Six ideas that shaped physics all units 3rd edition thomas a moore

Six Ideas That Shaped Physics - All Units 3rd Edition

Thomas A. Moore

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Six Ideas That Shaped Physics

Third Edition

Some Physical Constants

Commonly Used Physical Data

Gravitational field strength g = │ W g│ 9.80 N/kg = 9.80 m/s2 (near the earth’s surface)

Mass of the earth Me

Radius of the earth Re

× 1024 kg

km (equatorial) Mass of the sun M⊙

× 1030 kg Radius of the sun R⊙

km

†At

‡At

Useful Conversion Factors

1 meter = 1 m = 100 cm = 39.4 in = 3.28 ft

1 mile = 1 mi = 1609 m = 1.609 km = 5280 ft

1 inch = 1 in = 2.54 cm

1 light-year = 1 ly = 9.46 Pm = 0.946 × 1016 m

1 minute = 1 min = 60 s

1 hour = 1 h = 60 min = 3600 s

1 day = 1 d = 24 h = 86.4 ks = 86,400 s

1 year = 1 y = 365.25 d = 31.6 Ms = 3.16 × 107 s

1 newton = 1 N = 1 kg·m/s2 = 0.225 lb

1 joule = 1 J = 1 N m

1 radian = 1 rad = 57.3° = 0.1592 rev

1 revolution = 1 rev = 2π rad = 360°

1 cycle = 2π rad

1 hertz = 1 Hz = 1 cycle/s

°F

Standard Metric Prefixes

(for powers of 10)

1 m/s = 2.24 mi/h = 3.28 ft/s

1 mi/h = 1.61 km/h = 0.447 m/s = 1.47 ft/s

1 liter = 1 l = (10 cm)3 = 10-3 m3 = 0.0353 ft3

1 ft3 = 1728 in3 = 0.0283 m3

1 gallon = 1 gal = 0.00379 m3 = 3.79 l ≈ 3.8 kg H2O

Weight of 1-kg object near the earth = 9.8 N = 2.2 lb

1 pound = 1 lb = 4.45 N

1 calorie = energy needed to raise the temperature of 1 g of H2O by 1 K = 4.186 J

1 horsepower = 1 hp = 746 W

1 pound per square inch = 6895 Pa

1 food calorie = 1 Cal = 1 kcal = 1000 cal = 4186 J

1 electron volt = 1 eV = 1.602 × 10-19 J

Six Ideas That Shaped Physics

Unit C: Conservation Laws Constrain Interactions

Third Edition

SIX IDEAS THAT SHAPED PHYSICS, UNIT C: CONSERVATION LAWS CONSTRAIN INTERACTIONS, THIRD EDITION

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Dedication

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Names: Moore, Thomas A. (Thomas Andrew), author.

Title: Six ideas that shaped physics. Unit C, Conservation laws constrain interactions/Thomas A. Moore.

Other titles: Conservation laws constrain interactions

Description: Third edition. | New York, NY : McGraw-Hill Education, [2016] | 2017 | Includes index.

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About the Author

Thomas A. Moore graduated from Carleton College (magna cum laude with Distinction in Physics) in 1976. He won a Danforth Fellowship that year that supported his graduate education at Yale University, where he earned a Ph.D. in 1981. He taught at Carleton College and Luther College before taking his current position at Pomona College in 1987, where he won a Wig Award for Distinguished Teaching in 1991. He served as an active member of the steering committee for the national Introductory University Physics Project (IUPP) from 1987 through 1995. This textbook grew out of a model curriculum that he developed for that project in 1989, which was one of only four selected for further development and testing by IUPP.

He has published a number of articles about astrophysical sources of gravitational waves, detection of gravitational waves, and new approaches to teaching physics, as well as a book on general relativity entitled A General Relativity Workbook (University Science Books, 2013). He has also served as a reviewer and as an associate editor for American Journal of Physics. He currently lives in Claremont, California, with his wife Joyce, a retired pastor. When he is not teaching, doing research, or writing, he enjoys reading, hiking, calling contradances, and playing Irish traditional fiddle music.

Preface

Introduction

This volume is one of six that together comprise the text materials for Six Ideas That Shaped Physics, a unique approach to the two- or three-semester calculus-based introductory physics course. I have designed this curriculum (for which these volumes only serve as the text component) to support an introductory course that combines two elements that rarely appear together: (1) a thoroughly 21st-century perspective on physics (including a great deal of 20th-century physics), and (2) strong support for a student-centered classroom that emphasizes active learning both in and outside of class, even in situations where large-enrollment sections are unavoidable.

This course is based on the premises that innovative metaphors for teaching basic concepts, explicitly instructing students in the processes of constructing physical models, and active learning can help students learn the subject much more effectively. In the course of executing this project, I have completely rethought (from scratch) the presentation of every topic, taking advantage of research into physics education wherever possible. I have done nothing in this text just because “that is the way it has always been done.” Moreover, because physics education research has consistently underlined the importance of active learning, I have sought to provide tools for professors (both in the text and online) to make creating a coherent and selfconsistent course structure based on a student-centered classroom as easy and practical as possible. All of the materials have been tested, evaluated, and rewritten multiple times. The result is the culmination of more than 25 years of continual testing and revision.

I have not sought to “dumb down” the course to make it more accessible. Rather, my goal has been to help students become smarter. I have intentionally set higher-than-usual standards for sophistication in physical thinking, but I have also deployed a wide range of tools and structures that help even average students reach this standard. I don’t believe that the mathematical level required by these books is significantly different than that in most university physics texts, but I do ask students to step beyond rote thinking patterns to develop flexible, powerful, conceptual reasoning and modelbuilding skills. My experience and that of other users is that normal students in a wide range of institutional settings can (with appropriate support and practice) meet these standards.

Each of six volumes in the text portion of this course is focused on a single core concept that has been crucial in making physics what it is today. The six volumes and their corresponding ideas are as follows:

Unit C: Conservation laws constrain interactions

Unit N: The laws of physics are universal (Newtonian mechanics)

Unit R: The laws of physics are frame-independent (Relativity)

Unit E: Electric and Magnetic Fields are Unified

Unit Q: Particles behave like waves (Quantum physics)

Unit T: Some processes are irreversible (Thermal physics)

I have listed the units in the order that I recommend they be taught, but I have also constructed units R, E, Q, and T to be sufficiently independent so they can be taught in any order after units C and N. (This is why the units are lettered as opposed to numbered.) There are six units (as opposed to five or seven) to make it possible to easily divide the course into two semesters, three quarters, or three semesters. This unit organization therefore not only makes it possible to dole out the text in small, easily-handled pieces and provide a great deal of flexibility in fitting the course to a given schedule, but also carries its own important pedagogical message: Physics is organized hierarchically, structured around only a handful of core ideas and metaphors.

Another unusual feature of all of the texts is that they have been designed so that each chapter corresponds to what one might handle in a single 50-minute class session at the maximum possible pace (as guided by years of experience). Therefore, while one might design a syllabus that goes at a slower rate, one should not try to go through more than one chapter per 50-minute session (or three chapters in two 70-minute sessions). A few units provide more chapters than you may have time to cover. The preface to such units will tell you what might be cut.

Finally, let me emphasize again that the text materials are just one part of the comprehensive Six Ideas curriculum. On the Six Ideas website, at

you will find a wealth of supporting resources. The most important of these is a detailed instructor’s manual that provides guidance (based on Six Ideas users’ experiences over more than two decades) about how to construct a course at your institution that most effectively teaches students physics. This manual does not provide a one-size-fits-all course plan, but rather exposes the important issues and raises the questions that a professor needs to consider in creating an effective Six Ideas course at their particular institution. The site also provides software that allows professors to post selected problem solutions online where their students alone can see them and for a time period that they choose. A number of other computer applets provide experiences that support student learning in important ways. You will also find there example lesson plans, class videos, information about the course philosophy, evidence for its success, and many other resources.

There is a preface for students appearing just before the first chapter of each unit that explains some important features of the text and assumptions behind the course. I recommend that everyone read it.

Comments about the Current Edition

My general goals for the current edition have been to correct errors, enhance the layout, improve the presentation in many areas, make the book more flexible, and improve the quality and range of the homework problems as well as significantly increase their number. Users of previous editions will note that I have split the old “Synthetic” homework problem category into “Modeling” and “Derivations” categories. “Modeling” problems now more specifically focus on the process of building physical models, making appropriate approximations, and binding together disparate formulas. “Derivation” problems focus more on supporting or extending derivations presented in the text. I thought it valuable to more clearly separate these categories.

The “Basic Skills” category now includes a number of multipart problems specifically designed for use in the classroom to help students practice basic issues. The instructor ’s manual discusses how to use such problems.

I have also been more careful in providing instructors with more choices about what to cover, making it possible for instructors to omit chapters without a loss of continuity. See the unit-specific part of this preface for more details.

Users of previous editions will also note that I have dropped the menulike chapter location diagrams, as well as the glossaries and symbol lists that appeared at the end of each volume. I could find no evidence that these were actually helpful to students. Units C and N still instruct students very carefully on how to construct problem solutions that involve translating, modeling, solving, and checking, but examples and problem solutions for the remaining units have been written in a more flexible format that includes these elements implicitly but not so rigidly and explicitly. Students are rather guided in this unit to start recognizing these elements in more generally formatted solutions, something that I think is an important skill.

The only general notation change is that now I use │ W v│ exclusively and universally for the magnitude of a vector W v. I still think it is very important to have notation that clearly distinguishes vector magnitudes from other scalars, but the old mag(W v) notation is too cumbersome to use exclusively, and mixing it with using just the simple letter has proved confusing. Unit C contains some specific instruction about the notation commonly used in texts by other authors (as well as discussing its problems).

Finally, at the request of many students, I now include short answers to selected homework problems at the end of each unit. This will make students happier without (I think) significantly impinging on professors’ freedom.

Specific Comments About Unit C

This unit is the foundation on which a Six Ideas course rests. The current course structure assumes that unit C is taught first, immediately followed by unit N. Unit C contains core material that will be used in all the other units, as well as providing an introduction to the process of model building that is central to the course.

Why study conservation laws before Newtonian mechanics? The most important reasons are as follows: (1) Conservation of “stuff” is a concrete idea that is easy to understand. Beginning with such simple ideas helps build student confidence at the beginning of the course. (2) Using conservation laws does not really require calculus, and so helps students polish their algebra skills before getting involved with calculus. (3) Studying conservation of momentum and angular momentum does require vectors, allowing students to use vectors for several weeks in simple contexts before introducing vector calculus. (4) Conservation laws really are more fundamental than even Newtonian mechanics, so it is good to start the course with concepts that are central and will be used throughout the course.

I did not intuit these benefits at first: the earliest versions of Six Ideas presented mechanics in the standard order. Rather, this inversion emerged naturally as a consequence of observations of student learning and some reflection about the course’s logical flow.

Inverting the order can be a challenge (in both a positive and negative sense) for the student who already has some background in mechanics. Reviewing mechanics from a different perspective can be quite good for such a student because it makes her or him really think about the subject again. The instructor can play a key role in helping such students appreciate this and by emphasizing the power and breadth of the conservation law approach and its importance in contemporary physics, as well as celebrating with them the power one gains by being able to approach situations from multiple angles.

The momentum-transfer model of interactions (introduced now in chapter C2) is really what makes it possible to talk about conservation laws without starting with Newton’s laws. This model will be a new and challenging idea for almost everyone. Instructors should work carefully with students to give them enough practice with the model to ensure they understand it and can talk about it correctly. The payoff is that when students really grasp this model, it by its very nature helps them avoid many of the standard misconceptions that plague students in introductory courses.

I have substantially revised this unit from the second edition, focusing my attention on the following specific goals.

• I have enlarged the discussion of the model-building process, providing new examples and a more (literally) up-front discussion of tricks and techniques such as unit conversion and dimensional analysis.

• I have reorganized the first few chapters to provide a better logical flow.

• I have substantially rethought how to present expert problem-solving styles. The second edition’s experiment with cartoon balloons and interaction diagrams was not very successful with my students. I have replaced this approach with checklists that specify tasks to complete and a more flexible solution style that I think will be easier for students to emulate. Colored comments on many of the example solutions help students see the connection between the solutions and checklists, and there is also some opportunity for students to practice writing the comments themselves, and so become more self-reflective about the process.

• I have brought the two vector conservation laws (momentum and angular momentum) together, which has several pedagogical advantages (including highlighting how these quantities are similar and how they both contrast with energy).

• I have also reorganized the angular momentum chapters so that the basic idea (and what is necessary for later units) appears first and in its own chapter. All the complicated material (involving the cross product) appears in the second chapter, which may be postponed or even omitted.

• I have reorganized the material in the conservation of energy chapters to better even out the pace and improve the logical flow. In particular, I have separated the material on potential energy graphs from material on bonds, latent heat, chemical energy, and nuclear energy (this was all just too much for one chapter).

• Finally, and most importantly, I have reorganized the energy material to be more consistent with the approach that John Jewett outlined in his series of “Energy and the Confused Student” articles in various issues of The Physics Teacher in 2008. While I don’t completely agree with Jewett on every issue, his insights into student difficulties were consistent with what I have observed in the classroom, and I think his approach is superior pedagogically to what I had been doing. This has meant saying farewell to “k-work,” which now much more correctly appears as the requirement that conservation of momentum imposes on a system. I now also have a complete discussion of “work” that allows a consistent application to deformable systems (including human bodies).

Finally, I have sought to provide more flexibility for instructors. Most of the chapters are crucial and should be discussed in order, but, as noted earlier, chapter C7 on the hard parts of angular momentum may be omitted or delayed, because no other chapter depends on it.

I have also made chapter C14 optional. I think that is very valuable, particularly as a preparation for the last chapters of unit R, but it is not absolutely necessary.

Appreciation

A project of this magnitude cannot be accomplished alone. A list including everyone who has offered important and greatly appreciated help with this project over the past 25 years would be much too long (and such lists appear in the previous editions), so here I will focus for the most part on people who have helped me with this particular edition. First, I would like to thank Tom Bernatowicz and his colleagues at Washington University (particularly Marty Israel and Mairin Hynes) who hosted me for a visit to Washington University where we discussed this edition in detail. Many of my decisions about what was most important in this edition grew out of that visit. Bruce Sherwood and Ruth Chabay always have good ideas to share, and I appreciate their generosity and wisdom. Benjamin Brown and his colleagues at Marquette University have offered some great suggestions as well, and have been working hard on the important task of adapting some Six Ideas problems for computer grading. I’d like to thank Michael Lange at McGraw-Hill for having faith in the Six Ideas project and starting the push for this edition, and Thomas Scaife for continuing that push. Eve Lipton and Jolynn Kilburg has been superb at guiding the project at the detail level. Many others at McGraw-Hill, including Melissa Leick, Ramya Thirumavalavan, Kala Ramachandran, David Tietz, and Deanna Dausener, were instrumental in proofreading and producing the printed text. I’d also like to thank Dwight Whitaker of Pomona College and and his Physics 70 students (especially Nathaniel Roy, Eric Cooper, Neel Kumar, Milo Barisof, Sabrina Li, Samuel Yih, Asher Abrahms, Owen Chapman, Mariana Cisneros, Nick Azar, William Lamb, Wuyi Li, Errol Francis, and Cameron Queen) and my students in Physics 71 (especially Alex Hof, Gabrielle Mehta, Gail Gallaher and Jonah Grubb) for helping me track down errors in the manuscript. David Haley and Marilee Oldstone-Moore helped me with several crucial photographs and offered useful feedback. Finally a very special thanks to my wife, Joyce, who sacrificed and supported me and loved me during this long and demanding project. Heartfelt thanks to all!

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Why

active learning is crucial

Introduction for Students

Introduction

Welcome to Six Ideas That Shaped Physics! This text has a number of features that may be different from science texts you may have encountered previously. This section describes those features and how to use them effectively.

Why Is This Text Different?

Research into physics education consistently shows that people learn physics most effectively through activities where they practice applying physical reasoning and model-building skills in realistic situations. This is because physics is not a body of facts to absorb, but rather a set of thinking skills acquired through practice. You cannot learn such skills by listening to factual lectures any more than you can learn to play the piano by listening to concerts!

This text, therefore, has been designed to support active learning both inside and outside the classroom. It does this by providing (1) resources for various kinds of learning activities, (2) features that encourage active reading, and (3) features that make it as easy as possible to use the text (as opposed to lectures) as the primary source of information, so that you can spend class time doing activities that will actually help you learn.

The Text as Primary Source

Features that help you use the text as the primary source of information

To serve the last goal, I have adopted a conversational style that I hope you will find easy to read, and have tried to be concise without being too terse.

Certain text features help you keep track of the big picture. One of the key aspects of physics is that the concepts are organized hierarchically: some are more fundamental than others. This text is organized into six units, each of which explores the implications of a single deep idea that has shaped physics. Each unit’s front cover states this core idea as part of the unit’s title.

A two-page chapter overview provides a compact summary of that chapter’s contents to give you the big picture before you get into the details and later when you review. Sidebars in the margins help clarify the purpose of sections of the main text at the subpage level and can help you quickly locate items later. I have highlighted technical terms in bold type (like this) when they first appear: their definitions usually appear nearby.

A physics formula consists of both a mathematical equation and a conceptual frame that gives the equation physical meaning. The most important formulas in this book (typically, those that might be relevant outside the current chapter) appear in formula boxes, which state the equation, its purpose (which describes the formula’s meaning), a description of any limitations on the formula’s applicability, and (optionally) some other useful notes. Treat everything in a box as a unit to be remembered and used together.

What is active reading?

Active Reading

Just as passively listening to a lecture does not help you really learn what you need to know about physics, you will not learn what you need by simply

scanning your eyes over the page. Active reading is a crucial study skill for all kinds of technical literature. An active reader stops to pose internal questions such as these: Does this make sense? Is this consistent with my experience? Do I see how I might be able to use this idea? This text provides two important tools to make this process easier.

Use the wide margins to (1) record questions that arise as you read (so you can be sure to get them answered) and the answers you eventually receive, (2) flag important passages, (3) fill in missing mathematical steps, and (4) record insights. Writing in the margins will help keep you actively engaged as you read and supplement the sidebars when you review.

Each chapter contains three or four in-text exercises, which prompt you to develop the habit of thinking as you read (and also give you a break!). These exercises sometimes prompt you to fill in a crucial mathematical detail but often test whether you can apply what you are reading to realistic situations. When you encounter such an exercise, stop and try to work it out. When you are done (or after about 5 minutes or so), look at the answers at the end of the chapter for some immediate feedback. Doing these exercises is one of the more important things you can do to become an active reader.

SmartBook (TM) further supports active reading by continuously measuring what a student knows and presenting questions to help keep students engaged while acquiring new knowledge and reinforcing prior learning.

Class Activities and Homework

This book’s entire purpose is to give you the background you need to do the kinds of practice activities (both in class and as homework) that you need to genuinely learn the material. It is therefore ESSENTIAL that you read every assignment BEFORE you come to class. This is crucial in a course based on this text (and probably more so than in previous science classes you have taken).

The homework problems at the end of each chapter provide for different kinds of practice experiences. Two-minute problems are short conceptual problems that provide practice in extracting the implications of what you have read. Basic Skills problems offer practice in straightforward applications of important formulas. Both can serve as the basis for classroom activities: the letters on the book’s back cover help you communicate the answer to a two-minute problem to your professor (simply point to the letter!). Modeling problems give you practice in constructing coherent mental models of physical situations, and usually require combining several formulas to get an answer. Derivation problems give you practice in mathematically extracting useful consequences of formulas. Rich-context problems are like modeling problems, but with elements that make them more like realistic questions that you might actually encounter in life or work. They are especially suitable for collaborative work. Advanced problems challenge advanced students with questions that involve more subtle reasoning and/or difficult math.

Note that this text contains perhaps fewer examples than you would like. This is because the goal is to teach you to flexibly reason from basic principles, not slavishly copy examples. You may find this hard at first, but real life does not present its puzzles neatly wrapped up as textbook examples. With practice, you will find your power to deal successfully with realistic, practical problems will grow until you yourself are astonished at how what had seemed impossible is now easy. But it does take practice, so work hard and be hopeful!

Features that support developing the habit of active reading

Read the text BEFORE class!

Types of practice activities provided in the text

The Art of Model Building C1

Chapter Overview

Section

C1.1:

The Nature of Science

One of the main goals of science is the development of imaginative conceptual models of physical reality. A model deliberately simplifies a complex reality in such a way that it captures its essence and helps us think more clearly about it. This text’s main purpose is to teach you the art of scientific model building, by helping you not only understand and appreciate the grand models we call theories but also practice building the small-scale models one needs to apply a theory in a given situation.

Science is an unusually effective process for generating powerful models of reality that involves four crucial elements coming together:

1. A sufficiently large community of scholars, who share

2. A commitment to logical consistency as an essential feature of all models,

3. An agreement to use reproducible experiments to test models, and

4. A grand theory rich enough to provide a solid foundation for research.

In the case of physics, the Greek philosophical tradition created a community that valued logical reasoning. Early Renaissance thinkers championed the value of reproducible experiments as being crucial for testing models. But physics was not really launched until 1687, when Newton provided a theory of mechanics grand and compelling enough to unify the community and provide a solid context for research.

Section C1.2: The Development and Structure of Physics

Since the days of Newton, physicists have sought to create models able to embrace originally distinct areas of study and thus cover broader ranges of physical phenomena. The current conceptual structure of physics, illustrated in figure C1.1, rests on two grand theories: general relativity (GR) and the Standard Model (SM) of particle physics. In practice, though, physicists almost always use five simpler theories (which are approximations valid in various limited contexts): newtonian mechanics, special relativity, electromagnetic field theory, quantum mechanics, and statistical mechanics. This text focuses on these five models.

Physicists have recently come to appreciate the role that symmetries play in physics. Both GR and the SM acknowledge (as almost any imaginable theory must) certain symmetries (such as the time and position independence of physical laws) that give rise to conservation laws (such as the laws of conservation of energy and momentum). Such laws have a validity beyond the specific theories currently in vogue. Indeed, GR and the SM themselves are based on new, nonobvious symmetries.

Section C1.3: A Model-Building Example

To apply a grand theoretical model to any actual physical situation of interest, a scientist must construct an idealized model that simplifies the situation, bringing its essence into focus in such a way that one can easily connect it to the grand model. This course is designed to help you practice this process, which is really the only way to learn how to do it. This section illustrates what is involved in an example situation.

In the process, the section describes some useful tricks that can help you simplify situations and think about which simplifications are appropriate:

1. Lines or rays from a very distant point are nearly parallel.

2. The length of a gentle curve between two points is almost the same as that of a straight line between those points.

3. The fractional uncertainty of a result calculated by multiplication or division from uncertain quantities is roughly equal to that of the most uncertain quantity involved. The same is true for the sine or tangent of small angles.

Part of the art of model building is to develop a bag of such tricks that you can pull out when helpful. The only real way to learn these tricks is by practice, and also by making mistakes that you learn to correct. So be bold and learn from your mistakes!

Solutions to most physics problems involve three different sections:

1. A model section that describes the simplifications one makes to the situation

2. A math section where one does the mathematics implied by the model

3. A check section where one decides whether the result makes sense

Your earlier experience with more trivial problems may lead you to neglect the model and check sections, but I strongly recommend you do not. The model section is particularly important in this course. A good and sufficiently well-labeled diagram is often the core of a sufficient model for problem solutions you prepare.

Section C1.4: Trick Bag: Unit Awareness

One of the most powerful tricks you can put in your bag is being aware of units. Units give meaning to quantities and are essential for correctly communicating that meaning to others. Being constantly aware of units (even when working with symbolic equations) is one of the best and easiest ways to spot mistakes in your work. Here is a list of the things you should know to increase your unit awareness:

1. Know the basic and derived SI units and SI prefixes (see the inside front cover).

2. Know and/or refer to the SI unit benchmarks in figure C1.2.

3. Know that the units on both sides of an equation must match.

4. Know that you cannot add or subtract quantities with different units, but you can multiply or divide them.

5. Know that you should be aware of units even in symbolic equations.

6. Know that math functions take unitless arguments and yield unitless results.

Section C1.5: Trick Bag: Unit Conversions

In most physics problem solutions, you will need to convert units. My preferred technique for doing this is the unit operator method, where you convert unit equalities such as 1 mile  = 1609 meters into a ratio equal to 1 such as 1 = (1 mi/1609 m) or 1  = (1609 m/1 mi). Since anything can be multiplied by 1 without changing it, you can multiply any quantity by such a unit operator and cancel units top and bottom (as if they were algebraic symbols) until only the units you want are left over. For example: 23 mi = (23 @ mi)(1609 m/1 @ mi) = 37,000 m. This method is foolproof as long as you pay attention to unit consistency and to canceling units correctly.

Section C1.6: Trick Bag: Dimensional Analysis

Dimensional analysis is a surprisingly powerful trick that often yields good estimates of physics formulas and/or quantities without requiring anything more than the most basic knowledge of a situation. As such, it often represents the simplest model you can construct of a given situation.

This trick takes advantage of the facts that (1) units must agree on both sides of any equation, (2) that most formulas in physics are simple power laws, and (3) that most unitless constants appearing in such formulas are within a factor of 10 or so of one. The steps in applying dimensional analysis to a situation are as follows:

1. Decide what quantities your desired value might depend on.

2. Assume that these quantities appear in a power law formula (e.g., Q = KAmBn).

3. Find the powers by requiring units to be consistent on both sides of the formula.

4. Assume that the unitless constant K in front of the equation is 1.

Model building occurs at all levels of science

C1.1 The Nature of Science

By our nature, we humans strive to discern order in the cosmos and love to tell stories that use ideas from our collective experience to “explain” what we see. Science stands firmly in this ancient tradition: stories about how the gods guide the planets around the sky and modern stories about how spacetime curvature does the same have much in common. What distinguishes science from the rest of the human storytelling tradition are (1) the types of stories scientists tell, (2) the process that they use for developing and sifting these stories, and (3) the predictive success enjoyed by the surviving stories.

Scientists express their stories in the form of conceptual models, which bear a similar relation to the real world as a model airplane does to a real jet. A good scientific model captures a phenomenon’s essence while being small and simple enough for a human mind to grasp. Models are essential because reality is too complicated to understand fully; models distill complex phenomena into bite-sized chunks that finite minds can digest. Framing a model is less an act of discovery than of imagination: a good model is a compelling story about reality that creatively ignores just the right amount of complexity.

Model-making in science happens at all levels. Theories—grand models embracing a huge range of phenomena—are for science what great novels are for literature: soaring works of imagination that we study and celebrate for their insight. But applying such a grand model to a real-life situation requires building a smaller model of the situation itself, simplifying the situation and making appropriate approximations to help us connect it to the grand model. Scientists do this second kind of model-making daily, and one of the main goals of this course is to help you learn that art.

Because models are necessarily and consciously simpler than reality, all have limits: the full “truth” about any phenomenon can never be told. Pushing any model far enough eventually exposes its inadequacies. Even so, one can distinguish better from poorer models. Better models are more logical, more predictive in a broader range of cases, more elegantly constructed, and more productive in generating further research than poorer ones are.

Science is really a process for building, evaluating, and refining models, one that (since its beginnings in the 1600s) has proved to be an astonishingly prolific producer of powerful and trustworthy models. It owes part of its success to its focus on the natural world, whose orderly behavior at many levels makes finding and testing models easier than in the world of human culture.

Scholars of the philosophy and history of science suggest that a discipline becomes a science only when the following four elements come together:

1. A sufficiently large community of scholars, who share

2. A commitment to logical consistency as an essential feature of all models,

3. An agreement to use reproducible experiments to test models, and

4. A grand theory rich enough to provide a solid foundation for research.

How

In the case of physics, the Greek philosophical tradition founded a community of scholars who appreciated the power of logical reasoning: indeed, this community found logic’s power so liberating that it long imagined pure logic to be sufficient for knowing. The idea of using experiments to test one’s logic and assumptions was not even fully expressed until the 13th century, and was not recognized as necessary until the 17th. Eventually, though, the community recognized that the human desire to order experience is so strong that the core challenge facing a thinker is to distinguish real order from merely imagined patterns. Reproducible experiments make what would otherwise be individual experience available to a wider community, anchoring models more firmly to reality. Galileo Galilei (1564–1642) was a great champion of

this approach. His use of the newly invented telescope to display features of heavenly bodies unanticipated by models of the time underlined to his peers the inadequacy of pure reason and the importance of observation.

A prescientific community lacking a grand theory, however, tends to fragment into schools, each championing its own theory. Rapid progress is thwarted because each school sees any collected data through the lens of its cherished model, making arguments virtually impossible to resolve. This was the situation in physics during most of the 1600s. However, in 1687 Isaac Newton published an ingenious model of physics broad enough to embrace both terrestrial and celestial phenomena. His grand theory captured the imagination of the entire physics community, which turned away from arguing about partial models and toward working together to refine, test, and extend Newton’s basic theory, confident that it would be shown to be universally true and valid. At this moment, physics became a science.

The unified community now made rapid progress in constructing powerful subordinate models that greatly extended the reach of Newton’s grand vision, feeding the Industrial Revolution along the way. Ironically, the community that strove energetically to extend Newton’s model universally eventually amassed evidence proving it incomplete! Only a community devoted to a theory can collect the kind of detailed and careful evidence necessary to expose its inadequacies, and thus move on to better theories. This irony is the engine that drives science forward.

C1.2 The Development and Structure of Physics

Unification of apparently distinct models has been an important theme in the development of physics since Newton’s theory unified terrestrial and celestial physics. In the 1800s, work on electricity, magnetism, and light (initially described by distinct partial models) culminated in an “electromagnetic field model” embracing them all, and physicists found how to subsume thermal phenomena into Newton’s model. This process was going so well in the late 1800s that the physicist Lord Kelvin famously claimed that there was probably little left to learn about physics!

In the early 1900s, though, physicists began to see that certain experimental results were simply incompatible with Newton’s framework. After what amounted to a period of revolution, the community demoted Newton’s theory and coalesced around two new grand theories—general relativity (1915) and quantum mechanics (1926)—which embraced the new results but yielded the same results as Newton’s theory in the appropriate limits.

In the 1950s, physicists were able to unify quantum mechanics, electromagnetic field theory, and special relativity (the nongravitational part of general relativity) to create quantum electrodynamics (QED), the first example of a relativistic quantum field theory. In the 1970s, physicists extended this model to create relativistic quantum field theories to describe two new (subatomic-scale) interactions discovered in previous decades and integrated them with QED into a coherent theory of subatomic particle physics called the Standard Model. This model has been quite successful, predicting new phenomena and particles that have been subsequently observed. The model’s latest triumph was the discovery of the predicted “Higgs boson” in 2012.

Currently, general relativity, which covers gravity and other physical phenomena at distance scales larger than molecules, and the Standard Model, which works in principle at all distance scales but does not and cannot cover gravity, stand as the squabbling twin grand theories of physics. Though no known experimental result defies explanation by one or the other, physicists

The ironic paradox at the heart of science

The history of physics since Newton

The current structure of physics

**THE STANDARD MODEL**

Deep underlying SYMMETRIES in the nature of physical interactions

CONSERVATION LAWS

(interwoven relativistic quantum eld theories for everything but gravity)

(Classical) ELECTROMAGNETIC FIELD THEORY

(approximately valid for weak elds and large distances)

Figure C1.1

QUANTUM MECHANICS

(approximately valid at low particle energies)

STATISTICAL MECHANICS

(describes complex objects constructed of many interacting particles)

The current grand theories of physics (starred) and the five approximate models more often used in practice.

The importance of symmetries in physics

**GENERAL RELATIVITY**

(describes gravity and the motion of objects bigger than molecules)

SPECIAL RELATIVITY

(approximately valid in the presence of weak gravitational elds)

NEWTONIAN MECHANICS

(approximately valid for massive objects moving much slower than light in weak gravitational elds)

are dissatisfied with each theory (for different reasons) and especially distressed that we need two deeply incompatible theories instead of one. While many unifying models have been proposed (string theory and loop quantum gravity are examples), these models lack both the level of development and the firm experimental basis to inspire general acceptance. The physics community is thus presently in the curious position of being devoted to two grand theories we already know to be wrong (or at best incomplete).

In practice, however, physicists rarely use either to explain any but the most exotic phenomena. Instead, they use one of five simpler theories: newtonian mechanics, special relativity, electromagnetic field theory, quantum mechanics, and statistical mechanics. Each has a more limited range of applicability than the two grand theories, but is typically much easier to use within that range. These theories, their limitations, and their relationships to the grand theories and each other are illustrated in figure C1.1.

This diagram also emphasizes the importance of symmetry principles in physics. Early in the 1900s, mathematician Emmy Noether showed that, given plausible assumptions about the form that physical laws must have, a symmetry principle stating that “the laws of physics are unaffected if you do such-and-such” automatically implies an associated conservation law. For example, the time-independence of the laws of physics (whatever those laws might be) implies that a quantity that we call energy is conserved (that is, does not change in time) in an isolated set of objects obeying those laws.

Conservation laws, therefore, stand independently and behind the particular models of physics, as figure C1.1 illustrates. For example, conservation of energy is a feature of newtonian mechanics, electromagnetic field theory, special relativity, quantum mechanics, and statistical mechanics

because all these theories involve physical laws that (1) have forms consistent with Noether’s theorem and (2) are assumed to be time independent. Each theory has a different way of defining energy, but all agree that it is conserved.

Symmetry principles are also important because both our current grand models of physics (the Standard Model and general relativity) propose and unravel the consequences of new and nonobvious symmetry principles. The section of this text on special relativity illustrates this by displaying how relativity’s mind-blowing features are in fact simple logical consequences of the symmetry principle that “the laws of physics are unaffected by one’s state of (uniform) motion.” Linking other symmetry principles with their consequences is (unfortunately) not quite so simple (and is beyond the level of this course), but is not qualitatively different.

Now, given the structure of physics illustrated in figure C1.1, it might seem logical to begin studying physics by starting with the two fundamental theories (or even the symmetry principles) and then working downward to the five approximate theories. However, this is impractical because the fundamental theories, in spite of their awesome breadth and beauty, are (1) very sophisticated mathematically and conceptually, (2) unnecessarily complicated to use in most contexts, and (3) necessarily expressed using the language and concepts of the five simpler theories. One must therefore start by learning those simpler theories. The other five volumes of this textbook will provide you with a very basic introduction to all five of these simpler theories, as well as exploring many of the supporting models that help broaden their range. This unit begins the process by looking at the conservation laws (in the context of newtonian mechanics) that underlie all these theories.

In the remainder of this chapter, though, we will explore the modelbuilding process in more detail and develop some general tools that help us avoid errors and maximize what we can gain from even limited knowledge.

C1.3 A Model-Building Example

This book is designed partly to teach you the kind of creative model-building that working scientists do daily. The model-building process cannot be reduced to formulaic procedures that one can follow like a recipe. It is an art that requires knowledge, intelligence, creativity, and most of all, practice. You can no more learn this art simply by reading books or attending lectures than you can learn to play the piano simply by attending concerts.

So let’s practice! The exercise below poses a simple question you can answer using some basic trigonometry and geometry, grade-school science, and a bit of creative model building. Spend at least 10 minutes but no more than 15 minutes trying to answer the question before turning the page.

Exercise C1X.1

About 240 B.C.E., Eratosthenes made the first good estimate of the earth’s size as follows. Caravan travelers told him that in the village of Syene, one could see the sun reflected in a deep well at noon on the summer solstice, meaning that it was directly overhead. Eratosthenes noted that at the same time on the same day in Alexandria (5000 Greek stadia to the north, as estimated by camel travel time), a vertical stick cast a shadow about 1/8 of its length. What is the earth’s radius in stadia? (Hint: Draw a picture. In 240 B.C.E., the Greeks knew that the earth was spherical and the sun was very far away.)

Why one must learn the five simpler theories first

One learns the art of model building through practice

An example that illustrates the model-building process

Example C1.1

Model If the sun is sufficiently far from the earth, light rays traveling from the sun to Alexandria (point A) and Syene (point S) will be almost parallel. Let’s assume they are exactly parallel and that the earth is perfectly spherical. The figure below shows a cross-sectional view of the situation from the east.

earth

parallel rays from the sun

CLOSE-UP: r d

earth’s circumference = c S C A θ θ θ stick stick shadow

Because alternate interior angles are equal, the angle θ between the stick and the sun’s rays at Alexandria is the same as the angle θ between lines AC and SC, where C is the earth’s center. The stick is vertical by assumption, so it is perpendicular to its shadow. Thus, the stick, the shadow, and the ray passing the stick’s end form a right triangle (see the “close-up” above). If the shadow is 1/8 the stick’s length, then tan θ = 1/8, so θ = tan-1(1/8) = 7.1°. The distance d between Alexandria and Syene is to the earth’s circumference c as θ is to 360° , so c/d = 360°/θ. Note also that c = 2πr, where r is the earth’s radius.

Math Therefore

r = c 2π = d 2π ( 360° θ ) = (5000 stadia)(360°)

2π(7.1°) = 40,000 stadia (C1.1)

Check The exact length of Eratosthene’s stadion is historically ambiguous, but if he meant the “itinerary stadion” (the one used for road trips), then 1 stadion = 0.157 km and r is 6300 km, pretty close to the modern value.

If you got something like this, congratulations! If you had trouble getting a useful result in 15 minutes, that’s normal. Doing a moderately realistic problem like this is hard, not usually because the math or concepts are hard (both are pretty basic here), but because constructing the model is hard. How does one know what approximations to make? How does one create a schematic diagram of a situation (like the one shown) that usefully exposes its essential features? How do you frame things so that the mathematics is simple?

You may even be annoyed with my solution: “Well,” you might say, “if I had known it was acceptable to make the false assumption that rays from the sun are parallel, then the solution would have been easy!” That is precisely the point! It is not only acceptable but also usually necessary to make simplifications to solve a problem at all. The trick is simplifying just enough to make the problem tractable without making the result uselessly crude. There are no “correct” answers in such a case, only poorer and better models that yield poorer or better results (and if a poor result is the best one can do, it is still better than nothing!). This is where the creativity and artistry comes in. My goal is to help you learn to simplify (that is, to be productively and creatively lazy) imaginatively, boldly, and exuberantly!

With this in mind, let’s examine more closely the simplifications and assumptions behind the model in example C1.1. The solution assumes that the sun is sufficiently distant that light rays from it are parallel at the earth: if this is not so, the two angles marked θ in the diagram are not equal. No one knew the distance to the sun in Eratosthenes’s time, so his assumption was quite bold, but we now know that two rays from a single point on the sun that arrive at Syene and Alexandria, respectively, are not parallel, but actually make an angle of about 0.00035° with each other. Solving this problem “more correctly” by taking this into account yields an r that is smaller by about 0.005%.

However, this is truly insignificant compared with other simplifications we are making. The sun’s angular diameter when viewed from the earth is about 0.5°, so there is not just one ray that grazes the top of the stick and connects it with the shadow on the ground, but rather a bundle of rays that could make angles with each other of as much as 0.53°. This means that the end of the stick’s shadow will be blurred, making its length and thus the angle  θ uncertain by about ±0.26°. Also the hills and valleys between Syene and Alexandria make the road distance d longer than the distance that would be measured on a perfect sphere. Moreover, Alexandria is not due north of Syene, as the drawing assumes it is. The earth is also not exactly spherical (its polar radius is smaller than its equatorial radius by about 11 km).

I could state yet more subtle assumptions, but I think you get the point. Reality is complicated, and the model simply ignores those complications.

Now, it turns out if you multiply or divide uncertain or erroneous quantities, the result has (roughly) the same percent uncertainty as the most uncertain of the quantities. This weakest-link rule is also (roughly) true for tangents or sines of small angles. (Check it out for yourself: see problem C1D.1.) In this case, the uncertainty in θ is roughly ±4% (±0.26°/7.1°) because of the angular width of the sun, and the uncertainty in the distance d is likely to be more than ±10%, since it is determined by camel travel time! The weakestlink rule implies, therefore, that we are not going to know the radius of the earth to better than about ±10% no matter how good our model is. Making a far more complicated model to correct the approximations described above is not going to make the slightest bit of practical difference: we are simply not given good enough information to calculate the earth’s radius more precisely. The problem (as stated) therefore does not deserve a better model!

Part of the art of model building is knowing when a model is “good enough.” Eratosthenes’s model was not merely “good enough;” it was pure genius at the time, since no other method of determining the earth’s radius was remotely as good. (Sometimes even a crude result is a big step forward!) One learns the art of “good enough” mostly by practice. Indeed, I hope this course will give you (among other things) a bag of useful tricks that are often “good enough.” Treating lines from a distant point as parallel is one such trick. The weakest-link rule about the uncertainty of multiplied or divided quantities is another. Practice with tricks like these puts them into your bag.

Another important trick is recognizing the importance of a good diagram. Drawing (and carefully labeling) the drawing in example C1.1 was probably the single most important thing I did to solve the problem. Most of the “Model” in my solution merely restates the diagram verbally. A good diagram is often the most important trick for solving a physics problem.

Indeed, solutions to all but the most trivial problems will involve the three sections appearing in the example solution: a model section where one draws a schematic diagram and/or discusses approximations and assumptions, a math section where one does the mathematics implied by the model to solve for the desired result, and a check section where one checks the result to see if it makes sense.

The simplifications and assumptions involved in example C1.1’s model

The “weakest link rule” for uncertain quantities

The art of “good enough”

Good diagrams are essential

The three sections of almost any physics problem solution

Don’t neglect either the “check” or “model” parts

The value of making mistakes

Beginners often neglect the “check” section, but you will learn by experience not to (if only to avoid submitting embarrassingly knuckle-headed results). However, in this course (since it is about practicing model-building), the model section is the most crucial part of any solution you submit. A model need not be much more than a well-labeled diagram and/or a few comments about assumptions. I will provide some examples to emulate as we go on.

Note that making mistakes (and then correcting them) is an honorable part of the learning process. Werner Heisenberg, a great physicist of the early 1900s, said that “An expert is [someone] who knows… the worst errors that can be made in a subject… and [thus] how to avoid them.” This is true in my experience, and is something one can often only learn by making those mistakes.

C1.4 Trick Bag: Unit Awareness

Why unit awareness is important

SI units and prefixes

STANDARD SI PREFIXES

An essential item for your bag of tricks is being aware of units. Units attach physical meaning to bare numbers and communicating the magnitude of a measured quantity requires using an agreed-upon unit for that quantity.

The importance of this was starkly illustrated in 1999 when NASA’s $125-million Mars Climate Orbiter burned up in Mars’s atmosphere because the spacecraft’s builders had been sending thruster data in English units (pounds) to a NASA navigation team expecting data in metric units (newtons). One commentator to the Los Angeles Times stated, “[This] is going to be a cautionary tale to the end of time.” (He got that right.)

To help avoid such catastrophic confusion, in this text I will most often use SI units (from the French Système Internationale), the modern version of metric units. An international committee has worked since 1960 to provide clear and reproducible definitions for standard physical units. The entire system is based on seven base units, each with a standard abbreviation: the meter (abbreviation: m) for distance, the second (s) for time, the kilogram (kg) for mass, the kelvin (K) for temperature, the mole (mol) for counting molecules, the ampere (A) for electric current, and the candela (cd) for luminous intensity (we will not use the candela in this course). The committee has defined each unit (except the kilogram) in such a way that a scientist can in principle recreate the unit in his or her own laboratory.

The SI committee has also defined derived units that are combinations of the base units. The units we will use are the joule (1 J ≡ 1 kg·m2/s2) for energy, the watt (1 W ≡ 1 J/s) for power, the newton (1 N ≡ 1 kg·m/s2) for force, the pascal (1 Pa ≡ 1 N/m2) for pressure, the coulomb (1 C ≡ 1 A·s) for electric charge, the volt (1 V ≡ 1 J/C) for electrical energy per unit charge, the ohm (1 Ω ≡ 1 V/A) for electrical resistance, and the hertz (1 Hz ≡ 1 wave cycle/s). Future chapters will describe what these derived units mean.

Table C1.1

Standard SI prefixes for powers of 10. (You can also find this table on the inside front cover.)

The SI committee has also defined a set of standard prefixes and prefix abbreviations (see table C1.1) that one attaches to a unit to multiply it by selected powers of 10. Thus, 1 millimeter (abbreviation: mm) is equal to 10–3 m, 1 gigawatt (GW, pronounced with the g as in “get”) is 109 W, and 1 nanosecond (ns) = 10–9 s. You may already be familiar with some of the larger prefixes from computer terminology (e.g., TB for terabyte = 1012 bytes, GHz for gigahertz = 109 hertz). My experience is that English-speakers have the most trouble distinguishing milli (= 10–3) from micro (= 10–6) because “milli” sounds like “millionth” even though it means “thousandth.” I recommend memorizing the prefixes at least from pico to tera

Units for angles (the radian and the degree) are standard but (for historical reasons) are not considered formal SI units. I will sometimes also mention English units like the mile (mi), foot (ft), and pound (lb) (a unit of force).

edge of visible universe

diameter nearer stars solar system diameter

universe our galaxy the sun

hill

3.28

cell atom diameter

Some rough benchmarks for distances in meters, times in seconds, and masses in kilograms. (Note that the scales here are logarithmic: equal distances on the scale correspond to multiplication by equal powers of ten.)

Being aware of units is also valuable even when working with symbolic equations. One cannot add or subtract quantities with different units (though multiplying and dividing such quantities are fine). So an expression that (for example) contains 1 + m (where m is a mass) is absurd. If you find yourself writing such an equation, you should know that you have made a mistake.

Also, for the record, all mathematical functions [such as sin(x), tan–1(x), ex, ln x] all require a unitless argument and produce a unitless result. For historical reasons, angles in both degrees and radians are considered unitless.

One way to spot silly results is to know some SI benchmarks (see figure C1.2). For example, if a question asks you how high you can throw a ball and you get a distance larger than a galaxy, maybe something is wrong, hmm? Check figure C1.2 if a calculation’s results seem off. It is also cool to be able to say to your friends at the end of the school year “See you in about 9 megaseconds” and have them understand. (OK, maybe that is just a bit geeky.)

C1.5 Trick Bag: Unit Conversions

In many realistic physics problems, you are likely to have to change units. One of the most handy tools in your bag of tricks is a foolproof way to convert units. My favorite technique is the unit operator method. For example, suppose that you know that your hair grows at a rate of 6 inches per year and you’d like to know what this is in nanometers per second. We start by writing down the equations that define the relationships between the units (which you can find inside the front cover):

1

Being aware of units in symbolic expressions is also good

Know some benchmarks!

The “unit operator” method for unit conversion

Why this method is so great

Since 1 m is equivalent to 3.28 ft, the ratio of these quantities is 1:

( 1 m 3.28 ft ) = 1 and similarly, 1 = ( 1 nm 10-9 m ) = ( 1 ft 12 in ) = ( 1 y 3.17 × 107 s ) (C1.3)

Such a ratio is thus called a unit operator. Since we can multiply anything by 1 without changing it, we can multiply our original rate by 1 in the form of these unit operators. If you then rearrange factors and cancel the units that appear in both the numerator and denominator, we get the desired result:

( 6 in y ) ·1·1·1·1 = ( 6 in │ y ) ( 1 ⨉ ft 12 in ) ( 1 ⧹ m 3.28 ⨉ ft ) ( 1 nm 10-9 ⧹ m ) ( 1 │ y 3.16 × 107 s ) = 6 12(3.28)(10-9)(3.16 × 107) nm s = 4.8 nm s (C1.4)

Note that we can treat all of the units as if they were simply algebraic symbols that we can cancel if they appear in both the numerator and denominator! This unit operator method is great because if I didn’t have the right power for one of the unit operators or put it upside down by accident, the units would not cancel and I would have a mess of leftover units that would signal that I had done something wrong. This method is absolutely foolproof as long as you make sure that unwanted units really do cancel out. (Foolproof is good!)

How do I know when a unit operator is “right side up?” [After all, 1 = (1 m/3.28 ft) and 1 = (3.28 ft/1 m) both!] “Right side up” for a unit operator is whichever way gets the units to cancel as you need. It’s like the joke where a student asks a sculptor how to make a good statue of a person. The sculptor replies, “Just remove whatever parts of the rock don’t look like the person.” In the unit conversion task, just arrange your unit operators so that they remove any units that don’t look like the final units you need.

C1.6 Trick Bag: Dimensional Analysis

Dimensional analysis is a very cool trick because you can often use it to estimate an answer or guess a formula even if you know almost nothing about the physics involved. This will amaze your friends (convincing them that you know more than you do) and is a sign to other physicists that you belong.

The trick is based on the fact that the units on both sides of any equation must be consistent. We can often combine this with simple plausibility arguments to find a formula for a desired quantity even if we haven’t a clue about how to actually derive that formula. This often represents the simplest model one can construct (one based on only the most fundamental assumptions).

Consider the following problem as an example. The radius R of a black hole is the radius inside which all light is trapped by the black hole’s gravitational field. What is this radius for a black hole with a given mass M?

(1) Think about what the desired quantity might depend on

(2) Assume a power-law formula

Don’t panic just because we don’t know any general relativity! Think: what could R depend on? It could plausibly depend on the black hole’s mass  M, the speed of light c, and the universal gravitational constant G that characterizes the strength of the gravitational field created by a given mass (we will study this constant more later). No other physical quantity appears to be relevant. So let’s assume that only these quantities will appear in the formula for R. This is the first assumption in our model.

Secondly, let’s assume that the formula has the form

R = KG j Mk cn

(C1.5)

where K is some as-yet-unknown unitless constant and j, k, and n are as-yetunknown powers (not necessarily integers). This is a big assumption, but not completely bonkers. For example, as the black hole’s mass M goes to zero, we would expect the trapping radius R goes to zero. Similarly, if G W 0 (meaning that the gravitational field created by a given mass goes to zero), R should also go to zero. Both expectations are consistent with the formula.

Let {a} represent the SI units of the quantity a. Thus, {R} = m, {M}  = kg, {c} = m/s (since c is a speed), and {G} = m3/(kg·s2) (see the inside front cover). So if the units in equation C1.5 are to be consistent, we must have

m = {R} = {K}{G}j {M}k {c}n = ( m3 kg·s2 ) j (kg)k ( m s )n = m3j+nkgk-j s -2j-n (C1.6)

Since we do not have any units of kilograms on the left side of equation C1.6, we must have k – j = 0, or k = j. Since we also do not have units of seconds on the left, we must have –n – 2j = 0, or n = –2j. We have one power of meters on the left side, so 3j + n = 1 or (substituting n = –2j from above) 3j – 2j = 1, so j = 1. So our formula must be

R = KG jM kc n = KG1M1c -2 = K GM c2 (C1.7)

The constant K we have included (for greater generality) does not have any units, so this method cannot determine its value. If we assume that K = 1, then the radius of a black hole with a mass equal to that of the sun is

R = GM c2 = ( 6.67 × 10-11 m@ 3 kg · ⨉ s 2 ) 1.99 × 1030 kg (3.00 × 108 @ m/⨉ s)2 = 1470 m (C1.8)

Therefore, this super-simple model predicts that if our sun were to become a black hole, its radius would be about 1.5 km.

Now you might think this result completely untrustworthy because of all the assumptions we have made. Even granting the plausibility of the argument that the result must depend only on G, M, and c, what justifies the outrageous assumptions we made in equation C1.5 and in setting K = 1?

Such assumptions turn out to be surprisingly trustworthy. The universe seems to prefer simple formulas to complicated ones, and unitless constants appearing in physics formulas usually turn out to be like 1/3, π, or 5/8, not something very large or very small. Therefore taking the unitless constant to be equal to 1 has an uncannily good chance of yielding a reasonable estimate (within a factor of 10 or so). It doesn’t always work, but it often does, and when it does, one can get a good first estimate with spectacularly little effort.

In this particular case, the full derivation using general relativity shows that a black hole’s radius is really R = 2GM/c2. Thus, we got everything absolutely right except that K = 2, not 1. Not bad for such a simple model!

For historical reasons, this method is called dimensional analysis (long ago, but not today, a quantity’s units were called its “dimensions”).

Now you practice. (After making a good effort, check the chapter’s end.)

Exercise C1X.2

When the core of a massive star exceeds about 1.4 solar masses near the end of its life, reactions in its interior suddenly remove the very particles that have been supporting it against its own gravitational field. What is left of the core then falls basically freely inward from rest at a radius of about 10,000 km to a final radius that is negligible in comparison. (This violent collapse usually ignites a supernova explosion). How long does this collapse process take?

(3) Require unit consistency

(4) Assume that any unitless constant is approximately 1

TWO-MINUTE PROBLEMS

C1T.1 According to the definition of “science” given in this chapter, astrology is not a science. What does it lack?

A. A community of scholars devoted to its study

B. Agreement that models must be logically consistent

C. Use of reproducible experiments to test models

D. A grand theory embracing the discipline

C1T.2 According to the definition of “science” given in this chapter, which of the following do you think are sciences?

Choose the letter of the first discipline on the list that you think is not a science. (The answer is open to debate!)

A. Geology

B. Psychology

C. Economics

D. Anthropology

E. Political Science

F. Philosophy

T. All are sciences

C1T.3 Which of the following expressions gives the correct units for the volt in terms of base SI units?

A. 1 V = 1 kg·m2C–1s

B. 1 V = 1 kg·m2A–1s–3

C. 1 V = 1 kg·m·A–1s–1

D. 1 V = 1 kg·m2s–2C–2

E. 1 V = 1 J/C

F. Some other expression (specify).

C1T.4 Assume that D and R have units of meters, T has units of seconds, m and M have units of kilograms, v has units of meters per second, and g has units of m/s2. Which of the following equations has self-consistent units?

A. D = mR2

B. m = M[1 + R2]

C. D = [1 – m/M]gT2

D. g = mv2/R

E. D = v2/RT

F. None of these can be correct.

HOMEWORK PROBLEMS

Basic Skills

C1B.1 If you text to a friend “I’ll be over in 0.50 ks,” how many minutes will your friend wait? (Use unit operators.)

C1B.2 What is the speed of light in furlongs per fortnight? (One furlong = 1 8 mi = length of a medieval farm furrow, and one fortnight = 14 days. Use unit operators.)

C1B.3 A light year (1 ly) is the distance that light travels in one year. Find this distance in miles. (Use unit operators.)

C1T.5 One can raise a quantity q to a power a when a has units. T or F?

C1T.6 The following formulas are supposed to describe the speed v of a sphere sinking in a thick fluid. C is a unitless constant, ρ is the fluid’s density in kg/m3, A is the sphere’s cross-sectional area, m is its mass, and g is the gravitational field strength in N/kg. Which could be right?

A. v = CAρg

B. v = Cmg/ρA

C. v = (Cmg/ρA)2

D. v = (Cmg/ρA)1/2

E. None of these can be correct.

C1T.7 The speed v of sound waves in a gas like air might plausibly depend on the gas’s pressure P (which has units of N/m2), the gas’s density ρ (which has units of kg/m3) and its temperature T (which has units of K), and some unitless constant C. Assuming that no other quantities are relevant, which of the following formulas might possibly correctly give the speed of sound in a gas?

A. v = CPρT

B. v = CTP/ρ

C. v = CP/ρ

D. v = CP/ρ

E. v = Cρ/P

F. v = C(P/ρ)2

T. None of these can be correct.

C1T.8 The two stars in a binary star system revolve around each other with a certain period T. Which of the quantities listed below is not likely to be a part of the formula for this revolution period?

A. m1, m2 (the masses of the stars in the system)

B. r (the distance between the binary stars)

C. ℏ (Planck’s constant, which is generally associated with phenomena involving quantum mechanics)

D. G (the universal gravitational constant)

C1B.4 The speed of the earth in its orbit around the sun is 18 km/s. Find this speed (a) in miles per hour and (b) in knots, where 1 knot = 1 nautical mile per hour, and 1 nautical mile = 1852 m. (Use unit operators.)

C1B.5 What is a month in seconds (approximately)?

C1B.6 Water’s density is 1000 kg/m3. Use unit operators to show that a cube of water 10 cm on a side has a mass of 1 kg, and 1 cm3 of water has a mass of 1 g. (This used to be the definition of the kilogram, but the difficulty of precisely measuring volumes made this standard impractical.)

C1B.7 A friend says that the range D of a projectile fired at a speed v at an angle θ above the horizontal is D = v sin 2θ/g, where g has units of m/s2 (a) Explain why this can’t be right. (b) Propose a modification so that D still depends on v and g but could be right.

C1B.8 Do problem C1T.6 and explain your answer.

C1B.9 Do problem C1T.7 and explain your answer.

Modeling

C1M.1 Can “What is justice?” be investigated scientifically? If you think not, does this mean that “justice” does not exist or is not worth thinking about? If you think so, can you scientifically verify your belief? Defend your responses.

C1M.2 The text describes that historically, physics became a “science” when the physics community accepted Newton’s mechanics as its “grand theory.” What do you think is or was the corresponding grand theory that made biology a science? Chemistry? Geology? Defend your responses.

C1M.3 As a fraction of the stick’s length, what is the length of a vertical stick’s shadow in Saint Petersburg, Russia, at noon on the summer solstice? (Saint Petersburg is 3230 km almost due north from Alexandria.)

C1M.4 Consider an object of mass m moving in a circle of radius r with a constant speed v. What is a possible formula for the object’s acceleration a (in m/s2)?

C1M.5 A planet’s “escape speed” is the speed ve which an object must have at the planet’s surface to be able to escape the planet’s gravitational embrace and coast to an infinite distance. This speed depends on the planet’s mass M, its radius R, and the universal gravitational constant G. Up to an overall constant, what must the formula for ve be?

C1M.6 Imagine slicing a thick disk of radius R in half along its diameter. If you stand the half-disk on its curved edge and nudge it, it will rock back and forth. If the rocking is not too extreme, the time T required for a complete back-andforth oscillation turns out to be nearly independent of the angle through which the disk rocks. The only other things that T might plausibly depend on are the disk’s radius R, its mass M, and the local gravitational field strength g (in m/s2), since gravity is what is causing the rocking motion. (If you think about it, the disk’s thickness is only relevant in that a thicker disk has more mass than a thinner one, so we already have this covered if we consider dependence on M.) Use dimensional analysis to find a reasonable formula for this rocking time up to a unitless constant.

C1M.7 Consider various pendula, each consisting of a rod hanging in the earth’s gravitational field from a pivot at its end and that is free to swing around that pivot. The rods have identical shapes, but different lengths L (and diameters proportional to those lengths). The rods also might

have different densities and thus masses m that don’t scale simply with L. Experimentally, the period T of such a pendulum does not depend on the angle of swing for small oscillations. Find an approximate formula for the period T, and estimate T for a 1-m rod. (Hints: You should not assume, but rather show by dimensional analysis that the rod’s mass is irrelevant. The gravitational field strength g has units of m/s2.)

C1M.8 The radius r of a hydrogen atom (which consists of a proton and a comparatively lightweight electron) can only depend on the (equal) magnitudes e of the proton’s and the electron’s charge, the Coulomb constant 1/4πε0 that characterizes the strength of the electrostatic attraction between the proton and electron, the mass m of the orbiting electron, and (because quantum mechanics is likely involved) Planck’s constant in the form h/2π = ℏ. (Since the proton remains essentially at rest, its mass turns out to be irrelevant.)

(a) Assuming that r depends only on the stated quantities, use dimensional analysis and the information given inside the front cover of the text to find a plausible formula for the radius of a hydrogen atom.

(b) Calculate a numerical estimate of the radius.

(c) Explain why this is only an estimate.

C1M.9 Consider an object of mass m hanging from the end of a spring whose other end is attached to a fixed point. The object will oscillate vertically with some period T, which might depend on the object’s mass, the spring’s stiffness constant ks (in N/m), which expresses how much force the spring exerts when it is stretched a certain distance, and the gravitational field strength g (in m/s2) at the earth’s surface (since the object is moving up and down in the earth’s gravitational field). We find experimentally that the period does not depend on either the object’s maximum speed or how far it moves up and down.

(a) Up to a unitless constant, what is the formula for the period in terms of ks, m, and g?

(b) On the basis of your calculation, if the object oscillates with a period of 1.0 s on the earth’s surface, what is its period on the moon’s surface?

C1M.10 The critical density of the universe ρc is the density that the universe must have for the gravitational attraction between its parts to be strong enough to prevent it from expanding forever. This density must depend on the Hubble constant H, which specifies the universe’s current expansion rate (as a fractional expansion per unit time, so its units are 1/s). It might also depend on the speed of light c, which in combination with H tells us the radius of the universe we currently can receive light from, and thus how much of the universe might contribute to the attraction.

(a) What is a third quantity that plausibly appears in this formula and why?

(b) Up to a unitless constant, what is the formula for the universe’s critical density ρc?

(c) Ignoring the unitless constant, what is the critical density for our universe? (H = 2.28 × 10–18 s –1)