheory. hich of following the statements are hypotheses? (a The scientific method is an Themethod scientific iterative process ahy- inwhich (a),cand (d).Ahypothesis must experimentally be verifiable and therefore is ahypothesis. prediction test happens circumstances. under certain ) Distant planets harbor forms of life. observations deduce ) This statement that asserts the hypothesis induce remains tentative, and ) The planet ) Heavier some

- 07/10/13 2:41PM Concepts - - 07/10/13 2:41 PM 3 —

method c i f mind in this book, but in this book, mind but ienti —is a crucial—is in making step c - check to “Solutions volume, he s T 1.1 Principles I

1.1 30cents. worth pocket, together inmy coins two have Advertising agencies and magicians are masters at mak- at masters are magicians and agencies Advertising Magic, too, involves hidden assumptions. The trick The in hidden assumptions. involves too, Magic, - ob with agree to fails hypothesis a of the prediction If It may seem like a shaky proposition to build a hypothesis a hypothesis build to seem proposition a shaky like may It If one of them is not a nickel, what coins are they? are coins what a nickel, not them is of one If ing us fall into the trap of hidden assumptions. Imagine a a Imagine assumptions. hidden of the trap fall us into ing says, a new drug which in someone for radio commercial you If tremendously.” blood pressure my lowered “Baroxan - as of number a made have good, you sounds thinkthat words, them—in other of being aware without sumptions lower that instance, for says, Who hidden assumptions. ing blood pressure “tremendously” is a good is (dead thing “tremendously” blooding pressure the that or blood pressure) low tremendously have people too was high begin to with? blood pressure speaker’s - hap something that assume you make acts magic to is some A mind. in your a false assumption planting by pens, often to the ball here from did I move “How ask, might magician - know balls. two I won’t there?” using is while he in reality your into false assumptions put ingly un- may instructor) your and you I (or and you occasion on - dis a given during assumptions different make knowingly and confusion leads to unavoidably that a situation cussion, - care we that important is it Therefore misunderstanding. the assumptions for watch thinking and our fullyanalyze models. our into build we that several are there the hypothesis, test servations to made the rerun to is way One the discrepancy. address to ways producing keeps the test If reproducible. see is to if it test necessary- becomes it revise the hypoth to result, the same reexamine or it, into went that the rethink assumptions esis, the led hypothesis. to the that original observations - Some in a series observations. of patterns recognizing with must we sometimes but direct, times these are observations observe directly cannot (We observations. indirect settle for can a physicist but instance, for atom, an of the nucleus with behavior its and describe the nucleus of the structure the 1.2 indicates, Figure As accuracy.) and certainty great be often must observations our from emerge that patterns a model. build to assumptions simplifying with combined - consti what is assumptions model and of combination The a hypothesis. tutes making but proof, without accepted are that assumptions on consciouslythese assumptions— when for that, is required is All that the universe. sense of - these assump of be aware must we a hypothesis, mulating them if the predictions drop revise or to be ready and tions in partic- should, We validated. not are hypothesis our of calledassumptions hidden are what for out watch ular, an them. As of beingaware without make we assumptions the to (Turn question. try the following answering example, finalthesection of the answer.) for points,”

- reexamine, reexamine, data more gather special relativity special rethink assumptions observations FAIL repeat hypothesis revise which is a simplified conceptual conceptual a simplified which is and the theory and of model pattern prediction observations - the vol increases a TV on remote button test prediction test +

PASS ). Developing a scientific hypothesis often begins often a scientific hypothesis ). Developing Iterative process for developing a scientific hypothesis. developing for process Iterative

Let’s look at the iterative process of developing models models developing of process theiterative look at Let’s The formulation of a hypothesis almost always involves involves always almost a hypothesis of formulation The a theory developed mechanics, classical is A case in point observations Figure 1.2 Figure reuse and and reuse testing continue Figure 1.2 Figure

ume or the channel number. In everyday life, we base our base we our everyday life, In number. the channel or ume imagined, or real have, we knowledge whatever on models models build must science we In incomplete. or complete fill to in ways determine observationbased and careful on information. missing any determining eye toward an with in physics, hypotheses and be to avoided are pitfalls what needed and skills are what ( among them—are needed to describe the phenomena that that needed describe to them—are the phenomena among mechanics. classical of the range fall outside a model, developing to have don’t You phenomenon. some of representation de- Everyone models. develop to be a scientist as trained events how behave, people how of models mental velops we models, such Without work. things how and unfold, experiences, our decide understand to be able not would unexpected handle experiences. or take, to actions what door that useare in everyday we models life of Examples doors sides of opposite on are hinges door and handles that theand prove any scientific theory or law to be absolutely true (or be scientific to absolutely theory law or any prove will learn in you thematerial false). Thus even absolutely is truth”—it “ultimate some this book represent does not wrong. been proved not has it that the extent true to only everyday- ob of describe thein 17th to century themotion this this book). of Although most jects of thesubject (and - everyday phe most for results accurate theory produces orbiting satellites to in the air balls thrown from nomena, years hundred Earth, the last during made observations significant certain under circumstances, revealed that have - classi that clear now is It this theory from occur. deviations impor (albeit a limited only for applicable is cal mechanics physics— of new branches and phenomena, of range tant) quantum mechanics M01_MAZU0930_PRIN_Ch01_pp001-027.indd 3 M01_MAZU0930_PRIN_Ch01_pp001-027.indd 4

Concepts Exercise 1.2Deadmusicplayer 4 those principlesthose inexamples and exercises ( ples physicists presently agree on and for then you to apply is forisfying, me to simply tell you general the princi all - physics. One way, sat nor- useful very is which neither very sheds light fact this on howand you might want to learn ative of part doing physics involves inductive reasoning, tive which is arguing to from general,the specific the and in particular—involves of two types reasoning: inductive sibilities), there reason is agood to physics. take an engineer, or aresearch scientist (to name just- afew pos way. So, you whether becomeanalyst, afinancial adoctor, physics rigorous allows you inavery skills to these sharpen areof injust skills these useful about any context. Learning tions. It should not come as any to surprise you that many cally, developing models, and using models to make predic- terns, making and recognizing assumptions,- logi thinking doing interpreting science: observations, recognizing pat- scrap hypothesis the and anew. start universe has an underlying simplicity, it might better to be of increased accuracy. the we that like to Because think the some point complexity the at outweighs benefit the and ity, cess has itsalso limits. refinement Each adds complex- assumptions, or evencomplete The guesses. refining pro- are developed from incomplete information, vague ideas, Hypotheses donot always with start observations; some more complicated than suggested by Figures 1.1and 1.2. ment of physicsthe principles (Figure 1.3b). This approach observations and make you of and part discovery the refine- terns. Another way is for me to present you with data and likely to your benefit discovering career: underlying pat- you of opportunity the that to skill the learn is most the approach involvesThis deductive reasoning only and robs are hiddenassumptions? the twothe possible outcomes contains ahiddenassumption. What Solution modified or discarded.modified meanswhich hypothesis the is not supported and must either be player(2) the doesn’t work after batteries new the are installed, batteries are means which installed, hypothesis the is supported; work. Possible outcomes: (1)The player works once new the tion: If I replace batteries the withones, new playerthe should Here is one example. Hypothesis: The batteries are dead. Predic- at answer the below.) what you conclude from outcomes. the (Think hypothesis. two possible outcomes Describe of test the and play, and make then aprediction that you permits to test your is turned on. Develop a hypothesis explaining why it fails to A battery-operated portable music player fails to play it when

Figure shows 1.1 also that doing science—and physics Figure 1.2 gives an of idea that skills the are in useful The development of hypothesis ascientific is often , arguing from The most general the cre to specific. the - 1.2 Chapter 1 Foundations In Exercise 1.2,eachof conclusions the drawn from There are many reasons player the might not turn on. before you peek Figure 1.3a deduc ). - ,

Figure 1.3 Figure 1.4 for yourself before applying them by science (b) Learning discovering principles those by science (a) Learning applying established principles physics! is that and discovery refinement are at of heart the doing why I’m not just telling is more time-consuming, and sometimes you may wonder its appearance. Consider equilateral the in triangle operationscertain on performed can be it without changing as follows: when An exhibits object symmetry symmetry sociated with order, beauty, and harmony. We can define involves what physicists symmetry, call One of requirements basic the of any law of universe the 1.2 Symmetry 120° while you120° while have your appears eyes triangle closed, the If you yourclose eyes and someone rotates by triangle the (a) R two sidesofthetrianglearemirrorimages ofeachother (b) Re ectionsymmetry:Acrosseachre ection axis(labeledR), doesn’t changehowitlooks observations, data what I wrote. I what tions to checkpoints,” and compare what you have written with ofthat, Principles the section turn to final the why you would like to accomplish Once this. you have done course.this Write down what you would like to accomplish and R principles otat 1.3 io nal sy After reading this section, reflect on yourAfter reflect reading section, this goals for R e e mme R otat c t io t io n axis r y n axis : R otat discover apply to you outcome. final the The reason ing ane R re ection axis R R q e otat uilat e c io t io e n ab examples, exercises r n a al t aconcept often as - cr ou r principles iang oss t r volume, “Solu volume, otat le b io y 120° Figure n axis R re ne 120° 1.4a - . 23/08/13 12:03PM Concepts 07/10/13 2:41 PM 5 . - R R

R ). It therefore has has therefore ). It ymmetry RR S R (b) Re ection symmetry 1.2 Figure 1.7a Figure 60° ) Da Vinci’s Vitruvian Man Man Vitruvian (c) Da Vinci’s symmetry the re ection shows body the human of Exercise 1.3. Exercise shows a snowflake. Does the snowflake have rota Doesa snowflake. the have snowflake shows

I can rotate the snowflake by 60° or a multiple of 60° of a multiple 60° or the by snowflake rotate I can I can also fold the flake in half along any of the three the three of the flake any alsoI can fold in half along ) Rotational symmetry(a) Rotational The flake therefore has reflection has symmetry all six of flakeThe about therefore these axes. rotational symmetry. rotational 1.7b in Figure axes red the three of any along and axes blue Figure 1.6 Figure Solution 1.7 Figure Figure 1.6 Figure tional symmetry?tional describe in which the flake yes, the ways can If reflec- Does have it appearance. its changing without be rotated symmetry?tion describe in which the flake yes, the ways If the of image the half mirror is one so in two that becan split other. (120°, 180°, 240°, 300°, and 360°) in the plane of the photograph the photograph of in the 360°) plane (120°, 180°, 240°, 300°, and ( appearance its changing without xercise 1.3 Change is no change studying must therefore mathematically exhibit symmetry exhibit mathematically therefore must studying themathematical words, in time; in other translation under time. of be independent must these laws of expression E - - and in b) and ) Symmetrical arrangement (b) Symmetrical arrangement crystal in a salt atoms of Na Cl Figure 1.5a Figure it) and obtaining the same obtaining and it) . Any physical law that describes that this law physical . Any occurs when one half of an object is the the object is an half of occurs whenone

one of several of types geometrical symmetry. of one The symmetrical arrangement of atoms in a salt crystal in a salt these gives crystals atoms of shape. their cubic symmetricalThe arrangement ) Micrograph of salt crystals salt of (a) Micrograph

Another common type of geometrical symmetry, reflec geometrical symmetry, type of common Another The ideas of symmetry—that something appears un- symmetry—thatappears ideas of The something with make we measurements Likewise, expect we any influences along with it. For example, if Earth’s gravity is of importance, importance, of is gravity if Earth’s example, For it. with influences along Earth from does in space far not a location to the apparatus then moving result. theyield same *In moving our apparatus, we must take care to move any relevant external relevant any move to take care must we apparatus, our moving *In tion symmetry, triangle in equilateral The half. the other of mirror image 1.4 possessesFigure reflection symmetry the three about the trian- folding imagine you 1.4b. If in Figure shown axes are halves the two see can that you each axis, gle in half over Reflectionidentical. symmetry in the us: occurs all around in crystals ( atoms of arrangement the same when you open your eyes, and you can’t tell that that tell can’t you eyes, and your open when you the same rotational have to triangle said The is it beenhas rotated. symmetry, 1.5 Figure ), to name just just name 1.5c), to (Figure forms life most of the anatomy examples. two the to only not certain under operations—apply changed of realm abstract also the more to objects but of shape that experiment an do to can we things are there If physics. - then the phe unchanged, the experiment of the result leave possessto cer said is the experiment by tested nomenon carry out apparatus, an build we symmetries.tain Suppose the then move in a certain location, a certain measurement and the measurement, repeat location, another to apparatus - the appara moving By in both locations.* result the same get (translating newa location to tus has the observed that phenomenon shown have we result, symmetrytranslational - transla exhibit mathematically therefore must phenomenon this of symmetry;tional expression the mathematical is, that the location. of be independent must law earlier an at time as a later be at to the same apparatus our theeffect mea - no timein has on translation is, time; that are we thephenomenon describing laws The surements. M01_MAZU0930_PRIN_Ch01_pp001-027.indd 5 Self-quiz 13

Self-quiz

1. Two children in a playground swing on two swings of unequal length. The child on the shorter swing is considerably heavier than the child on the longer swing. You observe that the longer swing swings more slowly. Formulate a hypothesis that could explain your observation. How could you test your hypothesis? 2. What symmetries do you observe in the quilt patterns of Figure 1.14?

Figure 1.14

(a) (b) (c) (d)

3. Give the order of magnitude of these quantities in meters or seconds: (a) length of a football field, (b) height of a mature tree, (c) one week, (d) one year. 4. Starting from the first floor, an elevator stops at floors 5, 2, 4, 3, 6, and 4 before returning to the first floor. (a) Represent this motion visually. (b) If the distance between the first and sixth floors is 15 m, what is the total distance traveled by the elevator?

Answers 1. One hypothesis is that longer swings swing more slowly than shorter swings. You can test this hypothesis by adjusting the length of either swing until the two lengths are the same and then asking the children to remount their respective swings and swing again. If the originally longer swing is still the slower one, your hypothesis is not correct. If the two swings now have the same speed, your hypothesis is correct. Another hypothesis is that heavier children swing faster than lighter ones. You can test this hypothesis by asking the children to trade places. If the longer swing now swings faster than the shorter swing, your hypothesis is correct. If the longer swing still swings more slowly, your hypothesis is incorrect. 2. See Figure 1.15. (a) Reflection symmetry about a horizontal line through the center. (b) Rotational symmetry by multiples of 90°. (c) Rotational symmetry by 180°. (d) Rotational symmetry by multiples of 90° and reflection symmetry about a horizontal, vertical, or diagonal line through the center.

Figure 1.15

3. (a) 100 yards is about 100 m; the order of magnitude is thus 100 m = 102 m. (b) An average mature tree is be- self-quiz tween 5 and 20 m tall, for an average of 12 m = 1.2 × 101 m. The coefficient 1.2 rounds to 1, and so the order of magnitude is 1 × 101 m = 10 m. (c) 1 week = (1 week)(7 days week)(24 h day)(60 min h)(60 s min) = 604,800 s = 6 × 105 s; the coefficient 6 rounds to 10, and so the order of magnitude is 10 × 105 s = 106 s. (d) 1 year = 52 weeks = (52 weeks)(604,800 s week) = 31,449,600> s = 3.1 ×>107 s; the coefficient> 3.1> rounds to 10, and so the order of magnitude is 10 × 107 s = 108 s. > Figure 1.16 4. (a) See Figure 1.16 for one way to represent the motion. Note that the elevator itself is not represented because showing it would add nothing we need to the visual information. The only thing we are interested in is distances traveled, represented by the vertical lines. (b) If the distance between floors 1 and 6 is 15 m, one floor is (15 m) 5 = 3.0 m. From the figure, I see that the numbers of floors traveled between successive stops are 4, 3, 2, 1, 3, 2, and 3, for a total of 18 floors, or 18(3.0> m) = 54 m.

M01_MAZU0930_PRIN_Ch01_pp001-027.indd 13 07/10/13 2:41 PM 14 Chapter 1 Foundations

Table 1.1 Physical quantities and their 1.6 Physical quantities and units symbols Because physics is a quantitative science, statements must be expressed in num- Physical quantity Symbol bers, which requires either measuring or calculating numerical values for physical quantities. In this section we review some basic rules for dealing with physical length / quantities, which in this book are represented by italic symbols—typically letters time t from the Roman or Greek alphabet, such as t for time and s for electrical con- mass m ductivity. Table 1.1 gives the symbols for some of the physical quantities we use speed v throughout the book. volume V Physical quantities are expressed as the product of a numerical value and a energy E unit of measurement. For example, the length / of an object that is 1.2 m long temperature T can be expressed as / = 1.2 m. The unit system used in science and engineering throughout the world and in everyday life in most countries is the Système International (International System), and the units are collectively called SI units. This system consists of seven base units (Table 1.2) from which all other units currently in use can be derived. For example, the physical quantity speed, which we discuss in Chapter 2, is defined as the distance traveled divided by the time interval over which the travel takes place. Thus the SI derived unit of speed is meters per second (m s), the base unit of length divided by the base unit of time. A list of SI derived units and their relationship to the seven base units is given in Appendix C. > Be careful not to confuse abbreviations for units with symbols for physical quantities. Unit abbreviations are printed in roman (upright) type—m for meters, for instance—and symbols for physical quantities are printed in italic (slanted) type—t for time, say. Also, bear in mind that you can add and subtract quantities only if they have the same units; it is meaningless to add, say, 3 m to 4 kg. To produce multiples of any SI unit and conveniently work with very large or very small numbers, we modify the unit name with prefixes representing integer powers of ten (Table 1.3). For example, a billionth of a second is denoted by 1 ns (pronounced “one nanosecond”):

1 ns = 10-9 s. (1.1)

One thousand meters is denoted by 1 km, “one kilometer.” Prefixes are never used without a unit and are never combined into compound prefixes. The unit kilogram contains a prefix (kilo-) because it is derived from the non-SI unit gram (1 kg = 1000 g). Therefore 10-6 kg never becomes 1 mkg. Instead, the names and multiples of the kilogram are constructed by adding the appropriate prefix to the word gram and the abbreviation g. For example, 10-6 kg becomes 1 mg, “one milligram.” The standard practice in engineering is to use only the powers of ten that are multiples of three.

Table 1.2 The seven SI base units

Name of unit Abbreviation Physical quantity meter m length kilogram kg mass second s time tive tools

a ampere A electric current kelvin K thermodynamic temperature ntit

a mole mol amount of substance candela cd luminous intensity Qu

M01_MAZU0930_PRIN_Ch01_pp001-027.indd 14 07/10/13 2:41 PM 1.6 Physical quantities and units 15

Table 1.3 SI prefixes

10n Prefix Abbreviation 10n Prefix Abbreviation

100 — — 103 kilo- k 10-3 milli- m 106 mega- M 10-6 micro- m 109 giga- G 10-9 nano- n 1012 tera- T 10-12 pico- p 1015 peta- P 10-15 femto- f 1018 exa- E 10-18 atto- a 1021 zetta- Z 10-21 zepto- z 1024 yotta- Y 10-24 yocto- y

1.11 Use prefixes from Table 1.3 to remove either all or almost all of the zeros in each expression. (a) / = 150,000,000 m, (b) t = 0.000 000 000 012 s, (c) 1200 km s, (d) 2300 kg. > Of the seven SI base units, we have already discussed two, the meter and the second. We discuss the base unit for mass, the kilogram, in Chapter 4, the base unit for electric current, the ampere, in Chapter 27, and the base unit for temperature, the kelvin, in Chapter 20. The mole (abbreviated mol) is the SI base unit that measures the quantity of a given substance. A mole is currently defined as the number of atoms in 12 × 10-3 kg of carbon-12, the most common form of carbon. This number is called Avogadro’s number NA, after the 19th-century Italian physicist Amedeo Avogadro. The currently accepted experimental measurement of Avogadro’s number is 23 NA = 6.0221413 × 10 .

Note that the mole is simply a number: Just as one dozen means 12 of anything and one gross means 144 of anything, one mole means 6.022 × 1023 of anything. So 1 mol of helium atoms is 6.022 × 1023 helium atoms, and 1 mol of carbon dioxide molecules is 6.022 × 1023 carbon dioxide molecules. The final SI base unit, the candela, measures luminous intensity. One candela (1 cd) is roughly the amount of light generated by a single candle; the light emitted by a 100-watt light bulb is about 120 cd. The definition of the candela takes into account how the human eye perceives the intensity of various colors and is therefore rather unwieldy. For this reason we do not use the candela in this book in the chapters dealing with light, concentrating instead on the amount of energy carried by light. An important concept used throughout physics is density, the physical quantity that measures how much of some substance there is in a given volume. Depending on the quantity being measured, there are various types of density. Figure 1.17 Number density. For example, number density is the number of objects per unit volume. If there Same volume V

are N objects in a volume V, then the number density n of these objects is Quantitative tools N1 objects N2 objects N n K . (1.3) V (The symbol K means that the equality is either a definition or a convention.) If the objects in a given volume are packed together more tightly, the number density is higher (Figure 1.17). Mass density r (Greek rho) is the amount of mass m per unit volume: e greater the number N of objects in a given m r K . (1.4) space V, the higher the number density n = N V. V In this case N2 7 N1, so n2 7 n1. >

M01_MAZU0930_PRIN_Ch01_pp001-027.indd 15 11/15/13 11:24 AM 16 Chapter 1 Foundations

Exercise 1.6 Helium density

At room temperature and atmospheric pressure, 1 mol of helium gas has a volume of 24.5 × 10-3 m3. The same amount of liquid helium has a volume of 32.0 × 10-6 m3. What are the number and mass densities of (a) the gaseous helium and (b) the liquid helium? The mass of one helium atom is 6.647 × 10-27 kg. Solution (a) I know from Eq. 1.2 that 1 mol of helium contains 6.022 × 1023 atoms, and I can use this information in Eq. 1.3 to get the number density:

6.022 × 1023 atoms n = = 2.46 × 1025 atoms m3. 24.5 × 10-3 m3 > For the mass density, I must know the mass of 1 mol of helium atoms, and so I multiply the mass of one helium atom by the number of atoms in 1 mol:

m = (6.647 × 10-27 kg atom)(6.022 × 1023 atoms mol) > > = 4.003 × 10-3 kg mol. > Equation 1.4 then yields

4.003 × 10-3 kg r = = 0.163 kg m3. 24.5 × 10-3 m3 > (b) For the liquid helium, the same reasoning gives me

6.022 × 1023 atoms n = = 1.88 × 1028 atoms m3 32.0 × 10-6 m3 > 4.003 × 10-3 kg r = = 125 kg m3. 32.0 × 10-6 m3 >

Examples of non-SI units accepted for use along with SI units are the minute (1 min = 60 s), the hour (1 h = 3600 s), the liter (1 L = 10-3 m3), and the met- ric ton (1 t = 103 kg). A number of traditional, non-SI units are used in engineering; in various businesses, industries, sports, and trades; and in everyday life in the United States. Examples are inches, feet, yards, miles, acres, ounces, pints, gallons, and fluid ounces. These units are nondecimal, which makes it hard to interconvert them. When solving problems in this course, always begin by converting any quantities given in non-SI units to the SI equivalents. A conversion table is given in Appendix C. The simplest way to convert from one unit to another is to write the conversion factor for the two units as a ratio. For, example, in Appendix C, we see that 1 in. = 25.4 mm. By bringing the 25.4 mm to the left side of the equals sign, we can write this as either

1 in. 25.4 mm

tive tools = 1 or = 1 (1.5)

a 25.4 mm 1 in. ntit

a Note that you must write the units in these expressions because without them you obtain the incorrect expressions 1 = 1 and 25.4 = 1. Because multiplying

Qu 25.4 1 any number by 1 doesn’t change the number, you can use these ratios to convert

M01_MAZU0930_PRIN_Ch01_pp001-027.indd 16 07/10/13 2:41 PM 1.7 Significant digits 17

units. For example, to express 4.5 in. in millimeters, you multiply by the ratio on the right in Eq. 1.5 and cancel out the inches:

25.4 mm 4.5 in. = (4.5 in.) = 4.5 × 25.4 mm = 1.1 × 102 mm. (1.6) 1 in. a b

1.12 Why is the ratio on the left in Eq. 1.5 not suitable for converting inches to millimeters?

Exercise 1.7 Unit conversions

Convert each quantity to a quantity expressed either in meters or in meters raised to some power: (a) 4.5 in., (b) 3.2 acres, (c) 32 mi, (d) 3.0 pints. Solution I obtain my conversions factors from Appendix C. 2.54 × 10-2 m (a) (4.5 in.) = 1.1 × 10-1 m. 1 in. a b 4.047 × 103 m2 (b) (3.2 acres) = 1.3 × 104 m2. 1 acre a b 1.609 × 103 m (c) (32 mi) = 5.1 × 104 m. 1 mi a b 4.732 × 10-4 m3 (d) (3.0 pints) = 1.4 × 10-3 m3. 1 pint a b

1.13 (a) Using what you know about the diameters of atoms from Section 1.3, estimate the length of one side of a cube made up of 1 mol of closely packed carbon atoms. (b) The mass density of graphite (a form of carbon) is 2.2 × 103 kg m3. By how much does the length you calculated in part a change when you do your calcu- lation with this mass density value? Remember that 1 mol is the number of atoms> in 12 × 10-3 kg of carbon.

1.7 Significant digits The numbers we deal with in physics fall into two categories: exact numbers that are known with complete certainty (integers, such as the 14 in “I have 14 books on my desk”) and inexact numbers that result from measurements and are Figure 1.18 If you measure the width of a known to only within some finite precision. Consider, for example, using a ruler sticky note with the ruler shown, you can reliably read off two digits. to measure the width of a piece of paper (Figure 1.18). The width falls between the ruler marks for 21 mm and 22 mm and is closer to 21 mm than 22 mm. We might guess that the width is about 21.3 mm, but we cannot be sure about the last digit without a better measurement method. By recording the width as 21 mm, we are indicating that the actual value lies between 20.5 mm and 21.5 mm. The Quantitative tools value 21 mm is said to have two significant digits—digits that are known reliably. By expressing a value with the proper number of significant digits, we can convey the precision to which that value is known. For numbers that don’t contain any zeros, all digits shown are significant, which means that 21 has two significant digits, as just noted, and 21.3 has three significant digits (implying that the actual value lies between 21.25 and 21.35). millimeters With numbers that contain zeros, the situation is more complicated. Leading 20 zeros, which means any that come before the first nonzero digit, are never significant: 0.037 has two significant digits. Zeros that come between two nonzero digits,

M01_MAZU0930_PRIN_Ch01_pp001-027.indd 17 23/08/13 12:03 PM chapter glossary 11/20/13 10:22 AM 27 y Chapter Glossar Chapter W an Acauses event an henever A va the nearest to off rounded lue A A p Dig - ob from going for process iterative n S be can mea- that property hysical - reli are that value inanumerical its inatomic particles residing ubatomic of carbon-12, which is measured to be to measured whichis carbon-12, of . number See also Avogadro’s kg Th . Th 3

A - - defined time, the dura as of base unit e SI 23 A p symmetryn object exhibits when certain Th determine to us allows that quantity hysical with energy combined and matter of totality e 10 U 10 (s) A used System, inthe International measure of nits - phenom a natural of explanation well-tested (s) × × t (mol) - the quan measuring for base unit e SI 12 6.022 enon in terms of more basic processes and relationships. and basic processes more of in terms enon the sequence of related events. related the of sequence sured and that is expressed as the product of a numerical a numerical of the product as expressed is that and sured unit. a physical and value ably known. power of ten. of power tion of 9,192,631,770 periods of certain radiation emitted emitted 9,192,631,770 periods certain of radiation tion of atoms. cesium by in science and engineering. in science and Significant digits Symmetry Theory Time Universe Mole of magnitude Order Physical quantity Principle of causality Proton, neutron Scientific method Second SI units event B, all observers see event A happening first. all observers B, A happening seeevent event nuclei. tity of a given substance, defined as the number of atoms atoms of defined the number as substance, a given of tity in servations to theories validated by experiments. by theories validated servations to the space and time in which all events happen. time in which allthe events space and operations can be performed on it without changing its its changing without it be can performed on operations appearance. (1.2) (1.4) (1.3) of a of - the num is ) 3 - 299,792,458 (m > 1

. 23 10

. . × V V N m K K is the amount of mass per volume: mass of the amount is 6.022 n r ) 3 Th = m molecules or atoms of e number > A N (kg Th all- ob for the timesame is that e notion A di inspace. extent or stance . A t Th

be to observations of explanation entative defined length, - the dis as of base unit SI e A s Th A m (m) inacloud itself particle manifesting ubatomic another. to state one from e transition / there substance some of much how of easure A sim

(m) A b a of representation plified conceptual

matter. of block building asic A betweenobservable relationship a of description density mass phenomenon. servers in the universe, regardless of their location or motion. or their location of servers regardless in the universe, density number The volume. in a given is Model Electron Hypothesis Law Length See also mole Change Density The Glossary Chapter parentheses. in given are quantities physical of units SI time Absolute Atom Avogadro’s number Meter in 1 mol: second. around atomic nuclei. atomic around tance traveled by light in vacuum in in vacuum light by traveled tance used as a starting point for further investigation. for used point a starting as ber of objects perber volume: of quantities that manifests itself in recurring patterns of events. of in recurring patterns itself manifests that quantities M01_MAZU0930_PRIN_Ch01_pp001-027.indd 27 Concepts Quantitative tools 11/21/13 2:44 PM

macrostate on volume Dependence of entropy on energy Dependence of entropy ideal gas of a monatomic Properties Entropy 19.5 19.6 19.7 19.8 ideal gas of a monatomic entropy 19.1 States 19.2 equipartition of energy 19.3 equipartition of space 19.4 probable the most evolution toward 19 M19_MAZU0930_PRIN_Ch19_pp501-528.indd 501 M19_MAZU0930_PRIN_Ch19_pp501-528.indd 502

Concepts much pendulum the energy and air the have. molecules atomicthe Let’s scale. by begin getting some feel for how we to need atlook is never observed, molecular motion on why transfer this offrom energy air to molecules pendulum wouldmolecules supply energy. kinetic the To understand energy. The pendulum could swinging start again air ifthe conversionthe of back to that mechanical energy thermal atomic scale. ent nature of motion the of atoms and at molecules the to dissipateenergy appears related to be to incoher the universalpendulum. the for tendency So mechanical the motionthe of air inthe molecules surrounding the the formthe of incoherent associated with energy thermal sipated. is not This energy gone—it exists but still now in has come to rest, of all its dis- has mechanical been energy surroundingecules pendulum. the When pendulum the pendulum suspension and to collisions with air the mol- 7.3).This conversion Section (see is due inthe to friction of pendulum the is converted to incoherent energy thermal of part each swing coherent the asmall mechanical energy ing forever. energies remains constant and pendulum the keeps swing- .Ifthen is dissipated, no energy sum the of two these ergy, gets then which converted to energy, kinetic which gets converted energy kinetic to gravitational potential en- back and forth inside acontainer. As pendulum the swings, and up, never speeds letus consider apendulum swinging derstand why aswinging pendulum always slows down pendulums slow down and eventually come to rest. To un- swinging of its own accord? No; no when forces are applied, Have you a motionless everseen pendulum suddenly start 19.1 States T 502 irreversibility. entropy, in one direction only over time, Iintroduce aquantity, level. To help us quantify of tendency the systems to evolve to dowith randomness the of interactions the at atomic the causes irreversible the flowof toward time future? the do phenomena such never happen? as these What is it that frommolecules coming together into asingle drop. why So ofglass water, or that prohibits ink of acollection diffused “undamaged,” that prohibits from an icecube forming ina its cars damaged inacollision to spontaneously move apart Always, no exceptions. are throughout diffuses ink a ofglass water. phenomena These ofa cube icemelts and inadrink, placed adrop when of Not one of conservation the laws we have studied forbids time, however,Over swings the with get because smaller As chapter,in this we see shall underlying the reason has Yet there’s nothing conservation inthe laws that prohib- irreversible,

irreversibility range of phenomena, cannot yet they explain the he conservation laws allow us to analyze abroad whose change whose is strongly connected to our of sense Chapter 19 Entropy meaning happen they inone direction only. of time. Cars are damaged inacollision, -

the randomthe bombardment of grainsthe by surroundingthe zigzag motion due to collisions with surrounding water molecules. Exercise 19.1Pendulumenergy Figure 19.1 zigzag path like one the shown in Brown noticed that grains the bounced around, following a grainsmicroscope of suspended to inwater, observe pollen of random this effect the molecularbombardment. Using a Scottishthe Brown botanist Robert was first to the observe have to greatly reduce of pendulum. the inertia the In 1827 noticeable. To of such effect the collisions, see we would lisions air and between molecules pendulum the is hardly that of surrounding the air that of molecules col effect the - room temperature a typical nitrogen moves molecule at its maximum kinetic energy: its maximum energy: kinetic (b) The average of energy anitrogen kinetic is molecule why don’t of continuous this effects the we see bombardment? slow pendulum the down? (b)When pendulum the is at rest, pendulumthe from sides.(a)Why all bombardment this does siderably faster than pendulumthe and continuously bombard Solution molecules is molecules ( of individual the nitrogen ✔ molecules. (a billion billion greater times) than average the energy kinetic of pendulum energy the cal is more than 18orders of magnitude value with result the that Iobtained a,Isee mechani the inpart - cules. The masscules. of anitrogen is molecule the averagethe energies kinetic ofnitrogen the all molecules? oferage energy one kinetic nitrogen molecule,and (c What are (a Consider a0.10-kgpendulum swinging at amaximum of speed the pendulum.the ✔ four orders of magnitude greater than of mechanical the energy incoherent motion of nitrogen the all is more molecules) than associated with the energy kinetic (the that energy thermal this 2 1 3 0 c

. . ( This random motion, called The inertia ofThe apendulum inertia at rest much is so greater than ) The sum of average the energies kinetic ofnitrogen the all 2 8 4 0 . × 7

19.1 m × 1 > 0 s 1 - inside a box that contains 0 2 ( The nitrogen inExercise molecules 19.1move con- A grain of suspended inwater pollen undergoes arandom

- a J ) the mechanical energy of mechanical) the pendulum, energy the (b 2 . ) The of mechanical pendulum energy the isto equal ✔ 6

( k 1 g . ) 0 ( 5 × 0 0 1

m 0 2 3 > ) s ( ) 5 2 . 9 = start 2 1

m × 5 v . 1 9 Brownian motion, Brownian 2 0 = × Figure 19.1 - 1 2 .0 1 2 1 1

J ( 0 × ) 0 4 - . = .7 2 1 1 1 0

0 J × k 5 . 2 Comparing this . 3 g 9 nitrogen mole- 1 ) . ( × 0 0 - . 2 8 ) the sum) the of 1 6 nish 0

0 k m 5 2 is due to g J ) the av) the - 0 , . and at at and 0 > Note Note

s m ) 2 = > s .

11/21/13 2:44PM Concepts

11/21/13 2:44 PM 503 . Because microstate s State 19.1 . Because there are more basic states basic states more Because are there . 6 is used instead of basic of used is instead state 3 > 4 = ) - lan in spoken apart tell to hard often are alike and sound 4 The 36 possible combinations for two dice. two for combinations possible 36 The ( ) 6 3 > To calculate the probability of, say, the macrostate 9, the macrostate say, of, the probability calculate To 2 3 4 5 6 7 8 9 1 . ( 10 11 12 6 Before describing a gas, let us return to the simpler sys- the simpler to return us let a gas, describing Before 3 > and macrostate use the term basic state we’ll guage, sum of dots dots of sum both dice on Figure 19.2 19.2 Figure 1 dice after any given throw, or we can give a more detailed a more give can we or throw, given any dice after on showing dots specifying of by the number description particles, of numbers very with large systems For die. each the not properties, large-scale about only care often we describe we When each particle. of detailed specifications de- are we properties, large-scale its of in terms a system a detailed give we When macrostate. the system’s scribing the system, up all make the particles of specification that of a volume * For basic state. the system’s describing are we described is large-scale by the macrostate example, for gas, - tempera as such be measured, can readily that properties described is by basic state The volume. and pressure, ture, gas each individual of specifying velocity and the position particle. any after the system of macrostate The dice. two of tem given is the basicstate thrown; the sum by given is throw see can you As each die. on shown dots of the number by and the system for 36 basic states are 19.2, there in Figure to equally likely is basic state Because any 11 macrostates. occurring is basic state one any of the probability occur, we divide the number of basic states associated with this with associated basic states of the number divide we all with associated basic states of the number by macrostate correspond the 36 basic states of out Four the macrostates. a throwing of the probability 9. Therefore the macrostate to 9 is for the macrostate 7 than there are for the macrostate 9, you 9, you the macrostate for are there 7 than the macrostate for a 9. a 7 than average, on throw, to likely more are *Often the term microstate

start . 2 world we have described have we world 1 > 1 ✔

If you throw two dice 12 times, you dice 12 times, you two throw you If . . 2 2 means “large relative to the size of a the of size to relative “large means that a certain event willcertain a occur event the that is 1 1 > > 1 1 = ), three of which add to 4. On every throw, each of of each 4. On every which add to of ), three throw, 6

3 There are 36 combinations for the pair of dice of the pair for 36 combinations are There What sum of dots are you most likely to throw with with throw to likely most you are dots of sum What How would Brownian motion change if the mass of of if the mass change motion Brownian would How > 3 probability necessarily throw a 4 exactly once. The more times more The a 4 exactly once. throw necessarily

19.3 19.2 The The sev- are there a system, as dice the two consider we If An important conclusion we can draw from the obser from draw can we conclusion important An - the inco the of details treat to impossible Because is it Figure 19.2 Figure When you throw two dice, one red and one blue, many times, many blue, one and red one dice, two throw you When dice the two on which the dots for throws the fraction is of what 4? add to Solution two dice? two the grain in Figure 19.1 were increased? 19.1 were in Figure the grain the 36 combinations is equally likely to occur. If you throw the throw you If occur. to equally likely is the 36 combinations 4 approaches add to that times, the throws fraction of dice many the ratio ( fraction of times that event occurs in a large number of of number occurs in a large event times that fraction of of the probability that you tells 19.2 Exercise repetitions. a 4 is throwing can We this system. of describe can we the state eral ways the two on shown the dots of the sum by it characterize vation of Brownian motion in pollen grains suspended in suspended in pollen grains motion Brownian of vation of result a As incoherent. is the bombardment that is water over each grain of the displacement coherence, this of lack gains never the grain and zero time interval nearly is a long constantly were the grain If kinetic energy. loses any or accelerating keep would it direction, in the same kicked true is same The kinetic energy. gain and direction in that rest: at hanging and the in air suspended pendulum a for would it direction, one from hit constantly were it If swinging. need use a very to we quantitatively, motion atomic herent atoms of world the atomic connect to approach different molecules the macroscopic to and (macroscopicso far theory need use probability to we particular, In molecule”). taking place. the certain likelihood determine events to of a down slowing favor shall see, we the odds strongly As review the begin, let’s To up. speeding it over pendulum two of theory the throwing using probability of essentials example. an dice as won’t won’t the closer the fraction of however, the dice, throw you a 4 will be to throw times you hrowing dice Exercise 19.2 Throwing - small, the bom molecules. Because are thegrains water the grains all equalcausing sides, on always not is bardment up makes what are movements small The zigzag. random to motion. Brownian M19_MAZU0930_PRIN_Ch19_pp501-528.indd 503 M19_MAZU0930_PRIN_Ch19_pp501-528.indd 504

Concepts that macrostate increases (right). offraction states basic associated with pendulumin the the (left), decreases is 84.As number the ofunits energy states associated macrostates with all energy. The number of basic to onecorresponds of sixunits the of eachsolid-colored boxparticles; units among pendulum and three Figure 19.3.Distribution of sixenergy pendulumthe and of three particles units distributedenergy can be among Figure 19.4 504 Figure 19.3 gas particles gas( particles lum. Consider one swinging inside abox containing three fer energy units collidingfer the objects. energy between nature ofunits. energy The collisions box inthe thus trans quantum indivisible,small, discrete quanta units called of energy the confined systems can only change by very and with pendulum. the According to quantum mechanics, but elastically, with one another, of with walls the box,the that is, we can tell apart—andthem collide randomly, they particles. ticles. The pendulum slows down as itsis redistributed energy into the toss? three heads? (d)What is most the likely macrostate after any are there for system? this (c)What is probability the of tossing heads thrown. How many (a)macrostates and states (b)basic system’s the fine macrostate after any throw as number the of Let us nowLet how applies this see all to aswinging pendu-

19.4 Chapter 19 Entropy ), and manydiscrete experiments this confirm Suppose you are tossing four coins and to decide de- The ways six inwhich As apendulum inabox swings, it collides with three par Figure 19.3 ). The are particles

in pendulum units energy Number of 6 0 1 2 3 4 5 distinguishable (singular: (singular: 0 6 5 4 3 2 1 Number units of inparticles energy - 3 3 3 3 3 3 1 — - 6 6 6 6 3 with that macrostate increases. increases, numberparticles the of states basic associated Note threethe particles. that as amount the ofinthe energy units distributed energy canthe be over pendulum the and probability of any macrostate is number the of states basic Figure 19.4 energy units pendulum,energy inthe there are macrostates. seven If system’s the we describe macrostate by number the of tributed among pendulum the and three the gas particles. toleads abehavior that accurately our observations. describes Brownian motion but by that also fact the assumption the This assumption is not justified only by of observation the of and particles speeds the directions the move. they inwhich system:the that is, redistribution the jumbles up the both collisions distribute units energy the randomly throughout units are indistinguishable later.) We assume that further the (We’llthree particles. drop restriction the that energy the units distributedenergy to be among pendulum the and the that there are sixidentical (and therefore indistinguishable tion pendulum’s ofinthe energy suspension and we assume collide with pendulum,particles the we ignore any dissipa - given macrostate. The probability of any one state basic is we can statistics use to determine probability the of any distribution, state eachbasic is likely equally to occur, and 1 > ticles units? has two energy unitsthree energy over such three particles that one of par the 8 Let us investigateLet how units dis sixenergy can- the be To get afeel for how units energy are redistributed as the Because collisionsBecause completely randomize energy the 4 (because there (because are likely 84equally states). basic The 19.5 6 6 3 1 energy unitenergy inone particle three states basic having one shows how, for macrostates, eachof seven these What are sixways the you inwhich can distribute 3 3 3 3 3 6 1 basic statesbasic Number of 28 21 15 10 6 3 1 basic statesbasic Fraction of 84 84 10 84 84 21 84 15 84 28 84 6 1 3 - )

11/21/13 2:44PM Concepts

; - 11/25/13 11:20 AM 29 120 - 10 par 505 10 20 = 10 6

) 1 20 rgy (10 e n e If there are are there If . s. 40 17 - So time is the recurrence 10 10 s. containing 3 air molecules at molecules at 3 air containing artition of 15 3 = The age of the universe is only only is the universe of age The 10 of the system. of s) 1 m 1 s. 15 101 For a large number of particles, the of number large a For 10 fraction of energy in pendulum s. 3 particles 50 100 19.2 Equip 19.2 19 = - (84)(10 with 50 particles, it is on the order of of the order on is 50 particles, with it ; 10 s) so don’t hold your breath waiting for the pen- for waiting breath your hold so don’t 4

- As the number of particles increases from 3 to 100, the 3 to from particles increases of the number As ) Averaged over a long time interval, for what what time interval, for a long over (a) Averaged 19 1 probability relative 0 s,

- 10 quipartition of energy 18 E 10 )(10 19.7 120 Even with a very large number of particles, however, the particles, however, of number a very with large Even this take for would it long how for a feeling get can We volume of a box In raction of the time interval does the pendulum in Figure 19.3 the time interval in Figure doesraction of the pendulum f have more than 50% of the energy of the system? (b) Does the energy the system? this 50% of of than more have the increase we as the same stay or decrease, fractionincrease, particles? of number (10 number of basic states is equal to the number of particles of the number equal is to basic states of number energy of units: the number raised to relative probability of the pendulum containing all the en- containing the pendulum of probability relative that in a box a pendulum release if we So, ergy nonzero. is energy the the mechanical of particles and many contains is the particles, to there transferred gradually is pendulum - gain the pendulum of probability a very nonzero small but energy alling again. back its time interval between the average multiplying by happen to This the system. for basic states of the number by collisions cycle to average take on would time it the of length us gives timeinterval This is all the basic states. through randomly called the recurrence time about 19.2 gas among swinging a pendulum of description our In particles increases, of the number as that saw particles, we energy goesthein pendulum nonzero of theprobability Figure 19.7 19.7 Figure all 50 energy in the of units probability the relative and the on is particles,3 it With precipitously: drops pendulum of order probability of finding a certain fraction of the system’s 50 energy 50 units a certain finding the fraction system’s of of probability decreases energy in it zero finding of that to relative in the pendulum significantly. room temperature, the average time interval between- col the average temperature, room approximately is lisions approximately approximately dulum to recover all its energy! all its recover to dulum (Section 19.6). Thus the recurrence time is approximately approximately time is the recurrence (Section 19.6). Thus with 100 particles, it drops to to drops 100 particles,with it - ap is time interval between collisions theticles, average proximately

shows shows Figure 19.5 Figure 1 1 Figure 19.6 Figure 10 units 50 units 6 units fraction of energyfraction of in pendulum fraction of energyfraction of in pendulum

0 0

relative probability relative probability 1 0 0 As the energy is divided into more units, the probability of of the probability units, more the energy divided into is As As the fraction of energy in the pendulum increases, the increases, energy the fraction of in the pendulum As W shows that as the number of particles increases, particles increases, of the number as that shows 28 84

19.6 the (a) all en- finding of of the probability is hat We “digitized” the energy by arbitrarily dividing it into into it dividing arbitrarily the energy by “digitized” We What happens if we increase the number of particles? of the number increase if we happens What ) either zero or one unit of energy of unit one or zero (b) either ergy and in the pendulum in the pendulum? shows how this probability depends on the en- fraction depends on of this probability how shows ergy in the pendulum. six units. However, collisions could redistribute smaller redistribute could collisions However, six units. so energy the pen the mechanical - of and energy, of units smaller amounts. by change could dulum the spike in the relative probability near the origin sharpens the sharpens origin near probability in the relative the spike Figure 19.6 Figure Figure 19.7 Figure associated with that macrostate divided by the number of of the number divided by macrostate that with associated the which is all with macrostates, associated basic states 19.4. Figure of column in the last given number 19.5 Figure finding a certain fraction of the system’s energy in the pendulum relative to to energy relative in the pendulum a certainfinding the fraction system’s of line. the solid red energy approaches in it zero finding of that probability of that corresponding macrostate decreases. macrostate corresponding that of probability how the probability of finding a certain finding energy fraction of of the probability how energy in zero finding of that to relative in the pendulum Note smaller units. the energy divide we into as changes it into the division 19.6 that 19.5 and Figures comparing by In probabilities. effect the relative on little has finer units the energy dividing by obtain we the picture words, other - good approxi very a remarkably energy is coarse into units predicted fine realistic division more the much of mation mechanics. quantum by M19_MAZU0930_PRIN_Ch19_pp501-528.indd 505 Self-quiz 513

Self-quiz

1. Figure 19.4 shows that there are 15 basic states for four energy units distributed over three particles. List all 15. 2. Suppose a container holds three types of gas. The masses of the gas particles are m, 4m, and 9m. How do the average speeds of the particles compare? 3. In Figure 19.12, what is the probability of finding three particles in any compartment of the box? 4. The two partitions in the container shown in Figure 19.17 can slide freely. Where will they be posi- tioned when the space is equipartitioned?

Figure 19.17

partitions

0 1 2 3 4 5 6 possible positions of partitions

5. A box with a fixed partition through which energy can be exchanged contains 50 particles, 40 on the left and 10 on the right. The system contains 250 energy units. As the system approaches equilibrium, how many energy units end up on the left? On the right? 6. If a closed system tends to change in a specific direction, such as ice melting in an insulated glass of water, what can you say about the number0136150934 of basic states for this system? Mazur Answers Principles and Practice of Physics 1. 400, 040, 004, 310, 301, 031, 130, 013, 103, 220,Pearson 202, 022, 112, 121, 211. 2. According to the concept of equipartition of1509319017 energy, the energy needs to be shared equally among all the gas 1 2 1 2 1 2 PRIN_2 Fig.2 19_172 particles, 2 mv1 = 2 (4m)v2 = 2 (9m)v3, or v1 = 4v2 = 9v3. Thus v1 = 2v2 = 3v3. New 3. Figure 19.12, shows that there are ten basic statesRolin Graphicswith three particles in the top left compartment, regardless of the number of particles in the other compartments.bj 6/21/13 When 12p4 there x 7p11 are two particles in the top left compartment, there are six basic states with three particlesrev in any10/02/13 other compartment. With one particle in the top left compartment, there are 6 + 3 = 9 basic states with three particles in any other compartment. With no particles in the top left compartment, there are 6 + 3 = 9 basic states with three particles in any other compartment. Therefore there are 10 + 6 + 9 + 9 = 34 basic states with three particles in any compartment. The probability of finding three particles in any compartment is thus 34 84. 4. The partition shown at position 2 moves to position 4, and >the partition shown at 4 moves to 5, so that there is 1 particle per unit length of the box. 5. Because at equilibrium the energy must be equipartitioned, the 250 units are evenly distributed across the 50 particles; with five units/per particle. The 40 particles on the left have 200 units of energy, and the 10 particles on the right have 50 units. self-quiz 6. According to the second law of thermodynamics, the system evolves to the macrostate that has the maximum number of basic states. Because the process is irreversible, the final macrostate must have a greater number of basic states than the initial macrostate.

M19_MAZU0930_PRIN_Ch19_pp501-528.indd 513 11/21/13 2:45 PM 514 Chapter 19 Entropy

19.5 Dependence of entropy on volume In Sections 19.2 and 19.3, we saw that the equipartition of energy and space describes the most probable macrostate of any system—that is, the state that has the greatest number of basic states. The number of atomic parameters, such as the position and velocity of each particle, is intractably large for any real system, and so it is generally impossible to specify the basic state of the system. The macrostate, on the other hand, is readily specified by a few measurable macroscopic parameters, such as volume, pressure, or temperature. Figure 19.18 Each of the N particles inside To make the connection between macroscopic and atomic parameters, we con- the container can be placed in any one of the M sider a gas that consists of a very large number N of particles in a container of vol- equal-sized compartments. ume V. The particles occupy a negligible fraction of the volume V and interact with N particles in volume V one another only during collisions that randomize both the energy distribution and the spatial distribution. All collisions between the particles and the walls of the V container are elastic. A gas that satisfies these conditions is called an ideal gas. At high temperature and low pressure, an ideal gas is a good model for a real gas. Let us now calculate how the number of spatial basic states depends on V and N. The system is closed so that the energy of the gas remains constant. Through- out this section we shall take this energy to be equipartitioned over the gas particles. At equilibrium each particle has an equal probability of occupying any lo- volume V divided into M compartments c cation inside the container; the space is equipartitioned and each particle has its fair share of the volume V of the container. To count basic states, we divide the container into equal-sized compartments as we did in Section 19.3. If there are M equal-sized compartments and just one particle, that particle can be placed in any one of the M compartments. Thus there are M basic states for that one particle (Figure 19.18). A second particle can also be placed in any of the M compartments, and so, for two particles, there are M2 basic states. Continuing this ceach of which has volume dV process for N particles, we see that Ω, the number of basic states, is*

Ω = MN (closed system). (19.1)

Exercise 19.8 Number of basic states

Suppose ten distinguishable particles are equipartitioned in a container that is divided into 100 equal-sized compartments. (a) How many basic states are associated with this system? (b) What is the natural logarithm of this number of basic states? Solution (a) The number of basic states is Ω = MN = 10010 = (102)10 = 1020, an un- imaginably large number. ✔ (b) The natural logarithm of the number of basic states is ln Ω = ln(1020) = 46, a much more manageable number. ✔

19.17 If you decrease the size of the compartments in Exercise 19.8 by a factor of 10, what is the change (a) in the number of basic states and (b) in the natural logarithm of that number?

Exercise 19.8 and Checkpoint 19.17 illustrate that the natural logarithm of the number of basic states is much more manageable and less sensitive to a change in compartment size than the number itself. The natural logarithm of the number of basic states from Eq. 19.1 is

ln Ω = ln(MN) = N ln M (closed system at equilibrium), (19.2)

*We assume here that the particles are distinguishable from one another. In reality, the particles of an ideal gas are indistinguishable. In the case of indistinguishable particles and when M W N W 1, the number of basic states is found to be MN N! instead of the result shown in Eq. 19.1. For simplicity, Quantitative tools Quantitative we shall limit the treatment in this section to systems of distinguishable particles. >

M19_MAZU0930_PRIN_Ch19_pp501-528.indd 514 11/21/13 2:45 PM 19.5 Dependence of entropy on volume 515

where we have used the identity ln(ab) = b ln a. If the volume of the equal- sized compartments is dV, the number of compartments is M = V dV, and so Eq. 19.2 becomes > V ln Ω = N ln dV = N ln aV −bN ln dV (closed system at equilibrium), (19.3)

where we have used the identify ln(a b) = ln a − ln b. Because the quantity ln Ω—the natural logarithm of the number of basic states—is so important, we define a quantity> called entropy:*

S K ln Ω. (19.4)

Entropy is a unitless quantity that is a measure of the number of basic states in a system. The greater the number of basic states, the greater the entropy. Entropy allows us to quantify the spreading out of energy or particles over space. As we saw in Section 19.4, a closed system always evolves so as to maximize the number of basic states. As the number of basic states increases, so does the entropy: Sf 7 Si . Therefore,

∆S 7 0 (closed system evolving toward equilibrium). (19.5)

At equilibrium, the number of basic states reaches its maximum value Ωmax and the entropy of the system no longer changes:

∆S = 0 (closed system at equilibrium). (19.6)

Equations 19.5 and 19.6 constitute the entropy law, a mathematical expression of the second law of thermodynamics. The entropy law implies that the entropy of a closed system never decreases (it can only increase or remain constant). It is important to note that Eqs. 19.5 and 19.6 apply to only closed systems. For a system that is not closed, the entropy can increase, remain the same, or decrease. Back in Figure 19.16, for example, the particles in B transfer some of their energy to the particles in A, so B does not constitute a closed system. As we saw, ΩB decreases and so the entropy of B decreases: For systems that are not closed, the entropy can increase, decrease, or stay the same. For a system of N gas particles equipartitioned over a volume V, we can obtain an expression for how entropy depends on volume by substituting Eq. 19.3 into Eq. 19.4:

S = N ln V − N ln dV (closed system at equilibrium). (19.7) Quantitative tools

This expression describes a gas that is equipartitioned over a volume V—that is, at equilibrium. The value for S given by this expression thus represents the maximum value for the entropy of a gas of N particles in a volume V. If the gas is not at equilibrium, its entropy is smaller than that given by Eq. 19.7.

*Historically, entropy has been defined as S K kB ln Ω, where kB is a constant that makes the entropy a much smaller number and gives it units of J/K (see Eq. 19.39). In this book, however, we shall use the definition given in Eq. 19.3. This definition is called a statistical definition of entropy.

M19_MAZU0930_PRIN_Ch19_pp501-528.indd 515 11/21/13 2:45 PM 516 Chapter 19 Entropy

The first term on the right in Eq. 19.7 shows that for a fixed number of particles N, the entropy (that is, the natural logarithm of the number of basic states) increases as we increase the volume V of the container. The second term on the right depends on the size of the compartments: The smaller the compartments, the smaller this term and therefore the greater the entropy. That the entropy depends on the volume dV of the compartments may seem disturbing—after all, the partitioning is completely arbitrary. In general, however, dV V V, and so the second term on the right in Eq. 19.7 is much smaller than the first one and can be ignored. In general, we shall be concerned only with changes in entropy, in which case the second term in Eq. 19.7 drops out entirely. Consider, for example, the expansion of a gas made up of a fixed number of particles from an initial volume Vi to a final volume Vf 7 Vi . According to Eq. 19.7, the change in entropy is

∆S = Sf − Si = (N ln Vf − N ln dV) − (N ln Vi − N ln dV)

Vf = N ln Vf − N ln Vi = N ln (closed system at equilibrium). (19.8) Vi a b Because the second term on the right in Eq. 19.7 cancels out, the entropy change does not depend on the size of the compartments. Note that because Vf 7 Vi the change in entropy is positive.

19.18 (a) What does a positive entropy change in a closed system imply about the change in the number of basic states? (b) How does Eq. 19.8 show that a gas expands if given more room? (c) How does it show that the gas does not contract spontaneously into a subvolume?

Let us next examine how entropy changes when a system of two interacting gases evolves to equilibrium. Consider, for example, two gases inside a container with a partition separating the gases into two compartments, one of volume VA containing NA particles and the other of volume VB containing NB particles Figure 19.19 (a) Two gases of unequal densi- (Figure 19.19a).* If we let the partition move freely, the system evolves to the ties are placed on either side of a partition. (b) When the partition is free to move, the two macrostate that has the maximum number of basic states. If the partition moves gases evolve to the macrostate corresponding to to the right, VA increases, and so the number of basic states for gas A increases. the maximum number of basic states. As VA increases, VB decreases, and so the number of basic states for gas B decreases. Where does the partition come to rest? In other words, what value of gases at di erent densities VA maximizes the number of basic states for the system as a whole? In Section 19.4, we determined that the number of basic states for a system comprising two gases is the product of the number of basic states for the two

VA,i VB,i separate gases:

Ω = ΩAΩB, (19.9)

xed partition Using the definition of entropy (Eq. 19.4), we obtain

S = ln(ΩAΩB) = ln ΩA + ln ΩB = SA + SB, (19.10) When partition is free to move c where we have used the fact that ln(ab) = ln a + ln b. Note that, unlike Ω, the natural logarithm of Ω and thus entropy are extensive properties: V VA,f B,f The entropy of a combined system is the sum of the entropies of the individual systems.

cgases evolve to macrostate

Quantitative tools Quantitative corresponding to maximum number *We assume energy is equipartitioned over the system made up of both compartments and focus on of basic states. only volume changes here. In the next section we shall examine the equipartitioning of energy.

M19_MAZU0930_PRIN_Ch19_pp501-528.indd 516 11/21/13 2:45 PM 19.5 Dependence of entropy on volume 517

Example 19.9 Partitioned gas

A partition divides a container into two equal compartments, A ❸ Execute plan (a) Equation 19.8 gives and B. Initially compartment A contains a gas of distinguishable ∆S = N ln(2Vi Vi) = N ln 2. ✔ particles and compartment B is empty. (a) What is the change in (b) From Eq. 19.7, I obtain for the entropy before reinsertion of the entropy of the gas if the partition is removed and the gas is > the partition: Si = N ln V − N ln dV. Combining Eqs. 19.10 allowed to expand into the volume V of the container? (b) If the and 19.7, I obtain for the final entropy, partition is reinserted after the gas has reached equilibrium in the volume V, what is the change in the entropy of the gas due to the reinsertion? N V 2 V S = S + S = 2 ln = N ln − N ln dV f A B d ❶ Getting started When the partition is removed, the gas 2 >V 2 expands to fill the container, so the final volume occupied by the = N ln V − N lnc 2 − aN ln dbVd. gas is twice the initial volume: Vf = 2Vi . Reinserting the partition divides the container into two equal compartments A and B The change in entropy is thus ∆S = Sf − Si = -N ln 2. ✔ again. The only difference is that now B is not empty; it contains ❹ Evaluate result My answer to part a shows that the half of the particles, and A contains the other half. removal of the partition and subsequent expansion of the gas ❷ Devise plan To obtain the change in entropy due to the re- result in a positive change in entropy. This is what I expect given moval of the partition, I substitute Vf = 2Vi into Eq. 19.8. For that the system evolves toward equilibrium. My answer to part b is part b, the entropy change due to the reinsertion is equal to the the negative of the answer to part a, which tells me that the change difference between the entropy of the gas after the partition is in entropy due to the combined removal and reinsertion of the reinserted and its entropy before the partition is reinserted. To partition is zero: ∆S = N ln 2 − N ln 2 = 0. That is surprising determine the entropy before the partition is reinserted, I use because the gas has expanded into a larger volume. However, the Eq. 19.7. Equation 19.10 tells me that after reinsertion the en- particles in this problem are distinguishable, and in reality gas tropy of the gas is equal to the sum of the entropies in the two particles are indistinguishable. For distinguishable particles, the compartments. Because each compartment has half the volume reinsertion reduces the number of basic states (a particle in com- of the container and contains half the gas particles, the entropies partment B can no longer be in compartment A, and vice versa). in the two compartments are equal to each other. To determine If the particles are indistinguishable, reinserting the partition these entropies, I use Eq. 19.7 again. causes no entropy change.

Assuming that the gases on each side of the partition in Figure 19.19 remain at equilibrium, we have from Eq. 19.7

SA = NA ln VA − NA ln dV (19.11)

SB = NB ln VB − NB ln dV = NB ln(V − VA) − NB ln dV, (19.12)

where V = VA + VB is the combined (constant) volume of the system. Increasing Figure 19.20 VA thus increases SA but decreases SB. Figure 19.20 shows how the entropies As the partition in Figure 19.19 of the two compartments and the entropy of the combined system depend moves to the right and VA increases, SA increases and SB decreases until the entropy of the system on VA. When the number of basic states, and thus the entropy in Eq. 19.10, are is maximum. At this value of VA, the partition maximum, Eq. 19.6 applies, so the partition comes to rest and the combined comes to rest, and the system is at equilibrium. system is at equilibrium. We therefore would like to determine for what value of most probable VA the system’s entropy S is maximized. S macrostate At the maximum value of S, the slope of the S curve in Figure 19.20 is zero.

This slope is given by the derivative of S with respect to VA, and so at equilibrium, S = SA + SB slope = 0 Quantitative tools

dS dSA dSB dSA = + = 0 (equilibrium) (19.13) dVA dVA dVA dVA SB

dSB dSA dSB dVA or = - (equilibrium). (19.14) dV dV A A SA The quantities dS dV and dS dV represent the slopes of the S and S A A B A A B VA curves in Figure 19.20. Equation 19.14 thus tells us that these two curves must 0 > > dV V − dV have opposite slopes at the equilibrium value of VA.

M19_MAZU0930_PRIN_Ch19_pp501-528.indd 517 11/21/13 2:45 PM 528 Chapter 19 Entropy

Chapter Glossary

SI units of physical quantities are given in parentheses. Absolute temperature T (K) A quantity related to the rate Equilibrium state The macrostate of a system that has the of change of entropy with respect to thermal energy: greatest number of basic states and therefore is the most probable macrostate. 1 dS Equipartition of energy K , (19.38) The equal distribution of energy kBT dEth among all constituent parts of a system. Equipartition of space The equal distribution of particles where k is the Boltzmann constant. B over all available volume. Basic state The state of a system that is described using a Ideal gas A gas that consists of a very large number of complete specification of all the constituent particles. particles that move incoherently, occupy a negligible frac- Boltzmann constant kB (J K) A constant of proportional- tion of the volume occupied by the gas, and interact with ity that reconciles the units for absolute temperature with one another and the walls of the container only via elastic > common temperature measurements: collisions. Ideal gas law A relationship among the pressure, volume, -23 kB = 1.380 × 10 J K. (19.39) temperature, and number of atoms in an ideal gas system at equilibrium: > Entropy S (unitless) A quantity equal to the natural loga- N rithm of the number of basic states Ω for a system: P = k T (ideal gas system at equilibrium). (19.51) V B Ω S K ln . (19.4) Kelvin (K) The SI base unit of absolute temperature. For a system of N particles contained in a volume V at a Macrostate The state of a system that is described using temperature T at equilibrium, the entropy is given by only quantities that can be determined by measurements made on scales much larger than molecular sizes. S = N ln(T3/2V) + constant (equilibrium). (19.60) Probability The fraction of times that an event occurs in a large number of repetitions.

Entropy law (also called the second law of thermodynam- Root-mean-square speed vrms (m s) The square root of ics) A closed system always evolves so as to maximize the the average of the squares of the speeds of all the atoms in a number of basic states. As the number of basic states in- volume of gas. For an ideal gas made> up of atoms of mass m creases, so does the entropy: at temperature T, the rms speed is

∆S 7 0 (closed system evolving toward equilibrium). 3kBT v = . (19.53) (19.5) rms m At equilibrium, the number of basic states reaches its maxi- Second law of thermodynamicsB See Entropy law. mum value Ωmax and the entropy of the system no longer Thermal equilibrium The condition in which all parts of a changes: system are at the same temperature. In this condition thermal energy is equipartitioned over the system and the en- ∆S = 0 (closed system at equilibrium). (19.6) tropy of the system is at its maximum. chapter glossary

M19_MAZU0930_PRIN_Ch19_pp501-528.indd 528 11/21/13 2:45 PM VOLUME 1

MAZUR

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Practice Momentum Chapter Summary 52 Review Questions 53 Developing a Feel 54 55 and Guided Problems Worked 61 Questions and Problems Answers to Review Questions 68 68 Answers to Guided Problems 4 M04_MAZU0930_PRAC_Ch04_p051-068.indd 51 M04_MAZU0930_PRAC_Ch04_p051-068.indd 52

practice and conditions final of asystem. system. that not does on depend extent the of the of system. the quantity An intensive value is proportional to sizeor the “extent” system interactionsternal an isolated is called system throughout your analysis. tem, that must object remain of apart the todecide include sys- inthe object acertain systemthe however you want, but once you is not of part system. the You can choose ment. The environment that is everything our minds, from surrounding the environ- group of that objects separated, can be in Concepts Asystem of system the cannot change. another system, inthe but momentum the tum transferred can be from one to object isolated system is constant. The momen- tity, and therefore momentum the of an constant conserved quantity inan isolated system is conserved, to be cannot created be or destroyed is said Concepts Any extensive quantity that Conservation ofmomentum(Sections4.4,4.8) Systems andmomentum(Sections4.4,4.6,4.7,4.8) kilogram to represent it. The SI unit is of the inertia and for reason this we symbol the m use in object. Inertiathe is related to by amount the of that material contained material of object iswhich the made and is entirely determined ertia by of type the to resistdency achange inits velocity. In- track keep moving without slowing down. moving objects friction, along ahorizontal ing over another surface. In absence the of tion that one encounters surface mov when - Concepts Friction Inertia (Sections4.1–4.3,4.5) Chapter Summary 52

A An A system for there which are no ex- Inertia . system shows diagram initial the . Momentum quan is aconserved - extensive quantity extensive (kg). Chapter 4 practieMomentu is ameasure of an ten object's - and amount the of any is resistance the to mo- is any or object is one whose is one mass , of inertia of inertia

The momentum of asystem of is objects sum the of momenta the of its constituents: Quantitative Tools The impulse momentum:final Another way to say is this that for an isolated system, initial momentumthe is to equal the For an isolated system, Quantitative ToolsThe momentum of an isolated system is constant: Quantitative velocity: m s ,

the ratio the is of related inertias the to changes the velocities inthe by S J delivered to asystem isto equal change the inmomentum of system: the Tools If an of object unknown inertia The momentum S J = 0 S . p S K m m p S u s 1 S ∆ J p S p S K p S + i p S = K of an is object product the of and its inertia = - = p S m ∆ 2 ∆ ∆ p S v S 0 p S S f + v v . . . . (4.22) (4.6) (4.17) (4.18) u s

x x g .

m .

u collides with an inertial standard collides with an inertial (4.11, 4.23) (4.1) 10/24/13 3:58PM practice 10/24/13 3:58 PM 53 in physics? Questions Review . For a given system, what is the distinction between the distinction is external what system, a given For important? the distinction is why and interactions, internal and isolated use an is what of and system, isolated an is What system? of momentum The (a) equivalent? these statements two Are is (b) Momentum in time. constant remains system isolated an conserved. lost cylinder inertial were standard the 1-kg international Suppose momentum? of this affect conservation would How destroyed. or momentum? to related it is how and impulse, is What the and momentum of the conservation contrast and Compare law momentum Can the inertia of any object be negative? Can the inertia any of lead? 1 kg of or inertia: feathers of 1 kg greater has Which a sta- or a flying bumblebee momentum: greater has Which inertia? greater has tionary Which train? a easily as as knock wooden a can a fence A 3-g bullet off block this is possible? 140-g baseball How can. object be negative? any of Can the momentum interaction the word of the meaning is What 17. 18. 4.8 Conservation of momentum 19. 20. 21. 22. 11. 12. 4.6 Momentum 13. 14. 15. 4.7 Isolated systems 16.

Describe several ways of minimizing frictionDescribe minimizing betweenof a surface several ways a surface for possible it Is surface. that on object moving an and frictionless? beto completely low-friction track. a horizontal, carts on collide standard Two with cart compare one of in the velocity does the change How the other? of that Cart low-friction track. A a horizontal, on B collide Carts A and cart motionless. initially B is and twicehas the inertia cart B, of that with A compare of in the velocity does the change How of volume the same contain sphere iron an cube and iron An do their inertias compare? How material. differ of made are shape and volume identical objects of Two a system? is What and quantity extensive between an difference the key is What quantity? intensive an extensive an of the value change can processes four What in a system? quantity conserveda is quantity extensive an if mean does it What quantity? you would a baseball, carts and how standard of a pair Using the inertiadetermine the baseball? of of B? of their inertias wood. identical? Are and iron materials: ent 4.1 Friction 1. 4.2 Inertia 2. 3. 4.3 What determines inertia? 4. 5. 4.4 Systems 6. 7. 8. 9. 4.5 Inertial standard 10. Review Questions chapter. this of end the at found be can questions these to Answers M04_MAZU0930_PRAC_Ch04_p051-068.indd 53 M04_MAZU0930_PRAC_Ch04_p051-068.indd 54

practice P. O. N. M. L. K. J. I. H. G. F. E. D. C. B. A. Hints 5. 4. 3. 2. 1. thinking. your guide Make an order-of-magnitude estimate of each of the following quantities. Letters in parentheses refer to hints below. Use them asneeded to Developing aFeel 54

What is likely of the acar speed on aroad crossings? with deer What is required interval time amarathon? to run objects? What do you know about momentumthe of system this of two pins? What of moving isabowling typical speed the ball toward the What is volume the of atypical suitcase? What of atypical isdeer? inertia the What of on car the is speed freeway? the while What of abaseball? is inertia the What of asingle isphysics inertia the textbook? with less inertia? ject What is magnitude the of change the inmomentum of ob the - What of atennis is inertia the ball? What of atypical iscar? inertia the What of amarathon is inertia the runner? duringticed afew seconds of observation? What is largest the nonzero that no to- speed be issmall too an of object greater inertia? How object of the behave does less inertia after acollision with What of abowling is inertia the ball? major-league pitcher (I,S) The magnitude of momentum the pitched of abaseball by a toward pins the (A,M) The magnitude of momentum the of moving abowling ball netthe (F, R) The magnitude of momentum the of as it atennis ball crosses ably after hit being by (C,M,B) bowling the ball initiallyobject, at rest on abowling alley, doesn’t move notice- The ratio of of inertias an the and object ifthe abowling ball of with physicsThe asuitcase inertia filled (H,L,Q) textbooks

Chapter 4 practieMomentu object’s momentum; initial H.3kg;I.0.2J. motion the G. because reverses, about magnitude the twice of the constant; O. the car; V. car; the T. V. U. T. S. R. C. A. 7kg;B. Key Q. K. 1 10. 9. 8. 7. 6. 0;U.

1 5 What is length the of amarathon? usin 8.) question moving bouncing deer off astationary car to by analogy With what would rebound? speed deer the (First consider a What of acar is speed fueling? the when What of apitched is speed the baseball? What of atennis is speed the serve? What is volume the of asingle physics book? stationary (E,P, deer K,U, N) The magnitude of change the invelocity of acar that hits a leaves freeway the to stop for gas (E,J, T) The magnitude of change the inmomentum of acar that hitbowling by ball of tennis the question ball 3(B, G,N) The magnitude of change the in momentum of a stationary that reverses velocityof the ofquestion baseball the 5(G,N) The magnitude of change the inmomentum bat of abaseball ner (D, O, V) The magnitude of momentum the of atypical marathon- run 1 × × (all values approximate)(all 0 1 1 at somewhat aspeed slower of speed initial thanthe twice 1

m 0 0 1 -

> it bounces back, approximately reversing its velocity;

4 3 k s

3hor more; P. m g ; × Q. ; L. > 1 s ;

0 D. 3 0 1 k .1 × m

m 6 1 × 3 0 ; - M. 3 1

m 0 less than freewayperhaps half speed, 1 3 7

; k R.

m g ; E. >

5 s ; N.it remains approximately × 2 × 1 0 1 1

0 m 3

> k s g ;

; S. 3 F. ×

4 6 1 × × 0 1

1 1 m 0 0 1 > -

m 2 s k ;

> g s ; ;

10/24/13 3:58PM practice 10/24/13 3:58 PM 55 . f , c S v c m f , c + S p f , b a + S v f , ) b b a S p m = + i , a c S p m ( + = i , b S 0 S p + + i , i , b a S v S p b roble ms P and Guided Worked m + S 0 - vec the final momentum draw We WG4.2). (Figure x axis momentum is not zero before the catch because the ball in is the catch before zero not is momentum only a negligible effect on) the objects’ accelerations accelerations effect the objects’ a negligible on) only interest. the time interval of during objects the object or includes Choose that a system a cart example, (for the problem of the subject are that a such in in) interested are you whosemomentum the cross interactions theremaining of none that way - the ob around line dashed a Draw boundary. system the system represent to system of choice jects in your should interactions the remaining of None boundary. this line. cross final the initial and showing diagram a system Make environment. its and the system of states With this figure as a guide, we write an equation setting the ini- setting equation an write we guide, a as this figure With direction. 4. 5. Figure WG4.2 Figure must be equal to the system’s final momentum. We notice that that notice We final momentum. be the system’s equal to must the ball and begins, the athlete the dive after that, also note We motion. - com a with objectbecause a single they as move be can treated ab. the object subscript with label this composite We velocity. mon - mo the system’s to all the contributions for account to have We in the final and the dive) before (just in the initial state mentum draw best is to set It equal. the two and the dive), after (just state all the of track keep to representations finalinitial momentum and the choose for we the direction show to and objects in thesystem positive plus the athlete for the vector than larger the bit a canoe for tor be direction in the same must ball momentum because the system Note final pictures. thein initialand magnitude the same of and ball the positive as the incoming choose of the we direction that x tial and final momenta equal to each other: equal to tial final momenta and Because we know all the inertias, the three initial velocities, and the allthe and Because inertias, know initialvelocities, the three we all need we have we the athlete-ball combination, final of velocity the final get to speed the canoe. of

The The . s > m

.5 2 Because we are given the canoe’s inertia, we the canoe’s given Because are we The crucial point is that the momentum exchange exchange the momentum that is crucial The point Because momentum is a conserved quantity and and a conserved is quantity Because momentum . How fast is the canoe moving after he jumps off? jumps he after the moving canoe is fast How ) . s

m > > p m .2 = 1

Separate all objects named in the problem from one one from all in the problem objects named Separate another. Identify all possible interactions among these objects among all interactions possible Identify (the between their environment theseand objects and Earth, etc.). air, determine and individually interaction each Consider accelerate. objects to the interacting causes whether it has affect (or does not that interaction any Eliminate v ( Getting started evise plan Devise - the rel including WG4.1) (Figure diagram a system draw We When you analyze momentum changes in a problem, it a problem, in changes momentum analyze you When - momen which no for is choose to a system convenient isolated (an the system of out or into transferred is tum these steps: follow do so, To system). 1. 2. 3.

“slippery,” we choose to ignore any effect that resistance from the from resistance effect that any choose ignore to we “slippery,” Figure WG4.1 Figure ❷ because the system is isolated, the system’s initial momentum initial momentum the system’s isolated, is because the system These examples involve material from this chapter but are not associated with any particular section. section. particular any with associated not are but chapter this from material involve examples These provided. guidelines the following by out work should you others detail; in out worked are examples Some is both between the canoe and the athlete and between the ball and both the is athlete between the and canoe and betweenthe water interaction some However, the athlete. and - hap analyzing are we Because also the is canoe the possible. event water that know because we time interval a short and pens during is the this simplification, With motion. the canoe’s on have might water isolated. is athlete the ball, and canoe, comprising system of track keep to subscripts using and information velocity evant also use the We canoe. c for athlete, ball, a for objects: b for our keep to final quantities for f and initialquantities for i subscripts events. of order the temporal of track roblem 4.1 Jump ship 4.1 Problem Worked Guided Problems and Worked an isolated system Choosing Procedure: In a game of canoe-ball on a calm lake, an athlete stands at the front the front at stands athlete canoe-ball a calm an lake, of on a game In a throws shore on A player the shore. facing rest, at a canoe of end a speed the with canoe of at arrives 1.8-kg ball that - momen its of the magnitude know we speed its once determine can tum rules of the game demand that the athlete hold on to the ball to and on hold the athlete that demand rulesthe game of off does,whichhe horizontally, the water, into dive immediately inertia the inertia the 60 kg, is and of His the canoe. of the front speed his in the that determine shore on 80 kg. is Friends canoe is dive ❶ M04_MAZU0930_PRAC_Ch04_p051-068.indd 55 M04_MAZU0930_PRAC_Ch04_p051-068.indd 56

practice quantities we know: You are riding your bike at asteady Worked Problem 4.3Explodingbicyclepump Guided Problem 4.2Apieintheface We isolate our unknown, In terms becomes of this speeds, 56 ❸ the equationthe incomponent form: x direction, is which direction inthe of incoming the we write ball, quantities on signs depend our whose choice for positive the (Figure WG4.3). We are given position information for two of the ❶ should you for look spring? the body, 0.25kgfor handle, the and 0.20kg for spring, the where is going hard to be to spot. If are inertias the 0.40kgfor pump the pieces were to easy find,but third spring— the piece—a metal thin rock as 3.2m.Just rock the is beside pump the handle. two These your path, you measure distance the from pump the to the body andexploded, its to parallel road. the length was aligned Retracing stop 0.50 s after explosion.the The pump was horizontal it when watch pump the of ahead sail you, body hit then ground the and arockstrikes at road’s the edge, and into explodes three pieces. You to avoidswerve a pothole. Your bicycle pump from falls its bracket, The final speed of canoe, the speed toThe two final significant digits, is ❶ clownthe roll backward? skates. After pie the his strikes faceand sticks there, how fast does 2. 1. speed of At acarnival, asuper-sized, 1.0-kgcream pie is thrown with a

Getting started m Execute plan

Getting started c the collision?the What is object(s) (are) inmotion before collision? the After pie-clown collision. an isolatedSelect system and sketch asystem diagram for the v v v c c c , , , f f f

= = = = Chapter 4 practieMomentu 5 .0 m 7 ( m 1 m 8 b m b .8 m . v v 7 b

8 b > b b

k v k , v , 0 s i i g b g into faceof the a60-kgclown at rest on roller b

− + , ) k

i x Recalling thatRecalling vector components are signed ( # , g = i 2 m ( ( = .5 m m m > We with asketch begin of situation the ( v

a m m a s c ( c , m + f + a = > v = a + s c m , ) m + f 0 , 0 . + and substitute then values for the b m b 9 .9 ) ) m 8 ( v b 8 8 ( 3 ) - a b 6

( 0 b

) m m v 0 - , v

f k a (1)

a > k v b > g 4 b , s g a

f s x . ) b . 0 . , ✔ f , + f m ) + + 1 > m . s 8 m when suddenly when you c k v c g c v

x ) c , ( , f f 1 . . . 2 m > s )

tive answer for is asum of speed positivefinal terms. If Eq. 1allowed anega- behaves way inthe Equation we expect. 1shows that canoe’s the justified. resistance during of interval time neglected the interest can be is from pushing aboat off. For reason this assumption the that water would significantly take longer than jump, the as you may know resistance, it would reduce canoe’s the but speed, reduction this of ahuman-powered speed the boat. If there were significant water unreasonable; that is, it is of order the of magnitude for we expect pump disintegrates as our system. For between interval time the ❷ ❹ Figure WG4.3 pump before explosion the was same the as that of bike. the have of eachpiece, inertia the and we know that velocity the of the three pieces, but that not alone will locate third the piece. We also 5. ❹ ❸ 4. 3. athlete-ball combination.athlete-ball mentum of system the (negative) or final the momentum of the of changes these would increase either (positive) initial the mo- This matches speed. higher our qualitative understanding that each athlete ifthe speed, a higher had greater or ifhe inertia, dove at a would had greaterspeed increase ball ifit ifthe came inertia, inat left. the boat moves to right the inFigure WG4.1 as athlete the dives to bination is positive be (butalso momentum final the of com athlete-ball the - momentum of system the is positive, momentum final the must mentum for velocity components. The canoe’s final ❷

evise plan Devise

Evaluate result EXECUTE PLAN Devise plan We should our check whether also algebraic to see answer Furthermore, that we see Eq. 1implies that canoe’s the final Evaluate result Is your answer unreasonably large or small? unknowns.the Express your answer to question 3quantitatively and count What doyou know about momentum the of your system? p c negative). This is consistent with our intuition that the

x , f must positive be for initial reason:the this Because v c , f We consider three the pieces into the which , we would have to rethink our choice of signs

The numerical value of our result is not x component of mo- 10/24/13 3:58PM