CONTENTS vii Preface C Introduction Overview CLl Scientists are Model-Makers Cl.2 The Nature of Science Cl.3 The Current Structure of Physics Cl.4 SixIdeasThatShapedPhysics Cl.5 An Overview of UnitC Cl.6 How To Use This Text Effectively Cl.7 Summary Glossary Two-Minute Problems Homework Problems Answers to Exercises C2 Introduction to Momentum Overview Common-Sense Models of Motion Interactions and Motion Speed Some Collision Experiments Mass and Weight Momentum Also Involves Direction **** Physics Skills: Technical Terms **** Physics Skills: Units Summary Glossary Two-Minute Problems Homework Problems Answers to Exercises , C2.1 C2.2 C2.3 C2.4 C2.5 C2.6 C2.7 C3 Vectors Overview Displacement Vectors Reference Frames Components Symbols, Terms, and Conventions Vectors in One and Two Dimensions Vector Operations Vectors Have Units Math Skills: A Review of Basic **** Trigonometry Summary Glossary Two-Minute Problems Homework Problems Answers to Exercises C3.1 C3.2 C3.3 C3.4 C3.5 C3.6 C3.7 C3.8 C Particles and Systems C4.1 C4.2 C4.3 C4.4 C4.5 Overview Position, Velocity, and Momentum Interactions Transfer Momentum Systems of Particles A System's Center of Mass 11 12 12 12 12 13 13 14 15 17 18 19 20 21 22 24 25 25 26 26 27 27 28 28 30 31 33 35 37 38 39 40 40 41 42 43 43 44 46 48 48 C4 Particles and Systems (continued) How the Center of Mass Moves Application: Discovering Planets Math Skills: Illegal Vector Equations Summary Glossary Two-Minute Problems Homework Problems Answers to Exercises C4.6 C4.7 C5 Applying Momentum Conservation C5.1 Overview C5.2 Interactions With the Earth C5.3 When Multiple Interactions Act C5.4 What Counts as 'Isolated'? C5.5 A Framework for Solving Problems Airplanes and Rockets C5.6 Summary Glossary Two-Minute Problems Homework Problems Answers to Exercises C6 Introduction to Energy Overview C6.1 C6.2 The Idea of Conservation of Energy Defining Energy C6.3 C6.4 Interactions and Energy C6.5 Measuring Potential Energies C6.6 Negative Energy? C6.7 A Framework for Energy Problems Summary Glossary Two-Minute Problems Homework Problems Answers to Exercises C7 Potential Energy Functions Overview The Electromagnetic Interaction The Gravitational Interaction Gravitation Near the Earth The Potential Energy of a Spring Some Examples **** Physics Skills: Significant Digits **** Physics Skills: Formulas Summary Glossary Two-Minute Problems Homework Problems Answers to Exercjses C7.1 C7.2 C7.3 C7.4 C7.5 C7.6 51 52 54 55 56 56 57 58 59 59 60 61 63 64 69 71 72 72 73 74 75 75 76 76 80 82 83 84 86 87 87 87 88 89 89 90 91 92 94 96 100 100 101 102 102 103 104 vi Contents e Energy Transfers Overview Momentum and Kinetic Energy The Dot Product A Model For Energy Transfer The Earth's Kinetic Energy C8.6 Contact Interactions C8.7 Momentum, Force, and Energy Summary Glossary Two-Minute Problems Homework Problems Answers to Exercises -C8.1 C8.2 C8.3 C8.4 C8.5 e Rotational Energy C9.1 C9.2 C9.3 C9.4 C9.5 C9.6 C9.7 C9.8 Overview Introduction to Rotational Energy Measuring Angles Angular Velocity The Moment of Inertia Calculating Moments of Intertia Translation and Rotation Rolling Without Slipping Math Skills: Summation Notation Summary Glossary Two-Minute Problems Homework Problems Answers to Exercises C 10 Thermal Energy C 10.1 Overview C1O.2 The Case of the Disappearing Energy C1O.3 Caloric is Energy CIO.4 Thermal Energy C 10.5 Friction and Thermal Energy C1O.6 Heat and Work C I 0.7 Specific Heat* C1O.8 Keeping Track ofIntemal Energies Summary Glossary Two-Minute Problems Homework Problems Answers to Exercises 105 105 106 107 108 109 110 113 1I5 1I6 1I6 1I7 1I8 119 1I9 120 120 122 123 124 126 128 130 131 132 132 132 134 135 135 136 136 138 139 140 141 143 146 147 147 148 148 ell Energy in Bonds C 11 Overview C 11.2 Potential Energy Diagrams C 11.3 Bonds C 11.4 Latent Heat* Cl1.5 Chemical and Nuclear Energy Summary Glossary Two-Minute Problems Homework Problems Answers to Exercises e12 Applications C12.1 Overview C12.2 Power C12.3 Electrical Energy C12.4 Energy in Light and Sound C 12.5 Buoyancy C12.6 Flywheel Power C 12.7 Elastic and Inelastic Collisions Summary Glossary Two-Minute Problems Homework Problems Answers to Exercises e 13 Angular Momentum C 13.1 C13.2 C13.3 C 13.4 C13.5 Overview Introduction to Angular Momentum The Cross Product The Angular Momentum of a Particle The Angular Momentum of a Rigid Object C13.6 Conservation of Angular Momentum C13.7 Application: Neutron Stars Summary Glossary Two-Minute Problems Homework Problems Answers to Exercises Index 149 149 150 152 154 156 159 160 160 161 162 163 163 164 165 166 167 168 170 173 174 174 174 176 177 177 178 179 181 182 185 188 189 190 190 191 192 193 PREFACE INTRODUCTION This volume is one of sixthat together comprise the PRELIMINARY EDITION of the text materials for SixIdeasThatShaped Physics, a fundamentally new approach to the two- or three-semester calculus-based introductory physics course This course is still very much a work in progress We are publishing these volumes in preliminary form so that we can broaden the base of institutions using the course and gather the feedback that we need to better polish both the course and its supporting texts for a formal first edition in a few years Though we have worked very hard to remove as many of the errors and rough edges as possible for this edition, we would greatly appreciate your help in reporting any errors that remain and offering your suggestions for improvement I will tell you how you can contact us in a section near the end of this preface Much of this preface discusses features and issues that are common to all six volumes of the SixIdeas course For comments about this specific unit and how it relates to the others, see section SixIdeasThatShapedPhysics was created in response to a call for innovative curricula offered by the Introductory University Physics Project (IUPP), which subsequently supported its early development IUPP officially tested very early versions of the course at University of Minnesota during 1991/92 and at Amherst and Smith Colleges during 1992/93 In its present form, the course represents the culmination of over eight years of development, testing, and evaluation at Pomona College, Smith College, Amherst College, St Lawrence University, Beloit College, Hope College, UC-Davis, and other institutions We designed this course to be consistent with the three basic principles articulated by the IUPP steering committee in its call for model curricula: Opening comments about this preliminary edition The course's roots in the Introductory University Physics Project The three basic principles of the IUPP project The pace of the course should be reduced so that a broader range of students can achieve an acceptable level of competence and satisfaction physics to better show students what physics is like at the present The course should use one or more 'story lines' to help organize the ideas and motivate student interest There should be more 20th-century The design of SixIdeas was also strongly driven by two other principles: The course should seek to embrace the best of what educational research has taught us about conceptual and structural prob- My additional working principles lems with the standard course The course should stake out a middle ground between the standard introductory course and exciting but radical courses that require substantial investments in infrastructure and/or training This course should be useful in fairly standard environments and should be relatively easy for teachers to understand and adopt In its present form, SixIdeas course consists of a set of six textbooks (one for each 'idea'), a detailed instructor's guide, and a few computer programs that support the course in crucial places The texts have a variety of innovative features that are designed to (I) make them more clear and readable, (2) teach you explicitly about the processes of constructing models and solving complex problems, (3) confront well-known conceptual problems head-on, and (4) support the instructor in innovative uses of class time The instructor's manual is much A summary of the course's features distinctive viii SixIdeasThatShapedPhysics more detailed than is normal, offering detailed suggestions (based on many teacher-years of experience with the course at a variety of institutions) about how to structure the course and adapt it to various calendars and constituencies The instructor's manual also offers a complete description of effective approaches to class time that emphasize active and collaborative learning over lecture (and yet can still be used in fairly large classes), supporting this with day-by-day lesson plans that make this approach much easier to understand and adopt In the remainder of this preface, I will look in more detail at the structure and content of the course and briefly explore why we have designed the various features of the course the way that we have Problems with the traditional intro course The focus is more on skills than on specific content PHILOSOPHY apply basic physical principles to realistic situations, solve realistic problems, perceive or resolve contradictions involving their preconceptions, or organize the ideas of physics hierarchically What students in such courses effectively learn is how to solve highly contrived and patterned homework problems (either by searching for analogous examples in the text and then copying them without much understanding, or by d0ing a random search through the text for a formula that has the right variables.) The high pace of the standard course usually drives students to adopt these kinds of non-thinking behaviors even if they don't want to The goal of SixIdeas is to help students achieve a meaningful level of competence in each of the four thinking skills listed above We have rethought and restructured the course from the ground up so that students are goaded toward (and then rewarded for) behaviors that help them develop these skills We have designed texts, exams, homework assignments, and activity-based class sessions to reinforce each other in keeping students focused on these goals While (mostly for practical reasons) the course does span the most important fields of physics, the emphasis is not particularly on 'covering' material or providing background vocabulary for future study, but more on developing problem-solving, thinking, and modeling skills Facts and formulas evaporate quickly (particularly for those 32 out of 33 that will take no more physics) but if we can develop students' abilities to think like a physicist in a variety of contexts, we have given them something they can use throughout their lives TOPICS EXPLORED The six-unit structure OF THE COURSE The current standard introductory physics course has a number of problems that have been documented in recent years (1) There is so much material to 'cover' in the standard course that students not have time to develop a deep understanding of any part, and instructors not have time to use classroom techniques that would help students really learn (2) Even with all this material, the standard course, focused as it is on classical physics, does not show what physics is like today, and thus presents a skewed picture of the discipline to the 32 out of 33 students who will never take another physics course (3) Most importantly, the standard introductory course generally fails to teach physics Studies have shown that even students who earn high grades in a standard introductory physics course often cannot The goal: to help students become competent in using the skills listed above GENERAL IN THE COURSE SixIdeasThatShapedPhysics is divided into six units (normally offered three per semester) The purpose of each unit is to explore in depth a single idea that has changed the course of physics during the past three centuries The list below describes each unit's letter name, its length (l d = one day one 50minute class session), the idea, and the corresponding area of physics Topics Explored in the Course First UnitUnitUnit Semester C (14 d) N (14 d) R (9 d) (37 class days excluding test days): Conservation Laws Constrain Interactions The Laws of Physics are Universal Physics is Frame-Independent ix (conservation laws) (forces and motion) (special relativity) Second Semester (42 class days excluding test days): Unit E (17 d) Electromagnetic Fields are Dynamic (electrodynamics) Unit Q (16 d) Particles Behave Like Waves (basic quantum physics) (statistical physics) Unit T (9 d) Some Processes are Irreversible (Note that the spring semester is assumed to be longer than fall semester This is typically the case at Pomona and many other institutions, but one can adjust the length of the second semester to as few as 35 days by omitting parts of unit Q.) Dividing the course into such units has a number of advantages The core idea in each unit provides students with motivation and a sense of direction, and helps keep everyone focused But the most important reason for this structure is that it makes clear to students that some ideas and principles in physics are more important than others, a theme emphasized throughout the course The non-standard order of presentation has evolved in response to our observations in early trials [1] Conservation laws are presented first not only because they really are more fundamental than the particular theories of mechanics considered later but also because we have consistently observed that students understand them better and can use them more flexibly than they can Newton's laws It makes sense to have students start by studying very powerful and broadly applicable laws that they can also understand: this builds their confidence while developing thinking skills needed for understanding newtonian mechanics This also delays the need for calculus [2] Special relativity, which fits naturally into the first semester's focus on mechanics and conservation laws, also ends that semester with something both contemporary and compelling (student evaluations consistently rate this section very highly) [3] We found in previous trials that ending the second semester with the intellectually demanding material in unit Q was not wise: ending the course with Unit T (which is less demanding) and thus more practical during the end-of-year rush The suggested order also offers a variety of options for adapting the course to other calendars and paces One can teach these units in three IO-week quarters of two units each: note that the shortest units (R and 1) are naturally paired with longest units (E and Q respectively) ~hen the units are divided this way While the first four units essentially provide a core curriculum that is difficult to change substantially, omitting either Unit Q or Unit T (or both) can create a gentler pace without loss of continuity (since UnitC includes some basic thermal physics, a version of the course omitting unit T still spans much of what is in a standard introductory course) We have also designed unit Q so that several of its major sections can be omitted if necessary Many of these volumes can also stand alone in an appropriate context Units C and N are tightly interwoven, but with some care and in the appropriate context, these could be used separately Unit R only requires a basic knowledge of mechanics In addition to a typical background in mechanics, units E and Q require only a few very basic results from relativity, and Unit T requires only a very basic understanding of energy quantization Other orders are also possible: while the first four units form a core curriculum that works best in the designed order, units Q and T might be exchanged, placed between volumes of the core sequence, or one or the other can be omitted Superficially, the course might seem to involve quite a bit more material than a standard introductory physics course, since substantial amounts of time are devoted to relativity and quantum physics However, we have made substantial cuts in the material presented in the all sections of the course compared to a standard course We made these cuts in two different ways Comments about the nonstandard order Options for adapting to a different calendar Using the volumes alone or in different orders SixIdeasThatShapedPhysics TIle pace was reduced by cutting whole topics and by streamlining the presentation of the rest Choosing an appropriate pace First, we have omitted entire topics, such as fluid mechanics, most of rotational mechanics, almost everything about sound, many electrical engineering topics, geometric optics, polarization, and so on These cuts will no doubt be intolerable to some, but something has to go, and these topics did not fit as welI as others into this particular course framework Our second approach was to simplify and streamline the presentation of topics we discuss A typical chapter in a standard textbook is crammed with a variety of interesting but tangential issues, applications, and other miscellaneous factons The core idea of each SixIdeasunit provides an excellent filter for reducing the number density of factons: virtualIy everything that is not essential for developing that core idea has been eliminated This greatly reduces the 'conceptual noise' that students encounter, which helps them focus on learning the really important ideas Because of the conversational writing style adopted for the text, the total page count of the SixIdeas texts is actually similar to a standard text (about 1100 pages), but if you compare typical chapters discussing the same general material, you will find that the density of concepts in the SixIdeas text is much lower, leading to what I hope will be a more gentle perceived pace Even so, this text is not a 'dumbed-down' version of a standard text Instead of making the text dumber, I have tried very hard to challenge (and hopefully enable) students to become smarter The design pace of this course (one chapter per day) is pretty challenging considering the sophistication of the material, and really represents a maximum pace for fairly well-prepared students at reasonably selective colleges and universities However, I believe that the materials can be used at a much broader range of institutions and contexts at a lower pace (two chapters per three sessions, say, or one chapter per 75-minute class session) This means either cutting material or taking three semesters instead of two, but it can be done The instructor's manual discusses how cuts might be made Part of the point of arranging the text in a 'chapter-per-day' format is to bee clear about how the pace should be limited Course designs that require covering more than one chapter per day should be strictly avoided: if there are too few days to cover the chapters at the design pace, than chapters wilI have to be cut The texts are designed to serve as students' primary source of new information A list of some of the texts' important features FEATURES OF THE TEXT Studies have suggested that lectures are neither the most efficient nor most effective way to present expository material One of my most important goals was to develop a text that could essentially replace the lecture as the primary source of information, freeing up class time for activities that help students practice using those ideas I also wanted to create a text that not only presents the topics but goads students to develop model-building and problem-solving skills, helps them organize ideas hierarchically, encourages them to think qualitatively as well as quantitatively, and supports active learning both inside and outside of class In its current form, the text has a variety of features designed to address these needs, (many of which have evolved in response to early trials): The writing style is expansive and conversational, making the text more suitable to be the primary way students iearn new information Each chapter corresponds to one (SO-minute) class session, which helps guide instructors in maintaining an appropriate pace • Each chapter begins with a unit map and an overview that helps students see how the chapter fits into the general flow of the unit • Each chapter ends with a summary that presents the most important ideas and arguments in a hierarchical outline format • Each chapter has a glossary that summarizes technical terms, helping students realize that certain words have special meanings in physics .. CONTENTS vii Preface C Introduction Overview CLl Scientists are Model-Makers Cl.2 The Nature of Science Cl.3 The Current Structure of Physics Cl.4 Six Ideas That Shaped Physics Cl.5 An.. to Exercises , C2 .1 C2 .2 C2 .3 C2 .4 C2 .5 C2 .6 C2 .7 C3 Vectors Overview Displacement Vectors Reference Frames Components Symbols, Terms, and Conventions Vectors in One and Two Dimensions Vector Operations.. a core curriculum that is difficult to change substantially, omitting either Unit Q or Unit T (or both) can create a gentler pace without loss of continuity (since Unit C includes some basic