NewtonianPhysics

Benjamin Crowell

Book 1 in the Light and Matter series of free introductory physics textbookswww.lightandmatter.com

Useful Data

Metric PrefixesM- mega- 106

k- kilo- 103

m- milli- 10 –3

µ- (Greek mu) micro- 10 –6

n- nano- 10 –9

(Centi-, 10 –2, is used only in the centimeter.)

Notation and Unitsquantity unit symboldistance meter, m x, ∆xtime second, s t, ∆tmass kilogram, kg marea m2 (square meters) Avolume m3 (cubic meters) Vdensity kg/m3 ρforce Newton, 1 N=1 kg.m/s2 Fvelocity m/s vacceleration m/s2 a

symbol meaning∝ is proportional to≈ is approximately equal to~ on the order of

The Greek Alphabetα Α alpha ν Ν nuβ Β beta ξ Ξ xiγ Γ gamma ο Ο omicronδ ∆ delta π Π piε Ε epsilon ρ Ρ rhoζ Ζ zeta σ Σ sigmaη Η eta τ Τ tauθ Θ theta υ Υ upsilonι Ι iota φ Φ phiκ Κ kappa χ Χ chiλ Λ lambda ψ Ψ psiµ Μ mu ω Ω omega

ConversionsConversions between SI and other units:

1 inch = 2.54 cm (exactly)1 mile = 1.61 km1 pound = 4.45 N(1 kg)(g) = 2.2 lb1 gallon = 3.78x103 cm3

Conversions between U.S. units:1 foot = 12 inches1 yard = 3 feet1 mile = 5280 ft

Earth, Moon, and Sunbody mass (kg) radius (km)radius of orbit (km)earth 5.97x1024 6.4x103 1.49x108

moon 7.35x1022 1.7x103 3.84x105

sun 1.99x1030 7.0x105

The radii and radii of orbits are average values. Themoon orbits the earth and the earth orbits the sun.

Subatomic Particlesparticle mass (kg) radius (m)electron 9.109x10-31 ? – less than about 10-17

proton 1.673x10-27 about 1.1x10-15

neutron 1.675x10-27 about 1.1x10-15

The radii of protons and neutrons can only be givenapproximately, since they have fuzzy surfaces. Forcomparison, a typical atom is about 10-9 m in radius.

Fundamental Constantsspeed of light c=3.00x108 m/sgravitational constant G=6.67x10-11 N.m2.kg-2

Newtonian Physics

The Light and Matter series ofintroductory physics textbooks:

1 Newtonian Physics

2 Conservation Laws

3 Vibrations and Waves

4 Electricity and Magnetism

5 Optics

6 The Modern Revolution in Physics

Newtonian Physics

Benjamin Crowell

www.lightandmatter.com

© 1998 by Benjamin CrowellAll rights reserved.

To Paul Herrschaft and Rich Muller.

Brief Contents0 Introduction and Review ...............................15

1 Scaling and Order-of-Magnitude Estimates 35

Motion in One Dimension .............. 512 Velocity and Relative Motion ........................52

3 Acceleration and Free Fall ............................71

4 Force and Motion...........................................95

5 Analysis of Forces....................................... 111

Motion in Three Dimensions ....... 1316 Newton’s Laws in Three Dimensions ........131

7 Vectors..........................................................141

8 Vectors and Motion .....................................151

9 Circular Motion ............................................163

10 Gravity ........................................................177

Exercises ...........................................................195Solutions to Selected Problems ......................203Glossary .............................................................207Mathematical Review ........................................209Trig Tables..........................................................210Index ................................................................... 211

Contents

Motion in OneDimension 512 Velocity and Relative

Motion 522.1 Types of Motion ................................... 522.2 Describing Distance and Time ............. 552.3 Graphs of Motion; Velocity. .................. 582.4 The Principle of Inertia ......................... 622.5 Addition of Velocities ........................... 652.6 Graphs of Velocity Versus Time........... 672.7 ∫ Applications of Calculus .................... 67Summary ...................................................... 69Homework Problems .................................... 70

3 Acceleration and FreeFall 71

3.1 The Motion of Falling Objects .............. 713.2 Acceleration ......................................... 743.3 Positive and Negative Acceleration ..... 773.4 Varying Acceleration ............................ 803.5 The Area Under

the Velocity-Time Graph ................ 833.6 Algebraic Results for Constant

Acceleration .................................... 853.7* Biological Effects of Weightlessness .. 873.8 ∫ Applications of Calculus .................... 89Summary ...................................................... 90Homework Problems .................................... 91

1 Scaling and Order-of-Magnitude Estimates35

1.1 Introduction .......................................... 351.2 Scaling of Area and Volume ................ 371.3 Scaling Applied to Biology ................... 441.4 Order-of-Magnitude Estimates ............ 47Summary ...................................................... 50Homework Problems .................................... 50

Preface ......................................................... 13A Note to the Student Taking Calculus

Concurrently ........................................... 14

0 Introduction and Review15

0.1 The Scientific Method .......................... 150.2 What Is Physics? ................................. 170.3 How to Learn Physics .......................... 200.4 Self-Evaluation .................................... 220.5 Basics of the Metric System ................ 220.6 The Newton, the Metric Unit of Force .. 250.7 Less Common Metric Prefixes ............. 260.8 Scientific Notation ................................ 270.9 Conversions......................................... 280.10 Significant Figures ............................. 30Summary ...................................................... 32Homework Problems .................................... 33

4 Force and Motion 954.1 Force ................................................... 954.2 Newton’s First Law .............................. 984.3 Newton’s Second Law ....................... 1024.4 What Force Is Not .............................. 1044.5 Inertial and Noninertial

Frames of Reference ................... 106Summary .................................................... 108Homework Problems .................................. 109

5 Analysis of Forces 1115.1 Newton’s Third Law ............................ 1115.2 Classification and Behavior of Forces 1165.3 Analysis of Forces ............................. 1225.4 Transmission of Forces

by Low-Mass Objects .................. 1245.5 Objects Under Strain ......................... 1265.6 Simple Machines: The Pulley ............ 127Summary .................................................... 128Homework Problems .................................. 129 7 Vectors 141

7.1 Vector Notation .................................. 1417.2 Calculations with Magnitude

and Direction................................ 1447.3 Techniques for Adding Vectors .......... 1477.4* Unit Vector Notation ......................... 1487.5* Rotational Invariance ....................... 148Summary .................................................... 149Homework Problems .................................. 150

8 Vectors and Motion 1518.1 The Velocity Vector ............................ 1528.2 The Acceleration Vector ..................... 1538.3 The Force Vector

and Simple Machines .................. 1568.4 ∫ Calculus With Vectors ...................... 157Summary .................................................... 159Homework Problems .................................. 160Motion in Three

Dimensions 1316 Newton’s Laws in Three

Dimensions 1316.1 Forces Have No Perpendicular Effects1316.2 Coordinates and Components ........... 1346.3 Newton’s Laws in Three Dimensions 136Summary .................................................... 138Homework Problems .................................. 139

9 Circular Motion 1639.1 Conceptual Framework

for Circular Motion ....................... 1639.2 Uniform Circular Motion ..................... 1689.3 Nonuniform Circular Motion ............... 171Summary .................................................... 172Homework Problems .................................. 173

10 Gravity 17710.1 Kepler’s Laws .................................. 17810.2 Newton’s Law of Gravity .................. 17910.3 Apparent Weightlessness ................ 18310.4 Vector Addition

of Gravitational Forces................. 18410.5 Weighing the Earth .......................... 186Summary .................................................... 190Homework Problems .................................. 191

Exercises 195

Solutions to SelectedProblems 203

Glossary 207

Mathematical Review 209

Trig Tables 210

Index 211

13

PrefaceWhy a New Physics Textbook?

We assume that our economic system will always scamper to provide us with the products we want. Specialorders don’t upset us! I want my MTV! The truth is more complicated, especially in our education system, whichis paid for by the students but controlled by the professoriate. Witness the perverse success of the bloated sciencetextbook. The newspapers continue to compare our system unfavorably to Japanese and European education,where depth is emphasized over breadth, but we can’t seem to create a physics textbook that covers a manageablenumber of topics for a one-year course and gives honest explanations of everything it touches on.

The publishers try to please everybody by including every imaginable topic in the book, but end up pleasingnobody. There is wide agreement among physics teachers that the traditional one-year introductory textbookscannot in fact be taught in one year. One cannot surgically remove enough material and still gracefully navigatethe rest of one of these kitchen-sink textbooks. What is far worse is that the books are so crammed with topics thatnearly all the explanation is cut out in order to keep the page count below 1100. Vital concepts like energy areintroduced abruptly with an equation, like a first-date kiss that comes before “hello.”

The movement to reform physics texts is steaming ahead, but despite excellent books such as Hewitt’s Concep-tual Physics for non-science majors and Knight’s Physics: A Contemporary Perspective for students who knowcalculus, there has been a gap in physics books for life-science majors who haven't learned calculus or are learningit concurrently with physics. This book is meant to fill that gap.

Learning to Hate Physics?When you read a mystery novel, you know in advance what structure to expect: a crime, some detective work,

and finally the unmasking of the evildoer. When Charlie Parker plays a blues, your ear expects to hear certainlandmarks of the form regardless of how wild some of his notes are. Surveys of physics students usually show thatthey have worse attitudes about the subject after instruction than before, and their comments often boil down to acomplaint that the person who strung the topics together had not learned what Agatha Christie and Charlie Parkerknew intuitively about form and structure: students become bored and demoralized because the “march throughthe topics” lacks a coherent story line. You are reading the first volume of the Light and Matter series of introduc-tory physics textbooks, and as implied by its title, the story line of the series is built around light and matter: howthey behave, how they are different from each other, and, at the end of the story, how they turn out to be similarin some very bizarre ways. Here is a guide to the structure of the one-year course presented in this series:

1 Newtonian Physics Matter moves at constant speed in a straight line unless a force acts on it. (This seemsintuitively wrong only because we tend to forget the role of friction forces.) Objects made of matter can exertforces on each other, causing changes in their motion. A more massive object changes its motion more slowly inresponse to a given force.

2 Conservation Laws Newton’s matter-and-forces picture of the universe is fine as far as it goes, but it doesn’tapply to light, which is a form of pure energy without mass. A more powerful world-view, applying equally wellto both light and matter, is provided by the conservation laws, for instance the law of conservation of energy,which states that energy can never be destroyed or created but only changed from one form into another.

3 Vibrations and Waves Light is a wave. We learn how waves travel through space, pass through each other,speed up, slow down, and are reflected.

4 Electricity and Magnetism Matter is made out of particles such as electrons and protons, which are heldtogether by electrical forces. Light is a wave that is made out of patterns of electric and magnetic force.

5 Optics Devices such as eyeglasses and searchlights use matter (lenses and mirrors) to manipulate light.

6 The Modern Revolution in Physics Until the twentieth century, physicists thought that matter was madeout of particles and light was purely a wave phenomenon. We now know that both light and matter are made ofbuilding blocks that have both particle and wave properties. In the process of understanding this apparentcontradiction, we find that the universe is a much stranger place than Newton had ever imagined, and also learnthe basis for such devices as lasers and computer chips.

14

A Note to the Student Taking CalculusConcurrently

Learning calculus and physics concurrently is an excellent idea — it’s not a coincidence that the inventor ofcalculus, Isaac Newton, also discovered the laws of motion! If you are worried about taking these two demandingcourses at the same time, let me reassure you. I think you will find that physics helps you with calculus whilecalculus deepens and enhances your experience of physics. This book is designed to be used in either an algebra-based physics course or a calculus-based physics course that has calculus as a corequisite. This note is addressed tostudents in the latter type of course.

It has been said that critics discuss art with each other, but artists talk about brushes. Art needs both a “why”and a “how,” concepts as well as technique. Just as it is easier to enjoy an oil painting than to produce one, it iseasier to understand the concepts of calculus than to learn the techniques of calculus. This book will generallyteach you the concepts of calculus a few weeks before you learn them in your math class, but it does not discuss thetechniques of calculus at all. There will thus be a delay of a few weeks between the time when a calculus applicationis first pointed out in this book and the first occurrence of a homework problem that requires the relevant tech-nique. The following outline shows a typical first-semester calculus curriculum side-by-side with the list of topicscovered in this book, to give you a rough idea of what calculus your physics instructor might expect you to knowat a given point in the semester. The sequence of the calculus topics is the one followed by Calculus of a SingleVariable, 2nd ed., by Swokowski, Olinick, and Pence.

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15

0 Introduction and ReviewIf you drop your shoe and your keys side by side, they hit the ground at

the same time. Why doesn’t the shoe get there first, since gravity is pullingharder on it? How does the lens of your eye work, and why do your eye’smuscles need to squash its lens into different shapes in order to focus onobjects nearby or far away? These are the kinds of questions that physicstries to answer about the behavior of light and matter, the two things thatthe universe is made of.

0.1 The Scientific MethodUntil very recently in history, no progress was made in answering

questions like these. Worse than that, the wrong answers written by thinkerslike the ancient Greek physicist Aristotle were accepted without question forthousands of years. Why is it that scientific knowledge has progressed moresince the Renaissance than it had in all the preceding millennia since thebeginning of recorded history? Undoubtedly the industrial revolution is partof the answer. Building its centerpiece, the steam engine, required improvedtechniques for precise construction and measurement. (Early on, it wasconsidered a major advance when English machine shops learned to buildpistons and cylinders that fit together with a gap narrower than the thick-ness of a penny.) But even before the industrial revolution, the pace ofdiscovery had picked up, mainly because of the introduction of the modernscientific method. Although it evolved over time, most scientists todaywould agree on something like the following list of the basic principles ofthe scientific method:

(1)Science is a cycle of theory and experiment. Scientific theories arecreated to explain the results of experiments that were created under certainconditions. A successful theory will also make new predictions about newexperiments under new conditions. Eventually, though, it always seems tohappen that a new experiment comes along, showing that under certain

theory

experiment

The Mars Climate Orbiter is prepared for its mission.The laws of physics are the same everywhere, evenon Mars, so the probe could be designed based onthe laws of physics as discovered on earth.

There is unfortunately another reason why thisspacecraft is relevant to the topics of this chapter: itwas destroyed attempting to enter Mars’ atmospherebecause engineers at Lockheed Martin forgot toconvert data on engine thrusts from pounds into themetric unit of force (newtons) before giving theinformation to NASA. Conversions are important!

16

conditions the theory is not a good approximation or is not valid at all. Theball is then back in the theorists’ court. If an experiment disagrees with thecurrent theory, the theory has to be changed, not the experiment.

(2)Theories should both predict and explain. The requirement of predic-tive power means that a theory is only meaningful if it predicts somethingthat can be checked against experimental measurements that the theoristdid not already have at hand. That is, a theory should be testable. Explana-tory value means that many phenomena should be accounted for with fewbasic principles. If you answer every “why” question with “because that’s theway it is,” then your theory has no explanatory value. Collecting lots of datawithout being able to find any basic underlying principles is not science.

(3)Experiments should be reproducible. An experiment should be treatedwith suspicion if it only works for one person, or only in one part of theworld. Anyone with the necessary skills and equipment should be able toget the same results from the same experiment. This implies that sciencetranscends national and ethnic boundaries; you can be sure that nobody isdoing actual science who claims that their work is “Aryan, not Jewish,”“Marxist, not bourgeois,” or “Christian, not atheistic.” An experimentcannot be reproduced if it is secret, so science is necessarily a public enter-prise.

As an example of the cycle of theory and experiment, a vital step towardmodern chemistry was the experimental observation that the chemicalelements could not be transformed into each other, e.g. lead could not beturned into gold. This led to the theory that chemical reactions consisted ofrearrangements of the elements in different combinations, without anychange in the identities of the elements themselves. The theory worked forhundreds of years, and was confirmed experimentally over a wide range ofpressures and temperatures and with many combinations of elements. Onlyin the twentieth century did we learn that one element could be trans-formed into one another under the conditions of extremely high pressureand temperature existing in a nuclear bomb or inside a star. That observa-tion didn’t completely invalidate the original theory of the immutability ofthe ele-ments, but it showed that it was only an approximation, valid atordinary temperatures and pressures.

Self-Check

A psychic conducts seances in which the spirits of the dead speak to theparticipants. He says he has special psychic powers not possessed by otherpeople, which allow him to “channel” the communications with the spirits.What part of the scientific method is being violated here? [Answer below.]

The scientific method as described here is an idealization, and shouldnot be understood as a set procedure for doing science. Scientists have asmany weaknesses and character flaws as any other group, and it is verycommon for scientists to try to discredit other people’s experiments whenthe results run contrary to their own favored point of view. Successfulscience also has more to do with luck, intuition, and creativity than most

A satirical drawing of an alchemist’slaboratory. H. co*ck, after a drawingby Peter Brueghel the Elder (16thcentury).

If only he has the special powers, then his results can never be reproduced.

Chapter 0 Introduction and Review

17

people realize, and the restrictions of the scientific method do not stifleindividuality and self-expression any more than the fugue and sonata formsstifled Bach and Haydn. There is a recent tendency among social scientiststo go even further and to deny that the scientific method even exists,claiming that science is no more than an arbitrary social system thatdetermines what ideas to accept based on an in-group’s criteria. I thinkthat’s going too far. If science is an arbitrary social ritual, it would seemdifficult to explain its effectiveness in building such useful items as air-planes, CD players and sewers. If alchemy and astrology were no lessscientific in their methods than chemistry and astronomy, what was it thatkept them from producing anything useful?

Discussion QuestionsConsider whether or not the scientific method is being applied in the followingexamples. If the scientific method is not being applied, are the people whoseactions are being described performing a useful human activity, albeit anunscientific one?

A. Acupuncture is a traditional medical technique of Asian origin in which smallneedles are inserted in the patient’s body to relieve pain. Many doctors trainedin the west consider acupuncture unworthy of experimental study because if ithad therapeutic effects, such effects could not be explained by their theories ofthe nervous system. Who is being more scientific, the western or easternpractitioners?

B. Goethe, a famous German poet, is less well known for his theory of color.He published a book on the subject, in which he argued that scientificapparatus for measuring and quantifying color, such as prisms, lenses andcolored filters, could not give us full insight into the ultimate meaning of color,for instance the cold feeling evoked by blue and green or the heroic sentimentsinspired by red. Was his work scientific?

C. A child asks why things fall down, and an adult answers “because ofgravity.” The ancient Greek philosopher Aristotle explained that rocks fellbecause it was their nature to seek out their natural place, in contact with theearth. Are these explanations scientific?

D. Buddhism is partly a psychological explanation of human suffering, andpsychology is of course a science. The Buddha could be said to haveengaged in a cycle of theory and experiment, since he worked by trial anderror, and even late in his life he asked his followers to challenge his ideas.Buddhism could also be considered reproducible, since the Buddha told hisfollowers they could find enlightenment for themselves if they followed acertain course of study and discipline. Is Buddhism a scientific pursuit?

0.2 What Is Physics?Given for one instant an intelligence which could comprehend all the forcesby which nature is animated and the respective positions of the things whichcompose it...nothing would be uncertain, and the future as the past wouldbe laid out before its eyes.

Pierre Simon de LaplacePhysics is the use of the scientific method to find out the basic prin-

ciples governing light and matter, and to discover the implications of thoselaws. Part of what distinguishes the modern outlook from the ancient mind-set is the assumption that there are rules by which the universe functions,and that those laws can be at least partially understood by humans. Fromthe Age of Reason through the nineteenth century, many scientists began tobe convinced that the laws of nature not only could be known but, asclaimed by Laplace, those laws could in principle be used to predict every-

Physics is the studyof light and matter.

Science is creative.

Section 0.2 What Is Physics?

18

thing about the universe’s future if complete information was availableabout the present state of all light and matter. In subsequent sections, I’lldescribe two general types of limitations on prediction using the laws ofphysics, which were only recognized in the twentieth century.

Matter can be defined as anything that is affected by gravity, i.e. thathas weight or would have weight if it was near the Earth or another star orplanet massive enough to produce measurable gravity. Light can be definedas anything that can travel from one place to another through empty spaceand can influence matter, but has no weight. For example, sunlight caninfluence your body by heating it or by damaging your DNA and givingyou skin cancer. The physicist’s definition of light includes a variety ofphenomena that are not visible to the eye, including radio waves, micro-waves, x-rays, and gamma rays. These are the “colors” of light that do nothappen to fall within the narrow violet-to-red range of the rainbow that wecan see.

Self-checkAt the turn of the 20th century, a strange new phenomenon was discovered invacuum tubes: mysterious rays of unknown origin and nature. These rays arethe same as the ones that shoot from the back of your TV’s picture tube and hitthe front to make the picture. Physicists in 1895 didn’t have the faintest ideawhat the rays were, so they simply named them “cathode rays,” after the namefor the electrical contact from which they sprang. A fierce debate raged,complete with nationalistic overtones, over whether the rays were a form oflight or of matter. What would they have had to do in order to settle the issue?

Many physical phenomena are not themselves light or matter, but areproperties of light or matter or interactions between light and matter. Forinstance, motion is a property of all light and some matter, but it is notit*elf light or matter. The pressure that keeps a bicycle tire blown up is aninteraction between the air and the tire. Pressure is not a form of matter inand of itself. It is as much a property of the tire as of the air. Analogously,sisterhood and employment are relationships among people but are notpeople themselves.

Some things that appear weightless actually do have weight, and soqualify as matter. Air has weight, and is thus a form of matter even though acubic inch of air weighs less than a grain of sand. A helium balloon hasweight, but is kept from falling by the force of the surrounding more denseair, which pushes up on it. Astronauts in orbit around the Earth haveweight, and are falling along a curved arc, but they are moving so fast thatthe curved arc of their fall is broad enough to carry them all the way aroundthe Earth in a circle. They perceive themselves as being weightless becausetheir space capsule is falling along with them, and the floor therefore doesnot push up on their feet.

Optional TopicEinstein predicted as a consequence of his theory of relativity thatlight would after all be affected by gravity, although the effect wouldbe extremely weak under normal conditions. His prediction wasborne out by observations of the bending of light rays from stars asthey passed close to the sun on their way to the Earth. Einstein also

They would have had to weigh the rays, or check for a loss of weight in the object from which they were haveemitted. (For technical reasons, this was not a measurement they could actually do, hence the opportunity fordisagreement.)

This telescope picture shows twoimages of the same distant object, anexotic, very luminous object called aquasar. This is interpreted as evidencethat a massive, dark object, possiblya black hole, happens to be betweenus and it. Light rays that wouldotherwise have missed the earth oneither side have been bent by the darkobject’s gravity so that they reach us.The actual direction to the quasar ispresumably in the center of the image,but the light along that central line don’tget to us because they are absorbedby the dark object. The quasar isknown by its catalog number,MG1131+0456, or more informally asEinstein’s Ring.

Weight is whatdistinguishes

light from matter.

Chapter 0 Introduction and Review

19

predicted the existence of black holes, stars so massive andcompact that their intense gravity would not even allow light toescape. (These days there is strong evidence that black holesexist.)

Einstein’s interpretation was that light doesn’t really have mass, butthat energy is affected by gravity just like mass is. The energy in alight beam is equivalent to a certain amount of mass, given by thefamous equation E=mc2, where c is the speed of light. Because thespeed of light is such a big number, a large amount of energy isequivalent to only a very small amount of mass, so the gravitationalforce on a light ray can be ignored for most practical purposes.

There is however a more satisfactory and fundamental distinctionbetween light and matter, which should be understandable to you ifyou have had a chemistry course. In chemistry, one learns thatelectrons obey the Pauli exclusion principle, which forbids more thanone electron from occupying the same orbital if they have the samespin. The Pauli exclusion principle is obeyed by the subatomicparticles of which matter is composed, but disobeyed by theparticles, called photons, of which a beam of light is made.

Einstein’s theory of relativity is discussed more fully in book 6 of thisseries.

The boundary between physics and the other sciences is not alwaysclear. For instance, chemists study atoms and molecules, which are whatmatter is built from, and there are some scientists who would be equallywilling to call themselves physical chemists or chemical physicists. It mightseem that the distinction between physics and biology would be clearer,since physics seems to deal with inanimate objects. In fact, almost allphysicists would agree that the basic laws of physics that apply to moleculesin a test tube work equally well for the combination of molecules thatconstitutes a bacterium. (Some might believe that something more happensin the minds of humans, or even those of cats and dogs.) What differenti-ates physics from biology is that many of the scientific theories that describeliving things, while ultimately resulting from the fundamental laws ofphysics, cannot be rigorously derived from physical principles.

Isolated Systems and ReductionismTo avoid having to study everything at once, scientists isolate the things

they are trying to study. For instance, a physicist who wants to study themotion of a rotating gyroscope would probably prefer that it be isolatedfrom vibrations and air currents. Even in biology, where field work isindispensable for understanding how living things relate to their entireenvironment, it is interesting to note the vital historical role played byDarwin’s study of the Galápagos Islands, which were conveniently isolatedfrom the rest of the world. Any part of the universe that is considered apartfrom the rest can be called a “system.”

Physics has had some of its greatest successes by carrying this process ofisolation to extremes, subdividing the universe into smaller and smallerparts. Matter can be divided into atoms, and the behavior of individualatoms can be studied. Atoms can be split apart into their constituentneutrons, protons and electrons. Protons and neutrons appear to be madeout of even smaller particles called quarks, and there have even been someclaims of experimental evidence that quarks have smaller parts inside them.

virus

molecule

quarks

?

atom

neutronsand protons

Section 0.2 What Is Physics?

20

This method of splitting things into smaller and smaller parts and studyinghow those parts influence each other is called reductionism. The hope isthat the seemingly complex rules governing the larger units can be betterunderstood in terms of simpler rules governing the smaller units. Toappreciate what reductionism has done for science, it is only necessary toexamine a 19th-century chemistry textbook. At that time, the existence ofatoms was still doubted by some, electrons were not even suspected to exist,and almost nothing was understood of what basic rules governed the wayatoms interacted with each other in chemical reactions. Students had tomemorize long lists of chemicals and their reactions, and there was no wayto understand any of it systematically. Today, the student only needs toremember a small set of rules about how atoms interact, for instance thatatoms of one element cannot be converted into another via chemicalreactions, or that atoms from the right side of the periodic table tend toform strong bonds with atoms from the left side.

Discussion QuestionsA. I’ve suggested replacing the ordinary dictionary definition of light with amore technical, more precise one that involves weightlessness. It’s stillpossible, though, that the stuff a lightbulb makes, ordinarily called “light,” doeshave some small amount of weight. Suggest an experiment to attempt tomeasure whether it does.

B. Heat is weightless (i.e. an object becomes no heavier when heated), andcan travel across an empty room from the fireplace to your skin, where itinfluences you by heating you. Should heat therefore be considered a form oflight by our definition? Why or why not?

C. Similarly, should sound be considered a form of light?

0.3 How to Learn PhysicsFor as knowledges are now delivered, there is a kind of contract of errorbetween the deliverer and the receiver; for he that delivereth knowledgedesireth to deliver it in such a form as may be best believed, and not as maybe best examined; and he that receiveth knowledge desireth rather presentsatisfaction than expectant inquiry.

Sir Francis BaconMany students approach a science course with the idea that they can

succeed by memorizing the formulas, so that when a problem is assigned onthe homework or an exam, they will be able to plug numbers in to theformula and get a numerical result on their calculator. Wrong! That’s notwhat learning science is about! There is a big difference between memoriz-ing formulas and understanding concepts. To start with, different formulasmay apply in different situations. One equation might represent a defini-tion, which is always true. Another might be a very specific equation for thespeed of an object sliding down an inclined plane, which would not be trueif the object was a rock drifting down to the bottom of the ocean. If youdon’t work to understand physics on a conceptual level, you won’t knowwhich formulas can be used when.

Science is notabout plugginginto formulas.

Chapter 0 Introduction and Review

21

Most students taking college science courses for the first time also havevery little experience with interpreting the meaning of an equation. Con-sider the equation w=A/h relating the width of a rectangle to its height andarea. A student who has not developed skill at interpretation might viewthis as yet another equation to memorize and plug in to when needed. Aslightly more savvy student might realize that it is simply the familiarformula A=wh in a different form. When asked whether a rectangle wouldhave a greater or smaller width than another with the same area but asmaller height, the unsophisticated student might be at a loss, not havingany numbers to plug in on a calculator. The more experienced studentwould know how to reason about an equation involving division — if h issmaller, and A stays the same, then w must be bigger. Often, students fail torecognize a sequence of equations as a derivation leading to a final result, sothey think all the intermediate steps are equally important formulas thatthey should memorize.

When learning any subject at all, it is important to become as activelyinvolved as possible, rather than trying to read through all the informationquickly without thinking about it. It is a good idea to read and think aboutthe questions posed at the end of each section of these notes as you encoun-ter them, so that you know you have understood what you were reading.

Many students’ difficulties in physics boil down mainly to difficultieswith math. Suppose you feel confident that you have enough mathematicalpreparation to succeed in this course, but you are having trouble with a fewspecific things. In some areas, the brief review given in this chapter may besufficient, but in other areas it probably will not. Once you identify theareas of math in which you are having problems, get help in those areas.Don’t limp along through the whole course with a vague feeling of dreadabout something like scientific notation. The problem will not go away ifyou ignore it. The same applies to essential mathematical skills that you arelearning in this course for the first time, such as vector addition.

Sometimes students tell me they keep trying to understand a certaintopic in the book, and it just doesn’t make sense. The worst thing you canpossibly do in that situation is to keep on staring at the same page. Everytextbook explains certain things badly — even mine! — so the best thing todo in this situation is to look at a different book. Instead of college text-books aimed at the same mathematical level as the course you’re taking, youmay in some cases find that high school books or books at a lower mathlevel give clearer explanations. The three books listed on the left are, in myopinion, the best introductory physics books available, although they wouldnot be appropriate as the primary textbook for a college-level course forscience majors.

Finally, when reviewing for an exam, don’t simply read back over thetext and your lecture notes. Instead, try to use an active method of review-ing, for instance by discussing some of the discussion questions withanother student, or doing homework problems you hadn’t done the firsttime.

interpretingan equation

Section 0.3 How to Learn Physics

Other BooksPSSC Physics, Haber-Schaim etal., 7th ed., 1986. Kendall/Hunt,Dubuque, Iowa.

A high-school textbook at thealgebra-based level. This bookdistinguishes itself by giving aclear, careful, and honestexplanation of every topic, whileavoiding unnecessary details.

Physics for Poets, Robert H.March, 4th ed., 1996. McGraw-Hill, New York.

As the name implies, this book’sintended audience is liberal artsstudents who want to under-stand science in a broadercultural and historical context.Not much math is used, and thepage count of this little paper-back is about five times less thanthat of the typical “kitchen sink”textbook, but the intellectuallevel is actually pretty challeng-ing.

Conceptual Physics, Paul Hewitt.Scott Foresman, Glenview, Ill.

This is the excellent book usedfor Physics 130 here at FullertonCollege. Only simple algebra isused.

22

0.4 Self-EvaluationThe introductory part of a book like this is hard to write, because every

student arrives at this starting point with a different preparation. Onestudent may have grown up in another country and so may be completelycomfortable with the metric system, but may have had an algebra course inwhich the instructor passed too quickly over scientific notation. Anotherstudent may have already taken calculus, but may have never learned themetric system. The following self-evaluation is a checklist to help you figureout what you need to study to be prepared for the rest of the course.

If you disagree with this statement... you should study this section:

I am familiar with the basic metric units of meters,kilograms, and seconds, and the most common metricprefixes: milli- (m), kilo- (k), and centi- (c).

0.5 Basics of the Metric System

I know about the Newton, a unit of force 0.6 The Newton, the Metric Unit of Force

I am familiar with these less common metric prefixes:mega- (M), micro- (µ), and nano- (n). 0.7 Less Common Metric Prefixes

I am comfortable with scientific notation. 0.8 Scientific Notation

I can confidently do metric conversions. 0.9 Conversions

I understand the purpose and use of significant figures. 0.10 Significant Figures

It wouldn’t hurt you to skim the sections you think you already knowabout, and to do the self-checks in those sections.

0.5 Basics of the Metric SystemThe Metric System

Units were not standardized until fairly recently in history, so when thephysicist Isaac Newton gave the result of an experiment with a pendulum,he had to specify not just that the string was 37 7/

8 inches long but that it

was “37 7/8 London inches long.” The inch as defined in Yorkshire would

have been different. Even after the British Empire standardized its units, itwas still very inconvenient to do calculations involving money, volume,distance, time, or weight, because of all the odd conversion factors, like 16ounces in a pound, and 5280 feet in a mile. Through the nineteenthcentury, schoolchildren squandered most of their mathematical educationin preparing to do calculations such as making change when a customer in ashop offered a one-crown note for a book costing two pounds, thirteenshillings and tuppence. The dollar has always been decimal, and Britishmoney went decimal decades ago, but the United States is still saddled withthe antiquated system of feet, inches, pounds, ounces and so on.

Every country in the world besides the U.S. has adopted a system ofunits known in English as the “metric system.” This system is entirely

Chapter 0 Introduction and Review

23

decimal, thanks to the same eminently logical people who brought aboutthe French Revolution. In deference to France, the system’s official name isthe Système International, or SI, meaning International System. (Thephrase “SI system” is therefore redundant.)

The wonderful thing about the SI is that people who live in countriesmore modern than ours do not need to memorize how many ounces thereare in a pound, how many cups in a pint, how many feet in a mile, etc. Thewhole system works with a single, consistent set of prefixes (derived fromGreek) that modify the basic units. Each prefix stands for a power of ten,and has an abbreviation that can be combined with the symbol for the unit.For instance, the meter is a unit of distance. The prefix kilo- stands for 103,so a kilometer, 1 km, is a thousand meters.

The basic units of the metric system are the meter for distance, thesecond for time, and the gram for mass.

The following are the most common metric prefixes. You shouldmemorize them.

prefix meaning example

kilo- k 103 60 kg = a person’s mass

centi- c 10-2 28 cm = height of a piece of paper

milli- m 10-3 1 ms = time for one vibration of aguitar string playing thenote D

The prefix centi-, meaning 10-2, is only used in the centimeter; ahundredth of a gram would not be written as 1 cg but as 10 mg. The centi-prefix can be easily remembered because a cent is 10-2 dollars. The official SIabbreviation for seconds is “s” (not “sec”) and grams are “g” (not “gm”).

Although I advise you to memorize these prefixes because they are soimportant in all the sciences, it would still not be a bad idea to put them onyour 3x5 note card for use on quizzes and exams, just in case.

The SecondThe sun stood still and the moon halted until the nation had taken ven-geance on its enemies...

Joshua 10:12-14Absolute, true, and mathematical time, of itself, and from its own nature,flows equably without relation to anything external...

Isaac NewtonWhen I stated briefly above that the second was a unit of time, it may

not have occurred to you that this was not really much of a definition. Thetwo quotes above are meant to demonstrate how much room for confusionexists among people who seem to mean the same thing by a word such as“time.” The first quote has been interpreted by some biblical scholars asindicating an ancient belief that the motion of the sun across the sky wasnot just something that occurred with the passage of time but that the sunactually caused time to pass by its motion, so that freezing it in the sky

Section 0.5 Basic of the Metric System

24

would have some kind of a supernatural decelerating effect on everyoneexcept the Hebrew soldiers. Many ancient cultures also conceived of time ascyclical, rather than proceeding along a straight line as in 1998, 1999,2000, 2001,... The second quote, from a relatively modern physicist, maysound a lot more scientific, but most physicists today would consider ituseless as a definition of time. Today, the physical sciences are based onoperational definitions, which means definitions that spell out the actualsteps (operations) required to measure something numerically.

Now in an era when our toasters, pens, and coffee pots tell us the time,it is far from obvious to most people what is the fundamental operationaldefinition of time. Until recently, the hour, minute, and second weredefined operationally in terms of the time required for the earth to rotateabout its axis. Unfortunately, the Earth’s rotation is slowing down slightly,and by 1967 this was becoming an issue in scientific experiments requiringprecise time measurements. The second was therefore redefined as the timerequired for a certain number of vibrations of the light waves emitted by acesium atoms in a lamp constructed like a familiar neon sign but with theneon replaced by cesium. The new definition not only promises to stayconstant indefinitely, but for scientists is a more convenient way of calibrat-ing a clock than having to carry out astronomical measurements.

Self-Check

What is a possible operational definition of how strong a person is?

The MeterThe French originally defined the meter as 10-7 times the distance from

the equator to the north pole, as measured through Paris (of course). Even ifthe definition was operational, the operation of traveling to the north poleand laying a surveying chain behind you was not one that most workingscientists wanted to carry out. Fairly soon, a standard was created in theform of a metal bar with two scratches on it. This definition persisted until1960, when the meter was redefined as the distance traveled by light in avacuum over a period of (1/299792458) seconds.

Pope Gregory created our modern“Gregorian” calendar, with its systemof leap years, to make the length ofthe calendar year match the lengthof the cycle of seasons.Not until1752 did Protestant Englandswitched to the new calendar. Someless educated citizens believed thatthe shortening of the month byeleven days would shorten their livesby the same interval. In thisillustration by William Hogarth, theleaflet lying on the ground reads,“Give us our eleven days.”

The Time WithoutUnderwear

Unfortunately, the FrenchRevolutionary calendar nevercaught on. Each of its twelvemonths was 30 days long, withnames like Thermidor (the monthof heat) and Germinal (the monthof budding). To round out theyear to 365 days, a five-day periodwas added on the end of thecalendar, and named the sansculottides. In modern French, sansculottides means “time withoutunderwear,” but in the 18thcentury, it was a way to honor theworkers and peasants, who woresimple clothing instead of thefancy pants (culottes) of thearistocracy.

A dictionary might define “strong” as “posessing powerful muscles,” but that’s not an operational definition, becauseit doesn’t say how to measure strength numerically. One possible operational definition would be the number ofpounds a person can bench press.

107 m

Chapter 0 Introduction and Review

25

The KilogramThe third base unit of the SI is the kilogram, a unit of mass. Mass is

intended to be a measure of the amount of a substance, but that is not anoperational definition. Bathroom scales work by measuring our planet’sgravitational attraction for the object being weighed, but using that type ofscale to define mass operationally would be undesirable because gravityvaries in strength from place to place on the earth.

There’s a surprising amount of disagreement among physics textbooksabout how mass should be defined, but here’s how it’s actually handled bythe few working physicists who specialize in ultra-high-precision measure-ments. They maintain a physical object in Paris, which is the standardkilogram, a cylinder made of platinum-iridium alloy. Duplicates arechecked against this mother of all kilograms by putting the original and thecopy on the two opposite pans of a balance. Although this method ofcomparison depends on gravity, the problems associated with differences ingravity in different geographical locations are bypassed, because the twoobjects are being compared in the same place. The duplicates can then beremoved from the Parisian kilogram shrine and transported elsewhere in theworld.

Combinations of Metric UnitsJust about anything you want to measure can be measured with some

combination of meters, kilograms, and seconds. Speed can be measured inm/s, volume in m3, and density in kg/m3. Part of what makes the SI great isthis basic simplicity. No more funny units like a cord of wood, a bolt ofcloth, or a jigger of whiskey. No more liquid and dry measure. Just a simple,consistent set of units. The SI measures put together from meters, kilo-grams, and seconds make up the mks system. For example, the mks unit ofspeed is m/s, not km/hr.

Discussion questionIsaac Newton wrote, “...the natural days are truly unequal, though they arecommonly considered as equal, and used for a measure of time... It may bethat there is no such thing as an equable motion, whereby time may beaccurately measured. All motions may be accelerated or retarded...” Newtonwas right. Even the modern definition of the second in terms of light emitted bycesium atoms is subject to variation. For instance, magnetic fields could causethe cesium atoms to emit light with a slightly different rate of vibration. Whatmakes us think, though, that a pendulum clock is more accurate than asundial, or that a cesium atom is a more accurate timekeeper than a pendulumclock? That is, how can one test experimentally how the accuracies of differenttime standards compare?

0.6 The Newton, the Metric Unit of ForceA force is a push or a pull, or more generally anything that can change

an object’s speed or direction of motion. A force is required to start a carmoving, to slow down a baseball player sliding in to home base, or to makean airplane turn. (Forces may fail to change an object’s motion if they arecanceled by other forces, e.g. the force of gravity pulling you down rightnow is being canceled by the force of the chair pushing up on you.) Themetric unit of force is the Newton, defined as the force which, if applied forone second, will cause a 1-kilogram object starting from rest to reach a

Section 0.6 The Newton, the Metric Unit of Force

26

speed of 1 m/s. Later chapters will discuss the force concept in more detail.In fact, this entire book is about the relationship between force and motion.

In the previous section, I gave a gravitational definition of mass, but bydefining a numerical scale of force, we can also turn around and define ascale of mass without reference to gravity. For instance, if a force of twoNewtons is required to accelerate a certain object from rest to 1 m/s in 1 s,then that object must have a mass of 2 kg. From this point of view, masscharacterizes an object’s resistance to a change in its motion, which we callinertia or inertial mass. Although there is no fundamental reason why anobject’s resistance to a change in its motion must be related to how stronglygravity affects it, careful and precise experiments have shown that theinertial definition and the gravitational definition of mass are highlyconsistent for a variety of objects. It therefore doesn’t really matter for anypractical purpose which definition one adopts.

Discussion QuestionSpending a long time in weightlessness is unhealthy. One of the mostimportant negative effects experienced by astronauts is a loss of muscle andbone mass. Since an ordinary scale won’t work for an astronaut in orbit, whatis a possible way of monitoring this change in mass? (Measuring theastronaut’s waist or biceps with a measuring tape is not good enough, becauseit doesn’t tell anything about bone mass, or about the replacement of musclewith fat.)

0.7 Less Common Metric PrefixesThe following are three metric prefixes which, while less common than

the ones discussed previously, are well worth memorizing.

prefix meaning example

mega- M 106 6.4 Mm = radius of the earth

micro- µ 10-6 1 µm = diameter of a human hair

nano- n 10-9 0.154 nm = distance between carbonnuclei in an ethane molecule

Note that the abbreviation for micro is the Greek letter mu, µ — acommon mistake is to confuse it with m (milli) or M (mega).

There are other prefixes even less common, used for extremely large andsmall quantities. For instance, 1 femtometer=10-15 m is a convenient unitof distance in nuclear physics, and 1 gigabyte=109 bytes is used for comput-ers’ hard disks. The international committee that makes decisions about theSI has recently even added some new prefixes that sound like jokes, e.g. 1yoctogram = 10-24 g is about half the mass of a proton. In the immediatefuture, however, you’re unlikely to see prefixes like “yocto-” and “zepto-”used except perhaps in trivia contests at science-fiction conventions or othergeekfests.

10-9

10-6

10-3

103

106

Nine little

nano

micro

milli

kilo

mega

nuns

mix

milky

mugs.

This is a mnemonic to help youremember the most importantmetric prefixes. The word "little"is to remind you that the list startswith the prefixes used for smallquantities and builds upward. Theexponent changes by 3 with eachstep, except that of course we donot need a special prefix for 100,which equals one.

Chapter 0 Introduction and Review

27

Self-CheckSuppose you could slow down time so that according to your perception, abeam of light would move across a room at the speed of a slow walk. If youperceived a nanosecond as if it was a second, how would you perceive amicrosecond?

0.8 Scientific NotationMost of the interesting phenomena our universe has to offer are not on

the human scale. It would take about 1,000,000,000,000,000,000,000bacteria to equal the mass of a human body. When the physicist ThomasYoung discovered that light was a wave, it was back in the bad old daysbefore scientific notation, and he was obliged to write that the time requiredfor one vibration of the wave was 1/500 of a millionth of a millionth of asecond. Scientific notation is a less awkward way to write very large andvery small numbers such as these. Here’s a quick review.

Scientific notation means writing a number in terms of a product ofsomething from 1 to 10 and something else that is a power of ten. Forinstance,

32 = 3.2 x 101

320 = 3.2 x 102

3200 = 3.2 x 103 ...Each number is ten times bigger than the previous one.

Since 101 is ten times smaller than 102, it makes sense to use thenotation 100 to stand for one, the number that is in turn ten times smallerthan 101. Continuing on, we can write 10-1 to stand for 0.1, the numberten times smaller than 100. Negative exponents are used for small numbers:

3.2 = 3.2 x 100

0.32 = 3.2 x 10-1

0.032 = 3.2 x 10-2 ...A common source of confusion is the notation used on the displays of

many calculators. Examples:

3.2 x 106 (written notation)

3.2E+6 (notation on some calculators)

3.26 (notation on some other calculators)

The last example is particularly unfortunate, because 3.26 really standsfor the number 3.2x3.2x3.2x3.2x3.2x3.2 = 1074, a totally different numberfrom 3.2 x 106 = 3200000. The calculator notation should never be used inwriting. It’s just a way for the manufacturer to save money by making asimpler display.

A microsecond is 1000 times longer than a nanosecond, so it would seem like 1000 seconds, or about 20 minutes.

Section 0.8 Scientific Notation

28

Self-CheckA student learns that 104 bacteria, standing in line to register for classes atParamecium Community College, would form a queue of this size:

The student concludes that 102 bacteria would form a line of this length:

Why is the student incorrect?

0.9 ConversionsI suggest you avoid memorizing lots of conversion factors between SI

units and U.S. units. Suppose the United Nations sends its black helicoptersto invade California (after all who wouldn’t rather live here than in NewYork City?), and institutes water fluoridation and the SI, making the use ofinches and pounds into a crime punishable by death. I think you could getby with only two mental conversion factors:

1 inch = 2.54 cmAn object with a weight on Earth of 2.2 lb has a mass of 1 kg.

The first one is the present definition of the inch, so it’s exact. Thesecond one is not exact, but is good enough for most purposes. The poundis a unit of gravitational force, while the kg is a unit of mass, which mea-sures how hard it is to accelerate an object, not how hard gravity pulls on it.Therefore it would be incorrect to say that 2.2 lb literally equaled 1 kg, evenapproximately.

More important than memorizing conversion factors is understandingthe right method for doing conversions. Even within the SI, you may needto convert, say, from grams to kilograms. Different people have differentways of thinking about conversions, but the method I’ll describe here issystematic and easy to understand. The idea is that if 1 kg and 1000 grepresent the same mass, then we can consider a fraction like

103 g1 kg

to be a way of expressing the number one. This may bother you. Forinstance, if you type 1000/1 into your calculator, you will get 1000, notone. Again, different people have different ways of thinking about it, butthe justification is that it helps us to do conversions, and it works! Now ifwe want to convert 0.7 kg to units of grams, we can multiply 0.7 kg by thenumber one:

0.7 kg ×

103 g1 kg

If you’re willing to treat symbols such as “kg” as if they were variables asused in algebra (which they’re really not), you can then cancel the kg on topwith the kg on the bottom, resulting in

Chapter 0 Introduction and Review

Exponents have to do with multiplication, not addition. The first line should be 100 times longer than the second,not just twice as long.

29

0.7 kg/ ×103 g

1 kg/= 700 g .

To convert grams to kilograms, you would simply flip the fraction upsidedown.

One advantage of this method is that it can easily be applied to a seriesof conversions. For instance, to convert one year to units of seconds,

1 year/ ×

365 days/1 year/

×24 hours/

1 day/×

60 min/1 hour/

× 60 s

1 min/= 3.15 x 107 s .

Should that exponent be positive or negative?A common mistake is to write the conversion fraction incorrectly. For

instance the fraction

103 kg1 g

(incorrect)

does not equal one, because 103 kg is the mass of a car, and 1 g is the massof a raisin. One correct way of setting up the conversion factor would be

10– 3 kg1 g

. (correct)

You can usually detect such a mistake if you take the time to check youranswer and see if it is reasonable.

If common sense doesn’t rule out either a positive or a negative expo-nent, here’s another way to make sure you get it right. There are big prefixesand small prefixes:

big prefixes: k Msmall prefixes: m µ n

(It’s not hard to keep straight which are which, since “mega” and “micro” areevocative, and it’s easy to remember that a kilometer is bigger than a meterand a millimeter is smaller.) In the example above, we want the top of thefraction to be the same as the bottom. Since k is a big prefix, we need tocompensate by putting a small number like 10-3 in front of it, not a bignumber like 103.

Discussion QuestionEach of the following conversions contains an error. In each case, explainwhat the error is.

(a) 1000 kg x 1 kg1000 g = 1 g (b) 50 m x 1 cm

100 m = 0.5 cm

(c) "Nano" is 10-9, so there are 10-9 nm in a meter.(d) "Micro" is 10-6, so 1 kg is 106 µg.

checking conv ersionsusing common sense

checking conv ersions usingthe idea of “compensating ”

Section 0.9 Conversions

30

0.10 Significant FiguresAn engineer is designing a car engine, and has been told that the

diameter of the pistons (which are being designed by someone else) is 5 cm.He knows that 0.02 cm of clearance is required for a piston of this size, sohe designs the cylinder to have an inside diameter of 5.04 cm. Luckily, hissupervisor catches his mistake before the car goes into production. Sheexplains his error to him, and mentally puts him in the “do not promote”category.

What was his mistake? The person who told him the pistons were 5 cmin diameter was wise to the ways of significant figures, as was his boss, whoexplained to him that he needed to go back and get a more accurate num-ber for the diameter of the pistons. That person said “5 cm” rather than“5.00 cm” specifically to avoid creating the impression that the number wasextremely accurate. In reality, the pistons’ diameter was 5.13 cm. Theywould never have fit in the 5.04-cm cylinders.

The number of digits of accuracy in a number is referred to as thenumber of significant figures, or “sig figs” for short. As in the exampleabove, sig figs provide a way of showing the accuracy of a number. In mostcases, the result of a calculation involving several pieces of data can be nomore accurate than the least accurate piece of data. In other words, “garbagein, garbage out.” Since the 5 cm diameter of the pistons was not veryaccurate, the result of the engineer’s calculation, 5.04 cm, was really not asaccurate as he thought. In general, your result should not have more thanthe number of sig figs in the least accurate piece of data you started with.The calculation above should have been done as follows:

5 cm (1 sig fig)+ 0.04 cm (1 sig fig)= 5 cm (rounded off to 1 sig fig)

The fact that the final result only has one significant figure then alerts youto the fact that the result is not very accurate, and would not be appropriatefor use in designing the engine.

Note that the leading zeroes in the number 0.04 do not count assignificant figures, because they are only placeholders. On the other hand, anumber such as 50 cm is ambiguous — the zero could be intended as asignificant figure, or it might just be there as a placeholder. The ambiguityinvolving trailing zeroes can be avoided by using scientific notation, inwhich 5 x 101 cm would imply one sig fig of accuracy, while 5.0 x 101 cmwould imply two sig figs.

Dealing correctly with significant figures can save you time! Often,students copy down numbers from their calculators with eight significantfigures of precision, then type them back in for a later calculation. That’s awaste of time, unless your original data had that kind of incredible preci-sion.

Significant figur escommunicate the

accuracy of a number.

Chapter 0 Introduction and Review

31

The rules about significant figures are only rules of thumb, and are nota substitute for careful thinking. For instance, $20.00 + $0.05 is $20.05. Itneed not and should not be rounded off to $20. In general, the sig fig ruleswork best for multiplication and division, and we also apply them whendoing a complicated calculation that involves many types of operations. Forsimple addition and subtraction, it makes more sense to maintain a fixednumber of digits after the decimal point.

Self-CheckHow many significant figures are there in each of the following measurements?

(a) 9.937 m(b) 4.0 s(c) 0.0000037 kg

(a) 4; (b) 2; (c) 2

Section 0.10 Significant Figures

32

SummarySelected Vocabulary

matter ............................... Anything that is affected by gravity.light................................... Anything that can travel from one place to another through empty space

and can influence matter, but is not affected by gravity.operational definition ........ A definition that states what operations should be carried out to measure

the thing being defined.Système International ........ A fancy name for the metric system.mks system ........................ The use of metric units based on the meter, kilogram, and second. Ex-

ample: meters per second is the mks unit of speed, not cm/s or km/hr.mass .................................. A numerical measure of how difficult it is to change an object’s motion.significant figures .............. Digits that contribute to the accuracy of a measurement.

Notationm ...................................... symbol for mass, or the meter, the metric distance unitkg ...................................... kilogram, the metric unit of masss ........................................ second, the metric unit of timeM- ..................................... the metric prefix mega-, 106

k- ...................................... the metric prefix kilo-, 103

m- ..................................... the metric prefix milli-, 10-3

µ- ...................................... the metric prefix micro-, 10-6

n- ...................................... the metric prefix nano-, 10-9

SummaryPhysics is the use of the scientific method to study the behavior of light and matter. The scientific method

requires a cycle of theory and experiment, theories with both predictive and explanatory value, andreproducible experiments.

The metric system is a simple, consistent framework for measurement built out of the meter, the kilogram,and the second plus a set of prefixes denoting powers of ten. The most systematic method for doingconversions is shown in the following example:

370 ms × 10– 3 s1 ms = 0.37 s

Mass is a measure of the amount of a substance. Mass can be defined gravitationally, by comparing anobject to a standard mass on a double-pan balance, or in terms of inertia, by comparing the effect of a forceon an object to the effect of the same force on a standard mass. The two definitions are found experimentallyto be proportional to each other to a high degree of precision, so we usually refer simply to “mass,” withoutbothering to specify which type.

A force is that which can change the motion of an object. The metric unit of force is the Newton, defined asthe force required to accelerate a standard 1-kg mass from rest to a speed of 1 m/s in 1 s.

Scientific notation means, for example, writing 3.2x105 rather than 320000.

Writing numbers with the correct number of significant figures correctly communicates how accurate theyare. As a rule of thumb, the final result of a calculation is no more accurate than, and should have no moresignificant figures than, the least accurate piece of data.

Chapter 0 Introduction and Review

33

Homework Problems

1. Correct use of a calculator: (a) Calculate 7465853222 + 97554

on a calcula-

tor.

[Self-check: The most common mistake results in 97555.40.]

(b) Which would be more like the price of a TV, and which would be morelike the price of a house, $ 3.5x105 or $ 3.55?

2. Compute the following things. If they don't make sense because of units,say so.

(a) 3 cm + 5 cm (b) 1.11 m + 22 cm

(c) 120 miles + 2.0 hours (d) 120 miles / 2.0 hours

3. Your backyard has brick walls on both ends. You measure a distance of23.4 m from the inside of one wall to the inside of the other. Each wall is29.4 cm thick. How far is it from the outside of one wall to the outside ofthe other? Pay attention to significant figures.

4 . The speed of light is 3.0x108 m/s. Convert this to furlongs per fort-night. A furlong is 220 yards, and a fortnight is 14 days. An inch is 2.54cm.

5 . Express each of the following quantities in micrograms: (a) 10 mg, (b)104 g, (c) 10 kg, (d) 100x103 g, (e) 1000 ng.

6 S. Convert 134 mg to units of kg, writing your answer in scientificnotation.

7. In the last century, the average age of the onset of puberty for girls hasdecreased by several years. Urban folklore has it that this is because ofhormones fed to beef cattle, but it is more likely to be because modern girlshave more body fat on the average and possibly because of estrogen-mimicking chemicals in the environment from the breakdown of pesticides.A hamburger from a hormone-implanted steer has about 0.2 ng of estrogen(about double the amount of natural beef ). A serving of peas contains about300 ng of estrogen. An adult woman produces about 0.5 mg of estrogen perday (note the different unit!). (a) How many hamburgers would a girl haveto eat in one day to consume as much estrogen as an adult woman’s dailyproduction? (b) How many servings of peas?

S A solution is given in the back of the book. « A difficult problem. A computerized answer check is available. ∫ A problem that requires calculus.

Homework Problems

34

35

1 Scaling and Order-of-Magnitude Estimates

1.1 IntroductionWhy can’t an insect be the size of a dog? Some skinny stretched-out cells

in your spinal cord are a meter tall — why does nature display no singlecells that are not just a meter tall, but a meter wide, and a meter thick aswell? Believe it or not, these are questions that can be answered fairly easilywithout knowing much more about physics than you already do. The onlymathematical technique you really need is the humble conversion, appliedto area and volume.

Area and VolumeArea can be defined by saying that we can copy the shape of interest

onto graph paper with 1 cm x 1 cm squares and count the number ofsquares inside. Fractions of squares can be estimated by eye. We then say thearea equals the number of squares, in units of square cm. Although thismight seem less “pure” than computing areas using formulae like A=πr2 fora circle or A=wh/2 for a triangle, those formulae are not useful as definitionsof area because they cannot be applied to irregularly shaped areas.

Units of square cm are more commonly written as cm2 in science. Ofcourse, the unit of measurement symbolized by “cm” is not an algebrasymbol standing for a number that can be literally multiplied by itself. Butit is advantageous to write the units of area that way and treat the units as ifthey were algebra symbols. For instance, if you have a rectangle with an areaof 6 m2 and a width of 2 m, then calculating its length as (6 m2)/(2 m)=3 mgives a result that makes sense both numerically and in terms of units. Thisalgebra-style treatment of the units also ensures that our methods of

Amoebas this size are seldomencountered.

A normal red blood cell (left), a cell newly infected with the malaria parasite (middle), and an infected cell in which the parasitehas been developing for 12 hours (right). Red blood cells are so small compared to other human cells that only with veryrecent advances in x-ray microscopy has it become possible to make images as detailed as these. Why is a red blood cell sosmall? Its small size gives it both a small volume and a small surface area, but counterintuitively, the ratio of surface tovolume becomes greater for smaller objects. Red blood cells’ function is to transport oxygen, and their large surface-to-volume ratio makes them more efficient at picking up and delivering oxygen through their cell membranes.

36

1 yd2x(3 ft/1 yd)2=9 ft2. 1 yd3x(3 ft/1 yd)3=27 ft3.

converting units work out correctly. For instance, if we accept the fraction

100 cm1 m

as a valid way of writing the number one, then one times one equals one, sowe should also say that one can be represented by

100 cm1 m

× 100 cm1 m

which is the same as

10000 cm2

1 m2 .

That means the conversion factor from square meters to square centi-meters is a factor of 104, i.e. a square meter has 104 square centimeters init.

All of the above can be easily applied to volume as well, using one-cubic-centimeter blocks instead of squares on graph paper.

To many people, it seems hard to believe that a square meter equals10000 square centimeters, or that a cubic meter equals a million cubiccentimeters — they think it would make more sense if there were 100 cm2

in 1 m2, and 100 cm3 in 1 m3, but that would be incorrect. The examplesshown in the figure below aim to make the correct answer more believable,using the traditional U.S. units of feet and yards. (One foot is 12 inches,and one yard is three feet.)

1 ft 1 yd = 3 ft

1 ft2 1 yd2 = 9 ft2 1 ft3

1 yd3 = 27 ft3

Self-CheckBased on the figure, convince yourself that there are 9 ft2 in a square yard ,and 27 ft3 in a cubic yard, then demonstrate the same thing symbolically (i.e.with the method using fractions that equal one).

Discussion question

A. How many square centimeters are there in a square inch? (1 inch=2.54 cm)First find an approximate answer by making a drawing, then derive theconversion factor more accurately using the symbolic method.

Chapter 1 Scaling and Order-of-Magnitude Estimates

37

1.2 Scaling of Area and VolumeGreat fleas have lesser fleasUpon their backs to bite ‘em.And lesser fleas have lesser still,And so ad infinitum.

Jonathan SwiftNow how do these conversions of area and volume relate to the ques-

tions I posed about sizes of living things? Well, imagine that you are shrunklike Alice in Wonderland to the size of an insect. One way of thinkingabout the change of scale is that what used to look like a centimeter nowlooks like perhaps a meter to you, because you’re so much smaller. If areaand volume scaled according to most people’s intuitive, incorrect expecta-tions, with 1 m2 being the same as 100 cm2, then there would be noparticular reason why nature should behave any differently on your new,reduced scale. But nature does behave differently now that you’re small. Forinstance, you will find that you can walk on water, and jump to many timesyour own height. The physicist Galileo Galilei had the basic insight that thescaling of area and volume determines how natural phenomena behavedifferently on different scales. He first reasoned about mechanical struc-tures, but later extended his insights to living things, taking the then-radicalpoint of view that at the fundamental level, a living organism should followthe same laws of nature as a machine. We will follow his lead by firstdiscussing machines and then living things.

Galileo on the behavior of nature on large and small scalesOne of the world’s most famous pieces of scientific writing is Galileo’s

Dialogues Concerning the Two New Sciences. Galileo was an entertainingwriter who wanted to explain things clearly to laypeople, and he livened uphis work by casting it in the form of a dialogue among three people. Salviatiis really Galileo’s alter ego. Simplicio is the stupid character, and one of thereasons Galileo got in trouble with the Church was that there were rumorsthat Simplicio represented the Pope. Sagredo is the earnest and intelligentstudent, with whom the reader is supposed to identify. (The followingexcerpts are from the no longer copyrighted 1914 translation by Crew andde Salvio.)

A boat this large needs to have timbersthat are thicker compared to its size.

The small boat holds up just fine.

A larger boat built with the sameproportions as the small one willcollapse under its own weight.

Galileo Galilei (1564-1642) was a Renaissance Italian who brought the scientificmethod to bear on physics, creating the modern version of the science. Comingfrom a noble but very poor family, Galileo had to drop out of medical school atthe University of Pisa when he ran out of money. Eventually becoming a lecturerin mathematics at the same school, he began a career as a notorioustroublemaker by writing a burlesque ridiculing the university’s regulations — hewas forced to resign, but found a new teaching position at Padua. He inventedthe pendulum clock, investigated the motion of falling bodies, and discoveredthe moons of Jupiter. The thrust of his life’s work was to discredit Aristotle’sphysics by confronting it with contradictory experiments, a program which pavedthe way for Newton’s discovery of the relationship between force and motion. InChapter 3 we’ll come to the story of Galileo’s ultimate fate at the hands of theChurch.

Section 1.2 Scaling of Area and Volume

38

SALVIATI: ...we asked the reason why [shipbuilders] employed stocks,scaffolding, and bracing of larger dimensions for launching a big vessel thanthey do for a small one; and [an old man] answered that they did this in orderto avoid the danger of the ship parting under its own heavy weight, a danger towhich small boats are not subject?SAGREDO: Yes, that is what I mean; and I refer especially to his last assertionwhich I have always regarded as false...; namely, that in speaking of these andother similar machines one cannot argue from the small to the large, becausemany devices which succeed on a small scale do not work on a large scale.Now, since mechanics has its foundations in geometry, where mere size [ isunimportant], I do not see that the properties of circles, triangles, cylinders,cones and other solid figures will change with their size. If, therefore, a largemachine be constructed in such a way that its parts bear to one another thesame ratio as in a smaller one, and if the smaller is sufficiently strong for thepurpose for which it is designed, I do not see why the larger should not be ableto withstand any severe and destructive tests to which it may be subjected.

Salviati contradicts Sagredo:

SALVIATI: ...Please observe, gentlemen, how facts which at first seemimprobable will, even on scant explanation, drop the cloak which has hiddenthem and stand forth in naked and simple beauty. Who does not know that ahorse falling from a height of three or four cubits will break his bones, while adog falling from the same height or a cat from a height of eight or ten cubits willsuffer no injury? Equally harmless would be the fall of a grasshopper from atower or the fall of an ant from the distance of the moon.

The point Galileo is making here is that small things are sturdier inproportion to their size. There are a lot of objections that could be raised,however. After all, what does it really mean for something to be “strong”, tobe “strong in proportion to its size,” or to be strong “out of proportion to itssize?” Galileo hasn’t spelled out operational definitions of things like“strength,” i.e. definitions that spell out how to measure them numerically.

Also, a cat is shaped differently from a horse — an enlarged photographof a cat would not be mistaken for a horse, even if the photo-doctoringexperts at the National Inquirer made it look like a person was riding on itsback. A grasshopper is not even a mammal, and it has an exoskeletoninstead of an internal skeleton. The whole argument would be a lot moreconvincing if we could do some isolation of variables, a scientific term thatmeans to change only one thing at a time, isolating it from the othervariables that might have an effect. If size is the variable whose effect we’re

This plank is the longest it can be without collapsing under its own weight. If it was a hundredth of an inch longer, it would collapse.

This plank is made out of the same kind of wood. It is twice as thick, twice as long, and twice as wide. It will collapse under its own weight.

(After Galileo's original drawing.)

Chapter 1 Scaling and Order-of-Magnitude Estimates

39

interested in seeing, then we don’t really want to compare things that aredifferent in size but also different in other ways.

Also, Galileo is doing something that would be frowned on in modernscience: he is mixing experiments whose results he has actually observed(building boats of different sizes), with experiments that he could notpossibly have done (dropping an ant from the height of the moon).

After this entertaining but not scientifically rigorous beginning, Galileostarts to do something worthwhile by modern standards. He simplifieseverything by considering the strength of a wooden plank. The variablesinvolved can then be narrowed down to the type of wood, the width, thethickness, and the length. He also gives an operational definition of what itmeans for the plank to have a certain strength “in proportion to its size,” byintroducing the concept of a plank that is the longest one that would notsnap under its own weight if supported at one end. If you increased itslength by the slightest amount, without increasing its width or thickness, itwould break. He says that if one plank is the same shape as another but adifferent size, appearing like a reduced or enlarged photograph of the other,then the planks would be strong “in proportion to their sizes” if both werejust barely able to support their own weight.

He now relates how he has done actual experiments with such planks,and found that, according to this operational definition, they are not strongin proportion to their sizes. The larger one breaks. He makes sure to tell thereader how important the result is, via Sagredo’s astonished response:

SAGREDO: My brain already reels. My mind, like a cloud momentarily illuminatedby a lightning flash, is for an instant filled with an unusual light, which nowbeckons to me and which now suddenly mingles and obscures strange, crudeideas. From what you have said it appears to me impossible to build twosimilar structures of the same material, but of different sizes and have themproportionately strong.

In other words, this specific experiment, using things like woodenplanks that have no intrinsic scientific interest, has very wide implicationsbecause it points out a general principle, that nature acts differently ondifferent scales.

To finish the discussion, Galileo gives an explanation. He says that thestrength of a plank (defined as, say, the weight of the heaviest boulder youcould put on the end without breaking it) is proportional to its cross-sectional area, that is, the surface area of the fresh wood that would beexposed if you sawed through it in the middle. Its weight, however, isproportional to its volume.

How do the volume and cross-sectional area of the longer plankcompare with those of the shorter plank? We have already seen, whilediscussing conversions of the units of area and volume, that these quantitiesdon’t act the way most people naively expect. You might think that thevolume and area of the longer plank would both be doubled compared tothe shorter plank, so they would increase in proportion to each other, andthe longer plank would be equally able to support its weight. You would bewrong, but Galileo knows that this is a common misconception, so he has

Section 1.2 Scaling of Area and Volume

Galileo discusses planks made ofwood, but the concept may be easierto imagine with clay. All three clay rodsin the figure were originally the sameshape. The medium-sized one wastwice the height, twice the length, andtwice the width of the small one, andsimilarly the large one was twice asbig as the medium one in all its lineardimensions. The big one has fourtimes the linear dimensions of thesmall one, 16 times the cross-sectionalarea when cut perpendicular to thepage, and 64 times the volume. Thatmeans that the big one has 64 timesthe weight to support, but only 16 timesthe strength compared to the smallestone.

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Salviati address the point specifically:

SALVIATI: ...Take, for example, a cube two inches on a side so that each facehas an area of four square inches and the total area, i.e., the sum of the sixfaces, amounts to twenty-four square inches; now imagine this cube to besawed through three times [with cuts in three perpendicular planes] so as todivide it into eight smaller cubes, each one inch on the side, each face oneinch square, and the total surface of each cube six square inches instead oftwenty-four in the case of the larger cube. It is evident therefore, that thesurface of the little cube is only one-fourth that of the larger, namely, the ratioof six to twenty-four; but the volume of the solid cube itself is only one-eighth;the volume, and hence also the weight, diminishes therefore much morerapidly than the surface... You see, therefore, Simplicio, that I was not mistakenwhen ... I said that the surface of a small solid is comparatively greater thanthat of a large one.

The same reasoning applies to the planks. Even though they are notcubes, the large one could be sawed into eight small ones, each with half thelength, half the thickness, and half the width. The small plank, therefore,has more surface area in proportion to its weight, and is therefore able tosupport its own weight while the large one breaks.

Scaling of area and volume for irregularly shaped objectsYou probably are not going to believe Galileo’s claim that this has deep

implications for all of nature unless you can be convinced that the same istrue for any shape. Every drawing you’ve seen so far has been of squares,rectangles, and rectangular solids. Clearly the reasoning about sawing thingsup into smaller pieces would not prove anything about, say, an egg, whichcannot be cut up into eight smaller egg-shaped objects with half the length.

Is it always true that something half the size has one quarter the surfacearea and one eighth the volume, even if it has an irregular shape? Take theexample of a child’s violin. Violins are made for small children in lengthsthat are either half or 3/4 of the normal length, accommodating their smallhands. Let’s study the surface area of the front panels of the three violins.

Consider the square in the interior of the panel of the full-size violin. Inthe 3/4-size violin, its height and width are both smaller by a factor of 3/4,so the area of the corresponding, smaller square becomes 3/4x3/4=9/16 ofthe original area, not 3/4 of the original area. Similarly, the correspondingsquare on the smallest violin has half the height and half the width of theoriginal one, so its area is 1/4 the original area, not half.

The same reasoning works for parts of the panel near the edge, such asthe part that only partially fills in the other square. The entire square scalesdown the same as a square in the interior, and in each violin the samefraction (about 70%) of the square is full, so the contribution of this part tothe total area scales down just the same.

Since any small square region or any small region covering part of asquare scales down like a square object, the entire surface area of an irregu-larly shaped object changes in the same manner as the surface area of asquare: scaling it down by 3/4 reduces the area by a factor of 9/16, and soon.

In general, we can see that any time there are two objects with the sameshape, but different linear dimensions (i.e. one looks like a reduced photo ofthe other), the ratio of their areas equals the ratio of the squares of theirlinear dimensions:

full size

3/4 size

half size

Chapter 1 Scaling and Order-of-Magnitude Estimates

41

A 1

A 2=

L 1

L 2

2

.

Note that it doesn’t matter where we choose to measure the linear size, L, ofan object. In the case of the violins, for instance, it could have been mea-sured vertically, horizontally, diagonally, or even from the bottom of the leftf-hole to the middle of the right f-hole. We just have to measure it in aconsistent way on each violin. Since all the parts are assumed to shrink orexpand in the same manner, the ratio L1/L2 is independent of the choice ofmeasurement.

It is also important to realize that it is completely unnecessary to have aformula for the area of a violin. It is only possible to derive simple formulasfor the areas of certain shapes like circles, rectangles, triangles and so on, butthat is no impediment to the type of reasoning we are using.

Sometimes it is inconvenient to write all the equations in terms ofratios, especially when more than two objects are being compared. A morecompact way of rewriting the previous equation is

A∝L 2.

The symbol “∝” means “is proportional to.” Scientists and engineers oftenspeak about such relationships verbally using the phrases “scales like” or“goes like,” for instance “area goes like length squared.”

All of the above reasoning works just as well in the case of volume.Volume goes like length cubed:

V ∝ L3 .If different objects are made of the same material with the same density,ρ=m/V, then their masses, m=ρV, are proportional to L3, and so are theirweights. (The symbol for density is ρ, the lower-case Greek letter “rho”.)

An important point is that all of the above reasoning about scaling onlyapplies to objects that are the same shape. For instance, a piece of paper islarger than a pencil, but has a much greater surface-to-volume ratio.

One of the first things I learned as a teacher was that students were notvery original about their mistakes. Every group of students tends to comeup with the same goofs as the previous class. The following are someexamples of correct and incorrect reasoning about proportionality.

Section 1.2 Scaling of Area and Volume

42

Example: scaling of the area of a triangleQuestion: In fig. (a), the larger triangle has sides twice as long.How many times greater is its area?Correct solution #1: Area scales in proportion to the square of thelinear dimensions, so the larger triangle has four times more area(22=4).Correct solution #2: You could cut the larger triangle into four ofthe smaller size, as shown in fig. (b), so its area is four times greater.(This solution is correct, but it would not work for a shape like acircle, which can’t be cut up into smaller circles.)Correct solution #3: The area of a triangle is given by

A = 12 bh, where b is the base and h is the height. The areas of the

triangles are

A1 = 12 b1h1

A2 = 12 b2h2

= 12 (2b1)(2h1)

= 2b1h1

A2/A1 = (2b1h1)/(12 b1h1)

= 4(Although this solution is correct, it is a lot more work thansolution #1, and it can only be used in this case because a triangle isa simple geometric shape, and we happen to know a formula for itsarea.)

Correct solution #4: The area of a triangle is A = 12 bh. The

comparison of the areas will come out the same as long as the ratiosof the linear sizes of the triangles is as specified, so let’s just sayb1=1.00 m and b2=2.00 m. The heights are then also h1=1.00 mand h2=2.00 m, giving areas A1=0.50 m2 and A2=2.00 m2, so A2/A1=4.00.(The solution is correct, but it wouldn’t work with a shape forwhose area we don’t have a formula. Also, the numerical calculationmight make the answer of 4.00 appear inexact, whereas solution #1makes it clear that it is exactly 4.)

Incorrect solution: The area of a triangle is A = 12 bh, and if you

plug in b=2.00 m and h=2.00 m, you get A=2.00 m2, so the biggertriangle has 2.00 times more area. (This solution is incorrectbecause no comparison has been made with the smaller triangle.)

The big triangle has four times morearea than the little one.

(a)

(b)

Chapter 1 Scaling and Order-of-Magnitude Estimates

43

Example: scaling of the volume of a sphereQuestion: In figure (c), the larger sphere has a radius that is fivetimes greater. How many times greater is its volume?Correct solution #1: Volume scales like the third power of thelinear size, so the larger sphere has a volume that is 125 timesgreater (53=125).

Correct solution #2: The volume of a sphere is V=43 πr3, so

V1 = 43

πr 13

V2 = 43

πr 23

= 43

π(5r 1)3

= 5003

πr 13

V2/V1 = 5003

πr 13 / 4

3πr 1

3

= 125

Incorrect solution: The volume of a sphere is V=43 πr3, so

V1 = 43

πr 13

V2 = 43

πr 23

= 43

π ⋅ 5r 13

= 203

πr 13

V2/V1=( 203

πr 13 )/( 4

3πr 1

3 )

=5

(The solution is incorrect because (5r1)3 is not the same as 5r 1

3 .)

Example: scaling of a more complex shapeQuestion: The first letter “S” in fig. (d) is in a 36-point font, thesecond in 48-point. How many times more ink is required to makethe larger “S”?Correct solution: The amount of ink depends on the area to becovered with ink, and area is proportional to the square of the lineardimensions, so the amount of ink required for the second “S” isgreater by a factor of (48/36)2=1.78.Incorrect solution: The length of the curve of the second “S” islonger by a factor of 48/36=1.33, so 1.33 times more ink isrequired.(The solution is wrong because it assumes incorrectly that the widthof the curve is the same in both cases. Actually both the width andthe length of the curve are greater by a factor of 48/36, so the area isgreater by a factor of (48/36)2=1.78.)

S S(d) The 48-point “S” has 1.78 timesmore area than the 36-point “S.”

(c)

The big sphere has 125 times morevolume than the little one.

Section 1.2 Scaling of Area and Volume

44

Discussion questionsA. A toy fire engine is 1/30 the size of the real one, but is constructed from thesame metal with the same proportions. How many times smaller is its weight?How many times less red paint would be needed to paint it?B. Galileo spends a lot of time in his dialog discussing what really happenswhen things break. He discusses everything in terms of Aristotle’s now-discredited explanation that things are hard to break, because if somethingbreaks, there has to be a gap between the two halves with nothing in between,at least initially. Nature, according to Aristotle, “abhors a vacuum,” i.e. naturedoesn’t “like” empty space to exist. Of course, air will rush into the gapimmediately, but at the very moment of breaking, Aristotle imagined a vacuumin the gap. Is Aristotle’s explanation of why it is hard to break things anexperimentally testable statement? If so, how could it be testedexperimentally?

1.3 Scaling Applied to Biology

Organisms of different sizes with the same shapeThe first of the following graphs shows the approximate validity of the

proportionality m∝L3 for co*ckroaches (redrawn from McMahon andBonner). The scatter of the points around the curve indicates that someco*ckroaches are proportioned slightly differently from others, but in generalthe data seem well described by m∝L3. That means that the largest co*ck-roaches the experimenter could raise (is there a 4-H prize?) had roughly thesame shape as the smallest ones.

Another relationship that should exist for animals of different sizesshaped in the same way is that between surface area and body mass. If allthe animals have the same average density, then body mass should beproportional to the cube of the animal’s linear size, m∝L3, while surfacearea should vary proportionately to L2. Therefore, the animals’ surface areasshould be proportional to m2/3. As shown in the second graph, this relation-ship appears to hold quite well for the dwarf siren, a type of salamander.Notice how the curve bends over, meaning that the surface area does notincrease as quickly as body mass, e.g. a salamander with eight times morebody mass will have only four times more surface area.

This behavior of the ratio of surface area to mass (or, equivalently, theratio of surface area to volume) has important consequences for mammals,which must maintain a constant body temperature. It would make sense forthe rate of heat loss through the animal’s skin to be proportional to itssurface area, so we should expect small animals, having large ratios ofsurface area to volume, to need to produce a great deal of heat in compari-son to their size to avoid dying from low body temperature. This expecta-tion is borne out by the data of the third graph, showing the rate of oxygenconsumption of guinea pigs as a function of their body mass. Neither ananimal’s heat production nor its surface area is convenient to measure, butin order to produce heat, the animal must metabolize oxygen, so oxygenconsumption is a good indicator of the rate of heat production. Sincesurface area is proportional to m2/3, the proportionality of the rate of oxygenconsumption to m2/3 is consistent with the idea that the animal needs toproduce heat at a rate in proportion to its surface area. Although the smalleranimals metabolize less oxygen and produce less heat in absolute terms, theamount of food and oxygen they must consume is greater in proportion totheir own mass. The Etruscan pigmy shrew, weighing in at 2 grams as an

Chapter 1 Scaling and Order-of-Magnitude Estimates

45

250

500

750

1000

body

mas

s (m

g)

0 1 2 3

length of leg segment (mm)

Body mass, m, versus leglength, L, for the co*ckroachPeriplaneta americana.The data points rep-resent individualspecimens, and thecurve is a fit to thedata of the form

m=kL3, where k isa constant.

200

400

600

800

1000

surf

ace

are

a (

cm2

)0 500 1000

body mass (g)

Surfacearea versus

body mass fordwarf sirens, aspecies of sala-

mander (Pseudo-branchus striatus).

The data pointsrepresent individual

specimens, and the curve is

a fit of the form A=km2/3.

1

2

3

4

5

6

7

8

oxy

ge

n c

on

sum

ptio

n (

mL

/min

)

0 .0 0.2 0.4 0.6 0.8 1.0

body mass (kg)

Rate of oxygenconsumption versusbody mass for guineapigs at rest. Thecurve is a fit of the

form (rate)=km2/3.0

1

2

3

4

5

dia

me

ter

(cm

)

0 2 4 6

length (cm)

Diameter versus lengthof the third lumbarvertebrae of adultAfrican Bovidae(antelopes and oxen).The smallest animalrepresented is thecat-sized Gunther'sdik-dik, and thelargest is the850-kg gianteland. Thesolid curve isa fit of the form d=kL3/2,

and the dashedline is a linearfit. (AfterMcMahon andBonner, 1983.)

46

adult, is at about the lower size limit for mammals. It must eat continually,consuming many times its body weight each day to survive.

Changes in Shape to Accommodate Changes in SizeLarge mammals, such as elephants, have a small ratio of surface area to

volume, and have problems getting rid of their heat fast enough. Anelephant cannot simply eat small enough amounts to keep from producingexcessive heat, because cells need to have a certain minimum metabolic rateto run their internal machinery. Hence the elephant’s large ears, which addto its surface area and help it to cool itself. Previously, we have seen severalexamples of data within a given species that were consistent with a fixedshape, scaled up and down in the cases of individual specimens. Theelephant’s ears are an example of a change in shape necessitated by a changein scale.

Large animals also must be able to support their own weight. Returningto the example of the strengths of planks of different sizes, we can see that ifthe strength of the plank depends on area while its weight depends onvolume, then the ratio of strength to weight goes as follows:

strength/weight ∝ A/V ∝ 1/L .

Thus, the ability of objects to support their own weights decreasesinversely in proportion to their linear dimensions. If an object is to be justbarely able to support its own weight, then a larger version will have to beproportioned differently, with a different shape.

Since the data on the co*ckroaches seemed to be consistent with roughlysimilar shapes within the species, it appears that the ability to support itsown weight was not the tightest design constraint that Nature was workingunder when she designed them. For large animals, structural strength isimportant. Galileo was the first to quantify this reasoning and to explainwhy, for instance, a large animal must have bones that are thicker inproportion to their length. Consider a roughly cylindrical bone such as a legbone or a vertebra. The length of the bone, L, is dictated by the overalllinear size of the animal, since the animal’s skeleton must reach the animal’swhole length. We expect the animal’s mass to scale as L3, so the strength ofthe bone must also scale as L3. Strength is proportional to cross-sectionalarea, as with the wooden planks, so if the diameter of the bone is d, then

d 2 ∝ L 3

or

d ∝ L 3 / 2.

If the shape stayed the same regardless of size, then all linear dimensions,including d and L, would be proportional to one another. If our reasoningholds, then the fact that d is proportional to L3/2, not L, implies a change inproportions of the bone. As shown in the graph on the previous page, thevertebrae of African Bovidae follow the rule d ∝ L3/2 fairly well. Thevertebrae of the giant eland are as chunky as a coffee mug, while those of aGunther’s dik-dik are as slender as the cap of a pen.

Galileo’s original drawing, showinghow larger animals’ bones must begreater in diameter compared to theirlengths.

Chapter 1 Scaling and Order-of-Magnitude Estimates

47

Discussion questionsA. Single-celled animals must passively absorb nutrients and oxygen from theirsurroundings, unlike humans who have lungs to pump air in and out and aheart to distribute the oxygenated blood throughout their bodies. Even the cellscomposing the bodies of multicellular animals must absorb oxygen from anearby capillary through their surfaces. Based on these facts, explain why cellsare always microscopic in size.B. The reasoning of the previous question would seem to be contradicted bythe fact that human nerve cells in the spinal cord can be as much as a meterlong, although their widths are still very small. Why is this possible?

1.4 Order-of-Magnitude EstimatesIt is the mark of an instructed mind to rest satisfied with the degree of preci-sion that the nature of the subject permits and not to seek an exactnesswhere only an approximation of the truth is possible.

AristotleIt is a common misconception that science must be exact. For instance,

in the Star Trek TV series, it would often happen that Captain Kirk wouldask Mr. Spock, “Spock, we’re in a pretty bad situation. What do you thinkare our chances of getting out of here?” The scientific Mr. Spock wouldanswer with something like, “Captain, I estimate the odds as 237.345 toone.” In reality, he could not have estimated the odds with six significantfigures of accuracy, but nevertheless one of the hallmarks of a person with agood education in science is the ability to make estimates that are likely tobe at least somewhere in the right ballpark. In many such situations, it isoften only necessary to get an answer that is off by no more than a factor often in either direction. Since things that differ by a factor of ten are said todiffer by one order of magnitude, such an estimate is called an order-of-magnitude estimate. The tilde, ~, is used to indicate that things are only ofthe same order of magnitude, but not exactly equal, as in

odds of survival ~ 100 to one .

The tilde can also be used in front of an individual number to emphasizethat the number is only of the right order of magnitude.

Although making order-of-magnitude estimates seems simple andnatural to experienced scientists, it’s a mode of reasoning that is completelyunfamiliar to most college students. Some of the typical mental steps can beillustrated in the following example.

Section 1.4 Order-of-Magnitude Estimates

48

Example: Cost of transporting tomatoesQuestion: Roughly what percentage of the price of a tomato comesfrom the cost of transporting it in a truck?

The following incorrect solution illustrates one of the main ways you can gowrong in order-of-magnitude estimates.

Incorrect solution: Let’s say the trucker needs to make a $400profit on the trip. Taking into account her benefits, the cost of gas,and maintenance and payments on the truck, let’s say the total costis more like $2000. I’d guess about 5000 tomatoes would fit in theback of the truck, so the extra cost per tomato is 40 cents. Thatmeans the cost of transporting one tomato is comparable to the costof the tomato itself. Transportation really adds a lot to the cost ofproduce, I guess.

The problem is that the human brain is not very good at estimatingarea or volume, so it turns out the estimate of 5000 tomatoes fitting in thetruck is way off. That’s why people have a hard time at those contests whereyou are supposed to estimate the number of jellybeans in a big jar. Anotherexample is that most people think their families use about 10 gallons ofwater per day, but in reality the average is about 300 gallons per day. Whenestimating area or volume, you are much better off estimating lineardimensions, and computing volume from the linear dimensions. Here’s abetter solution:

Better solution: As in the previous solution, say the cost of the tripis $2000. The dimensions of the bin are probably 4 m x 2 m x 1 m,for a volume of 8 m3. Since the whole thing is just an order-of-magnitude estimate, let’s round that off to the nearest power of ten,10 m3. The shape of a tomato is complicated, and I don’t know anyformula for the volume of a tomato shape, but since this is just anestimate, let’s pretend that a tomato is a cube, 0.05 m x 0.05 m x0.05, for a volume of 1.25x10-4 m3. Since this is just a roughestimate, let’s round that to 10-4 m3. We can find the total numberof tomatoes by dividing the volume of the bin by the volume of onetomato: 10 m3 / 10-4 m3 = 105 tomatoes. The transportation costper tomato is $2000/105 tomatoes=$0.02/tomato. That means thattransportation really doesn’t contribute very much to the cost of atomato.

Approximating the shape of a tomato as a cube is an example of anothergeneral strategy for making order-of-magnitude estimates. A similar situa-tion would occur if you were trying to estimate how many m2 of leathercould be produced from a herd of ten thousand cattle. There is no point intrying to take into account the shape of the cows’ bodies. A reasonable planof attack might be to consider a spherical cow. Probably a cow has roughlythe same surface area as a sphere with a radius of about 1 m, which wouldbe 4π(1 m)2. Using the well-known facts that pi equals three, and fourtimes three equals about ten, we can guess that a cow has a surface area ofabout 10 m2, so the herd as a whole might yield 105 m2 of leather.

1 m

Chapter 1 Scaling and Order-of-Magnitude Estimates

49

The following list summarizes the strategies for getting a good order-of-magnitude estimate.

(1) Don’t even attempt more than one significant figure of precision.(2) Don’t guess area or volume directly. Guess linear dimensions and

get area or volume from them.(3) When dealing with areas or volumes of objects with complex

shapes, idealize them as if they were some simpler shape, a cube or asphere, for example.

(4) Check your final answer to see if it is reasonable. If you estimatethat a herd of ten thousand cattle would yield 0.01 m2 of leather,then you have probably made a mistake with conversion factorssomewhere.

Section 1.4 Order-of-Magnitude Estimates

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SummaryNotation

∝ ....................................... is proportional to~ ........................................ on the order of, is on the order of

SummaryNature behaves differently on large and small scales. Galileo showed that this results fundamentally from

the way area and volume scale. Area scales as the second power of length, A∝L2, while volume scales aslength to the third power, V∝L3.

An order of magnitude estimate is one in which we do not attempt or expect an exact answer. The mainreason why the uninitiated have trouble with order-of-magnitude estimates is that the human brain does notintuitively make accurate estimates of area and volume. Estimates of area and volume should be approachedby first estimating linear dimensions, which one’s brain has a feel for.

Homework Problems1 . How many cubic inches are there in a cubic foot? The answer is not12.

2. Assume a dog's brain is twice is great in diameter as a cat's, but eachanimal's brain cells are the same size and their brains are the same shape. Inaddition to being a far better companion and much nicer to come home to,how many times more brain cells does a dog have than a cat? The answer isnot 2.

3 . The population density of Los Angeles is about 4000 people/km2.That of San Francisco is about 6000 people/km2. How many times fartheraway is the average person's nearest neighbor in LA than in San Francisco?The answer is not 1.5.

4. A hunting dog's nose has about 10 square inches of active surface. Howis this possible, since the dog's nose is only about 1 in x 1 in x 1 in = 1 in3?After all, 10 is greater than 1, so how can it fit?

5. Estimate the number of blades of grass on a football field.

6. (a) Estimate the number of times a person's heart beats in a lifetime. (b)The universe is approximately 1010 years old. Which is greater, the ratio ofthe length of a person's life to the duration of a heartbeat, or the ratio of theage of the universe to a person's lifetime?

7. Suppose someone built a gigantic apartment building, measuring 10 kmx 10 km at the base. Estimate how tall the building would have to be tohave space in it for the entire world's population to live.

8. A hamburger chain advertises that it has sold 10 billion Bongo Burgers.Estimate the total mass of hay required to feed the cows used to make theburgers.

9. Estimate the volume of a human body, in cm3.

10 S. How many cm2 is 1 mm2?

S A solution is given in the back of the book. « A difficult problem. A computerized answer check is available. ∫ A problem that requires calculus.

Chapter 1 Scaling and Order-of-Magnitude Estimates

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Motion in OneDimension

I didn’t learn until I was nearly through with college that I could understand a book much better if Imentally outlined it for myself before I actually began reading. It’s a technique that warns mybrain to get little cerebral file folders ready for the different topics I’m going to learn, and as I’mreading it allows me to say to myself, “Oh, the reason they’re talking about this now is becausethey’re preparing for this other thing that comes later,” or “I don’t need to sweat the details of thisidea now, because they’re going to explain it in more detail later on.”

At this point, you’re about to dive in to the main subjects of this book, which are force and motion.The concepts you’re going to learn break down into the following three areas:

kinematics — how to describe motion numericallydynamics — how force affects motionvectors — a mathematical way of handling the three-dimensional nature of force and motion

Roughly speaking, that’s the order in which we’ll cover these three areas, but the earlier chaptersdo contain quite a bit of preparation for the later topics. For instance, even before the presentpoint in the book you’ve learned about the Newton, a unit of force. The discussion of forceproperly belongs to dynamics, which we aren’t tackling head-on for a few more chapters, but I’vefound that when I teach kinematics it helps to be able to refer to forces now and then to showwhy it makes sense to define certain kinematical concepts. And although I don’t explicitlyintroduce vectors until ch. 8, the groundwork is being laid for them in earlier chapters.

Here’s a roadmap to the rest of the book:

preliminarieschapters 0-1

motion in onedimension

chapters 2-6

motion in threedimensions

chapters 7-9

gravity:chapter 10

kinematics dynamics vectors

52

Rotation.

Simultaneous rotation andmotion through space.

One person might say that thetipping chair was only rotatingin a circle about its point ofcontact with the floor, butanother could describe it ashaving both rotation andmotion through space.

2 Velocity and RelativeMotion

2.1 Types of MotionIf you had to think consciously in order to move your body, you would

be severely disabled. Even walking, which we consider to be no great feat,requires an intricate series of motions that your cerebrum would be utterlyincapable of coordinating. The task of putting one foot in front of theother is controlled by the more primitive parts of your brain, the ones thathave not changed much since the mammals and reptiles went their separateevolutionary ways. The thinking part of your brain limits itself to generaldirectives such as “walk faster,” or “don’t step on her toes,” rather thanmicromanaging every contraction and relaxation of the hundred or somuscles of your hips, legs, and feet.

Physics is all about the conscious understanding of motion, but we’reobviously not immediately prepared to understand the most complicatedtypes of motion. Instead, we’ll use the divide-and-conquer technique.We’ll first classify the various types of motion, and then begin our campaignwith an attack on the simplest cases. To make it clear what we are and arenot ready to consider, we need to examine and define carefully what typesof motion can exist.

Rigid-body motion distinguished from motion that changesan object’s shape

Nobody, with the possible exception of Fred Astaire, can simply glideforward without bending their joints. Walking is thus an example in whichthere is both a general motion of the whole object and a change in the shapeof the object. Another example is the motion of a jiggling water balloon asit flies through the air. We are not presently attempting a mathematicaldescription of the way in which the shape of an object changes. Motionwithout a change in shape is called rigid-body motion. (The word “body”is often used in physics as a synonym for “object.”)

Center-of-mass motion as opposed to rotationA ballerina leaps into the air and spins around once before landing. We

feel intuitively that her rigid-body motion while her feet are off the groundconsists of two kinds of motion going on simultaneously: a rotation and amotion of her body as a whole through space, along an arc. It is notimmediately obvious, however, what is the most useful way to define thedistinction between rotation and motion through space. Imagine that youattempt to balance a chair and it falls over. One person might say that theonly motion was a rotation about the chair’s point of contact with the floor,but another might say that there was both rotation and motion down andto the side.

Chapter 2 Velocity and Relative Motion

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It turns out that there is one particularly natural and useful way to makea clear definition, but it requires a brief digression. Every object has abalance point, referred to in physics as the center of mass. For a two-dimensional object such as a cardboard cutout, the center of mass is thepoint at which you could hang the object from a string and make it balance.In the case of the ballerina (who is likely to be three-dimensional unless herdiet is particularly severe), it might be a point either inside or outside herbody, depending on how she holds her arms. Even if it is not practical toattach a string to the balance point itself, the center of mass can be definedas shown in the figure on the left.

Why is the center of mass concept relevant to the question of classifyingrotational motion as opposed to motion through space? As illustrated inthe figure above, it turns out that the motion of an object’s center of mass isnearly always far simpler than the motion of any other part of the object.The ballerina’s body is a large object with a complex shape. We mightexpect that her motion would be much more complicated that the motionof a small, simply-shaped object, say a marble, thrown up at the same angleas the angle at which she leapt. But it turns out that the motion of theballerina’s center of mass is exactly the same as the motion of the marble.That is, the motion of the center of mass is the same as the motion theballerina would have if all her mass was concentrated at a point. By restrict-ing our attention to the motion of the center of mass, we can thereforesimplify things greatly.

The motion of an object’s center ofmass is usually much simpler than the

motion of any other point on it.

No matter what point you hang thepear from, the string lines up with thepear’s center of mass. The center ofmass can therefore be defined as theintersection of all the lines made byhanging the pear in this way. Note thatthe X in the figure should not beinterpreted as implying that the centerof mass is on the surface — it isactually inside the pear.

We can now replace the ambiguous idea of “motion as a whole throughspace” with the more useful and better defined concept of “center-of-massmotion.” The motion of any rigid body can be cleanly split into rotationand center-of-mass motion. By this definition, the tipping chair does haveboth rotational and center-of-mass motion. Concentrating on the center of

center of mass

The leaping dancer’s motion iscomplicated, but the motion of hercenter of mass is simple.

The same leaping dancer, viewed fromabove. Her center of mass traces astraight line, but a point away from hercenter of mass, such as her elbow,traces the much more complicatedpath shown by the dots.

Section 2.1 Types of Motion

54

mass motion allows us to make a simplified model of the motion, as if acomplicated object like a human body was just a marble or a point-likeparticle. Science really never deals with reality; it deals with models ofreality.

Note that the word “center” in “center of mass” is not meant to implythat the center of mass must lie at the geometrical center of an object. A carwheel that has not been balanced properly has a center of mass that doesnot coincide with its geometrical center. An object such as the human bodydoes not even have an obvious geometrical center.

It can be helpful to think of the center of mass as the average location ofall the mass in the object. With this interpretation, we can see for examplethat raising your arms above your head raises your center of mass, since the

fixed point on dancer's body

center of mass

A fixed point on the dancer’s bodyfollows a trajectory that is flatter thanwhat we expect, creating an illusionof flight.

center of mass

geometricalcenter

An improperly balanced wheel has acenter of mass that is not at itsgeometric center. When you get a newtire, the mechanic clamps little weightsto the rim to balance the wheel.

higher position of the arms’ mass raises the average.

Ballerinas and professional basketball players can create an illusion offlying horizontally through the air because our brains intuitively expectthem to have rigid-body motion, but the body does not stay rigid whileexecuting a grand jete or a slam dunk. The legs are low at the beginningand end of the jump, but come up higher at the middle. Regardless of whatthe limbs do, the center of mass will follow the same arc, but the lowposition of the legs at the beginning and end means that the torso is highercompared to the center of mass, while in the middle of the jump it is lowercompared to the center of mass. Our eye follows the motion of the torsoand tries to interpret it as the center-of-mass motion of a rigid body. Butsince the torso follows a path that is flatter than we expect, this attemptedinterpretation fails, and we experience an illusion that the person is flyinghorizontally. Another interesting example from the sports world is the highjump, in which the jumper’s curved body passes over the bar, but the centerof mass passes under the bar! Here the jumper lowers his legs and upperbody at the peak of the jump in order to bring his waist higher compared tothe center of mass.

Later in this course, we’ll find that there are more fundamental reasons(based on Newton’s laws of motion) why the center of mass behaves in sucha simple way compared to the other parts of an object. We’re also postpon-ing any discussion of numerical methods for finding an object’s center ofmass. Until later in the course, we will only deal with the motion of objects’

The high-jumper’s body passes overthe bar, but his center of mass passesunder it.Photo by Dunia Young.

centerof mass

Chapter 2 Velocity and Relative Motion

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centers of mass.

Center-of-mass motion in one dimensionIn addition to restricting our study of motion to center-of-mass motion,

we will begin by considering only cases in which the center of mass movesalong a straight line. This will include cases such as objects falling straightdown, or a car that speeds up and slows down but does not turn.

Note that even though we are not explicitly studying the more complexaspects of motion, we can still analyze the center-of-mass motion whileignoring other types of motion that might be occurring simultaneously .For instance, if a cat is falling out of a tree and is initially upside-down, itgoes through a series of contortions that bring its feet under it. This isdefinitely not an example of rigid-body motion, but we can still analyze themotion of the cat’s center of mass just as we would for a dropping rock.

Self-CheckConsider a person running, a person pedaling on a bicycle, a person coastingon a bicycle, and a person coasting on ice skates. In which cases is thecenter-of-mass motion one-dimensional? Which cases are examples of rigid-body motion?

2.2 Describing Distance and TimeCenter-of-mass motion in one dimension is particularly easy to deal

with because all the information about it can be encapsulated in twovariables: x, the position of the center of mass relative to the origin, and t,which measures a point in time. For instance, if someone supplied you witha sufficiently detailed table of x and t values, you would know pretty muchall there was to know about the motion of the object’s center of mass.

A Point in Time as Opposed to DurationIn ordinary speech, we use the word “time” in two different senses,

which are to be distinguished in physics. It can be used, as in “a short time”or “our time here on earth,” to mean a length or duration of time, or it canbe used to indicate a clock reading, as in “I didn’t know what time it was,”or “now’s the time.” In symbols, t is ordinarily used to mean a point intime, while ∆t signifies an interval or duration in time. The capital Greekletter delta, ∆, means “the change in...,” i.e. a duration in time is the changeor difference between one clock reading and another. The notation ∆t doesnot signify the product of two numbers, ∆ and t, but rather one singlenumber, ∆t. If a matinee begins at a point in time t=1 o’clock and ends att=3 o’clock, the duration of the movie was the change in t,

∆t = 3 hours - 1 hour = 2 hours .

To avoid the use of negative numbers for ∆t, we write the clock reading“after” to the left of the minus sign, and the clock reading “before” to theright of the minus sign. A more specific definition of the delta notation istherefore that delta stands for “after minus before.”

Even though our definition of the delta notation guarantees that ∆t ispositive, there is no reason why t can’t be negative. If t could not be nega-tive, what would have happened one second before t=0? That doesn’t mean

Coasting on a bike and coasting on skates give one-dimensional center-of-mass motion, but running and pedalingrequire moving body parts up and down, which makes the center of mass move up and down. The only example ofrigid-body motion is coasting on skates. (Coasting on a bike is not rigid-body motion, because the wheels twist.)

Section 2.2 Describing Distance and Time

56

that time “goes backward” in the sense that adults can shrink into infantsand retreat into the womb. It just means that we have to pick a referencepoint and call it t=0, and then times before that are represented by negativevalues of t.

Although a point in time can be thought of as a clock reading, it isusually a good idea to avoid doing computations with expressions such as“2:35” that are combinations of hours and minutes. Times can instead beexpressed entirely in terms of a single unit, such as hours. Fractions of anhour can be represented by decimals rather than minutes, and similarly if aproblem is being worked in terms of minutes, decimals can be used insteadof seconds.

Self-Check

Of the following phrases, which refer to points in time, which refer to timeintervals, and which refer to time in the abstract rather than as a measurablenumber?

(a) “The time has come.”(b) “Time waits for no man.”(c) “The whole time, he had spit on his chin.”

Position as Opposed to Change in PositionAs with time, a distinction should be made between a point in space,

symbolized as a coordinate x, and a change in position, symbolized as ∆x.

As with t, x can be negative. If a train is moving down the tracks, notonly do you have the freedom to choose any point along the tracks and callit x=0, but it’s also up to you to decide which side of the x=0 point ispositive x and which side is negative x.

Since we’ve defined the delta notation to mean “after minus before,” itis possible that ∆x will be negative, unlike ∆t which is guaranteed to bepositive. Suppose we are describing the motion of a train on tracks linkingTucson and Chicago. As shown in the figure, it is entirely up to you todecide which way is positive.

Enid

x=0

Chicago

Tucson

x>0

x<0

∆x>0

Joplin

Enid

x=0

Chicago

Tucson

x<0

x>0

∆x<0

Joplin

Two equally valid ways of describing the motion of a train from Tucson toChicago. In the first example, the train has a positive ∆x as it goes from Enid toJoplin. In the second example, the same train going forward in the samedirection has a negative ∆x.

(a) a point in time; (b) time in the abstract sense; (c) a time interval

Chapter 2 Velocity and Relative Motion

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Note that in addition to x and ∆x, there is a third quantity we coulddefine, which would be like an odometer reading, or actual distancetraveled. If you drive 10 miles, make a U-turn, and drive back 10 miles,then your ∆x is zero, but your car’s odometer reading has increased by 20miles. However important the odometer reading is to car owners and usedcar dealers, it is not very important in physics, and there is not even astandard name or notation for it. The change in position, ∆x, is more usefulbecause it is so much easier to calculate: to compute ∆x, we only need toknow the beginning and ending positions of the object, not all the informa-tion about how it got from one position to the other.

Self-Check

A ball hits the floor, bounces to a height of one meter, falls, and hits the flooragain. Is the ∆x between the two impacts equal to zero, one, or two meters?

Frames of referenceThe example above shows that there are two arbitrary choices you have

to make in order to define a position variable, x. You have to decide whereto put x=0, and also which direction will be positive. This is referred to aschoosing a coordinate system or choosing a frame of reference. (The two termsare nearly synonymous, but the first focuses more on the actual x variable,while the second is more of a general way of referring to one’s point ofview.) As long as you are consistent, any frame is equally valid. You justdon’t want to change coordinate systems in the middle of a calculation.

Have you ever been sitting in a train in a station when suddenly younotice that the station is moving backward? Most people would describe thesituation by saying that you just failed to notice that the train was moving— it only seemed like the station was moving. But this shows that there isyet a third arbitrary choice that goes into choosing a coordinate system:valid frames of reference can differ from each other by moving relative toone another. It might seem strange that anyone would bother with acoordinate system that was moving relative to the earth, but for instance theframe of reference moving along with a train might be far more convenientfor describing things happening inside the train.

Zero, because the “after” and “before” values of x are the same.

Section 2.2 Describing Distance and Time

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2.3 Graphs of Motion; Velocity.Motion with constant velocity

In example (a), an object is moving at constant speed in one direction.We can tell this because every two seconds, its position changes by fivemeters.

In algebra notation, we’d say that the graph of x vs. t shows the samechange in position, ∆x=5.0 m, over each interval of ∆t=2.0 s. The object’svelocity or speed is obtained by calculating v=∆x/∆t=(5.0 m)/(2.0 s)=2.5 m/s. In graphical terms, the velocity can be interpreted as the slope of the line.Since the graph is a straight line, it wouldn’t have mattered if we’d taken alonger time interval and calculated v=∆x/∆t=(10.0 m)/(4.0 s). The answerwould still have been the same, 2.5 m/s.

Note that when we divide a number that has units of meters by anothernumber that has units of seconds, we get units of meters per second, whichcan be written m/s. This is another case where we treat units as if they werealgebra symbols, even though they’re not.

In example (b), the object is moving in the opposite direction: as timeprogresses, its x coordinate decreases. Recalling the definition of the ∆notation as “after minus before,” we find that ∆t is still positive, but ∆xmust be negative. The slope of the line is therefore negative, and we saythat the object has a negative velocity, v=∆x/∆t=(-5.0 m)/(2.0 s)=-2.5 m/s.We’ve already seen that the plus and minus signs of ∆x values have theinterpretation of telling us which direction the object moved. Since ∆t isalways positive, dividing by ∆t doesn’t change the plus or minus sign, andthe plus and minus signs of velocities are to be interpreted in the same way.In graphical terms, a positive slope characterizes a line that goes up as we goto the right, and a negative slope tells us that the line went down as we wentto the right.

Motion with changing velocityNow what about a graph like example (c)? This might be a graph of a

car’s motion as the driver cruises down the freeway, then slows down to lookat a car crash by the side of the road, and then speeds up again, disap-pointed that there is nothing dramatic going on such as flames or babiestrapped in their car seats. (Note that we are still talking about one-dimen-sional motion. Just because the graph is curvy doesn’t mean that the car’spath is curvy. The graph is not like a map, and the horizontal direction ofthe graph represents the passing of time, not distance.)

Example (c) is similar to example (a) in that the object moves a total of25.0 m in a period of 10.0 s, but it is no longer true that it makes the sameamount of progress every second. There is no way to characterize the entiregraph by a certain velocity or slope, because the velocity is different at everymoment. It would be incorrect to say that because the car covered 25.0 min 10.0 s, its velocity was 2.5 m/s . It moved faster than that at the begin-ning and end, but slower in the middle. There may have been certaininstants at which the car was indeed going 2.5 m/s, but the speedometerswept past that value without “sticking,” just as it swung through variousother values of speed. (I definitely want my next car to have a speedometercalibrated in m/s and showing both negative and positive values.)

5

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0 2 4 6 8 10t (s)

x(m)

(c) Motion with changing velocity.

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x(m)

∆t

∆x

(b) Motion that decreases x isrepresented with negative values of ∆xand v.

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x(m)

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∆x

(a) Motion with constant velocity.

Chapter 2 Velocity and Relative Motion

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We assume that our speedometer tells us what is happening to the speedof our car at every instant, but how can we define speed mathematically in acase like this? We can’t just define it as the slope of the curvy graph, becausea curve doesn’t have a single well-defined slope as does a line. A mathemati-cal definition that corresponded to the speedometer reading would have tobe one that attached a different velocity value to a single point on the curve,i.e. a single instant in time, rather than to the entire graph. If we wish todefine the speed at one instant such as the one marked with a dot, the bestway to proceed is illustrated in (d), where we have drawn the line throughthat point called the tangent line, the line that “hugs the curve.” We canthen adopt the following definition of velocity:

definition of velocityThe velocity of an object at any given moment is the slope ofthe tangent line through the relevant point on its x-t graph.

One interpretation of this definition is that the velocity tells us how manymeters the object would have traveled in one second, if it had continuedmoving at the same speed for at least one second. To some people thegraphical nature of this definition seems “inaccurate” or “not mathemati-

cal.” The equation v=∆x/∆t by itself, however, is only valid if the velocity isconstant, and so cannot serve as a general definition.

ExampleQuestion : What is the velocity at the point shown with a dot onthe graph?Solution : First we draw the tangent line through that point. Tofind the slope of the tangent line, we need to pick two points onit. Theoretically, the slope should come out the same regardlessof which two points we picked, but in practical terms we’ll be ableto measure more accurately if we pick two points fairly far apart,such as the two white diamonds. To save work, we pick pointsthat are directly above labeled points on the t axis, so that ∆t=4.0s is easy to read off. One diamond lines up with x≈17.5 m, theother with x≈26.5 m, so ∆x=9.0 m. The velocity is ∆x/∆t=2.2 m/s.

Conventions about graphingThe placement of t on the horizontal axis and x on the upright axis may

seem like an arbitrary convention, or may even have disturbed you, sinceyour algebra teacher always told you that x goes on the horizontal axis and ygoes on the upright axis. There is a reason for doing it this way, however.In example (e), we have an object that reverses its direction of motion twice.It can only be in one place at any given time, but there can be more thanone time when it is at a given place. For instance, this object passedthrough x=17 m on three separate occasions, but there is no way it couldhave been in more than one place at t=5.0 s. Resurrecting some terminol-ogy you learned in your trigonometry course, we say that x is a function oft, but t is not a function of x. In situations such as this, there is a usefulconvention that the graph should be oriented so that any vertical line passesthrough the curve at only one point. Putting the x axis across the page andt upright would have violated this convention. To people who are used tointerpreting graphs, a graph that violates this convention is as annoying as

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0 2 4 6 8 10t (s)

x(m)

Example: finding the velocity at thepoint indicated with the dot.

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x(m)

∆t=4.0 s

∆x

5

10

15

20

25

30

0 2 4 6 8 10t (s)

x(m)

∆t

∆x

(d) The velocity at any given momentis defined as the slope of the tangentline through the relevant point on thegraph.

Section 2.3 Graphs of Motion; Velocity.

(e) Reversing the direction ofmotion.

60

fingernails scratching on a chalkboard. We say that this is a graph of “xversus t.” If the axes were the other way around, it would be a graph of “tversus x.” I remember the “versus” terminology by visualizing the labels onthe x and t axes and remembering that when you read, you go from left toright and from top to bottom.

Discussion questions

A. An ant walks forward, pauses, then runs quickly ahead. It then suddenlyreverses direction and walks slowly back the way it came. Which graph couldrepresent its motion?

x x x

x x x

t t t

t t t

1 2 3

4 5 6

B. The figure shows a sequence of positions for two racing tractors. Comparethe tractors’ velocities as the race progresses.

t=0 s t=1 s t=2 s t=3 s t=4 s t=5 s t=6 s t=7 s

t=0 s t=1 s t=2 s t=3 s t=4 s t=5 s t=6 s t=7 s

C. If an object had a straight-line motion graph with ∆x=0 and ∆t≠0, what wouldbe true about its velocity? What would this look like on a graph? What about∆t=0 and ∆x≠0?D. If an object has a wavy motion graph like the one in example (e) on theprevious page, which are the points at which the object reverses its direction?What is true about the object’s velocity at these points?E. Discuss anything unusual about the following three graphs.

x

t

x

t

x

t

1 2 3

Chapter 2 Velocity and Relative Motion

61

F. I have been using the term “velocity” and avoiding the more commonEnglish word “speed,” because some introductory physics texts define them tomean different things. They use the word “speed,” and the symbol “s” to meanthe absolute value of the velocity, s=|v|. Although I have thrown in my lot withthe minority of books that don’t emphasize this distinction in technicalvocabulary, there are clearly two different concepts here. Can you think of anexample of a graph of x vs. t in which the object has constant speed, but notconstant velocity?G. In the graph on the left, describe how the object’s velocity changes.H. Two physicists duck out of a boring scientific conference to go get beer. Onthe way to the bar, they witness an accident in which a pedestrian is injured bya hit-and-run driver. A criminal trial results, and they must testify. In hertestimony, Dr. Transverz Waive says, “The car was moving along pretty fast, I’dsay the velocity was +40 mi/hr. They saw the old lady too late, and eventhough they slammed on the brakes they still hit her before they stopped.Then they made a U turn and headed off at a velocity of about -20 mi/hr, I’dsay.” Dr. Longitud N.L. Vibrasheun says, “He was really going too fast, maybehis velocity was -35 or -40 mi/hr. After he hit Mrs. Hapless, he turned aroundand left at a velocity of, oh, I’d guess maybe +20 or +25 mi/hr.” Is theirtestimony contradictory? Explain.

Discussion question G.

x

t

Section 2.3 Graphs of Motion; Velocity.

62

2.4 The Principle of InertiaPhysical effects relate only to a change in velocity

Consider two statements that were at one time made with the utmostseriousness:

People like Galileo and Copernicus who say the earth is rotating must becrazy. We know the earth can’t be moving. Why, if the earth was reallyturning once every day, then our whole city would have to be movinghundreds of leagues in an hour. That’s impossible! Buildings would shakeon their foundations. Gale-force winds would knock us over. Trees would falldown. The Mediterranean would come sweeping across the east coasts ofSpain and Italy. And furthermore, what force would be making the worldturn?

All this talk of passenger trains moving at forty miles an hour is sheerhogwash! At that speed, the air in a passenger compartment would all beforced against the back wall. People in the front of the car would suffocate,and people at the back would die because in such concentrated air, theywouldn’t be able to expel a breath.

Some of the effects predicted in the first quote are clearly just based ona lack of experience with rapid motion that is smooth and free of vibration.But there is a deeper principle involved. In each case, the speaker is assum-ing that the mere fact of motion must have dramatic physical effects. Moresubtly, they also believe that a force is needed to keep an object in motion:the first person thinks a force would be needed to maintain the earth’srotation, and the second apparently thinks of the rear wall as pushing onthe air to keep it moving.

Common modern knowledge and experience tell us that these people’spredictions must have somehow been based on incorrect reasoning, but it isnot immediately obvious where the fundamental flaw lies. It’s one of thosethings a four-year-old could infuriate you by demanding a clear explanationof. One way of getting at the fundamental principle involved is to considerhow the modern concept of the universe differs from the popular concep-tion at the time of the Italian Renaissance. To us, the word “earth” implies aplanet, one of the nine planets of our solar system, a small ball of rock anddirt that is of no significance to anyone in the universe except for membersof our species, who happen to live on it. To Galileo’s contemporaries,however, the earth was the biggest, most solid, most important thing in allof creation, not to be compared with the wandering lights in the sky knownas planets. To us, the earth is just another object, and when we talk looselyabout “how fast” an object such as a car “is going,” we really mean the car-object’s velocity relative to the earth-object.

Motion is relativeAccording to our modern world-view, it really isn’t that reasonable to

expect that a special force should be required to make the air in the trainhave a certain velocity relative to our planet. After all, the “moving” air inthe “moving” train might just happen to have zero velocity relative to someother planet we don’t even know about. Aristotle claimed that things“naturally” wanted to be at rest, lying on the surface of the earth. Butexperiment after experiment has shown that there is really nothing so

There is nothing special about motionor lack of motion relative to the planet

earth.

Chapter 2 Velocity and Relative Motion

63

special about being at rest relative to the earth. For instance, if a mattressfalls out of the back of a truck on the freeway, the reason it rapidly comes torest with respect to the planet is simply because of friction forces exerted bythe asphalt, which happens to be attached to the planet.

Galileo’s insights are summarized as follows:

The Principle of InertiaNo force is required to maintain motion with constant velocityin a straight line, and absolute motion does not cause anyobservable physical effects.

There are many examples of situations that seem to disprove theprinciple of inertia, but these all result from forgetting that friction is aforce. For instance, it seems that a force is needed to keep a sailboat inmotion. If the wind stops, the sailboat stops too. But the wind’s force is notthe only force on the boat; there is also a frictional force from the water. Ifthe sailboat is cruising and the wind suddenly disappears, the backwardfrictional force still exists, and since it is no longer being counteracted bythe wind’s forward force, the boat stops. To disprove the principle of inertia,we would have to find an example where a moving object slowed downeven though no forces whatsoever were acting on it.

Section 2.4 The Principle of Inertia

(a) (b) (c)

(d) (e) (f)

This Air Force doctor volunteered to ride a rocket sled as a medical experiment. The obvious effects onhis head and face are not because of the sled's speed but because of its rapid changes in speed: increasingin (b) and (c), and decreasing in (e) and (f).In (d) his speed is greatest, but because his speed is notincreasing or decreasing very much at this moment, there is little effect on him.

64

Self-CheckWhat is incorrect about the following supposed counterexamples to theprinciple of inertia?

(1) When astronauts blast off in a rocket, their huge velocity does causea physical effect on their bodies — they get pressed back into theirseats, the flesh on their faces gets distorted, and they have a hard timelifting their arms.(2) When you’re driving in a convertible with the top down, the wind inyour face is an observable physical effect of your absolute motion.

Discussion questionsA. A passenger on a cruise ship finds, while the ship is docked, that he canleap off of the upper deck and just barely make it into the pool on the lowerdeck. If the ship leaves dock and is cruising rapidly, will this adrenaline junkiestill be able to make it?B. You are a passenger in the open basket hanging under a helium balloon.The balloon is being carried along by the wind at a constant velocity. If you areholding a flag in your hand, will the flag wave? If so, which way? [Based on aquestion from PSSC Physics.]C. Aristotle stated that all objects naturally wanted to come to rest, with theunspoken implication that “rest” would be interpreted relative to the surface ofthe earth. Suppose we could transport Aristotle to the moon, put him in aspace suit, and kick him out the door of the spaceship and into the lunarlandscape. What would he expect his fate to be in this situation? If intelligentcreatures inhabited the moon, and one of them independently came up withthe equivalent of Aristotelian physics, what would they think about objectscoming to rest?

pool

ship's directionof motion

Discussion question A. Discussion question B.

(1) The effect only occurs during blastoff, when their velocity is changing. Once the rocket engines stop firing, theirvelocity stops changing, and they no longer feel any effect. (2) It is only an observable effect of your motion relativeto the air.

Chapter 2 Velocity and Relative Motion

65

2.5 Addition of VelocitiesAddition of velocities to describe relative motion

Since absolute motion cannot be unambiguously measured, the onlyway to describe motion unambiguously is to describe the motion of oneobject relative to another. Symbolically, we can write vPQ for the velocity ofobject P relative to object Q.

Velocities measured with respect to different reference points can becompared by addition. In the figure below, the ball’s velocity relative to thecouch equals the ball’s velocity relative to the truck plus the truck’s velocityrelative to the couch:

v BC = v BT + v TC

= 5 cm/s + 10 cm/sThe same equation can be used for any combination of three objects, justby substituting the relevant subscripts for B, T, and C. Just remember towrite the equation so that the velocities being added have the same sub-script twice in a row. In this example, if you read off the subscripts goingfrom left to right, you get BC...=...BTTC. The fact that the two “inside”subscripts on the right are the same means that the equation has been set upcorrectly. Notice how subscripts on the left look just like the subscripts onthe right, but with the two T’s eliminated.

Relative velocitiesadd together.

These two highly competent physicists disagree on absolute velocities, but they would agree on relativevelocities. Purple Dino considers the couch to be at rest, while Green Dino thinks of the truck as being at rest.They agree, however, that the truck’s velocity relative to the couch is vTC=10 cm/s, the ball’s velocity relativeto the truck is vBT=5 cm/s, and the ball’s velocity relative to the couch is vBC=vBT+vTC=15 cm/s.

In one second, Green Dino and thetruck both moved forward 10 cm, so their

velocity was 10 cm/s. The ball movedforward 15 cm, so it had v=15 cm/s.

Purple Dino and the couch bothmoved backward 10 cm in 1 s, so they

had a velocity of -10 cm/s. During the sameperiod of time, the ball got 5 cm closer to

me, so it was going +5 cm/s.

Section 2.5 Addition of Velocities

66

Negative velocities in relative motionMy discussion of how to interpret positive and negative signs of velocity

may have left you wondering why we should bother. Why not just makevelocity positive by definition? The original reason why negative numberswere invented was that bookkeepers decided it would be convenient to usethe negative number concept for payments to distinguish them fromreceipts. It was just plain easier than writing receipts in black and paymentsin red ink. After adding up your month’s positive receipts and negativepayments, you either got a positive number, indicating profit, or a negativenumber, showing a loss. You could then show the that total with a high-tech “+” or “-” sign, instead of looking around for the appropriate bottle ofink.

Nowadays we use positive and negative numbers for all kinds of things,but in every case the point is that it makes sense to add and subtract thosethings according to the rules you learned in grade school, such as “minus aminus makes a plus, why this is true we need not discuss.” Adding velocitieshas the significance of comparing relative motion, and with this interpreta-tion negative and positive velocities can used within a consistent framework.For example, the truck’s velocity relative to the couch equals the truck’svelocity relative to the ball plus the ball’s velocity relative to the couch:

v TC = v TB + v BC

= –5 cm/s + 15 cm/s= 10 cm/s

If we didn’t have the technology of negative numbers, we would havehad to remember a complicated set of rules for adding velocities: (1) if thetwo objects are both moving forward, you add, (2) if one is moving forwardand one is moving backward, you subtract, but (3) if they’re both movingbackward, you add. What a pain that would have been.

Discussion questions

A. Interpret the general rule v AB =–v BA in words.

B. If we have a specific situation where v AB+ v BC = 0 , what is going on?

If you consistently label velocities as positiveor negative depending on their directions,then adding velocities will also give signs

that consistently relate to direction.

Chapter 2 Velocity and Relative Motion

67

2.6 Graphs of Velocity Versus TimeSince changes in velocity play such a prominent role in physics, we need

a better way to look at changes in velocity than by laboriously drawingtangent lines on x-versus-t graphs. A good method is to draw a graph ofvelocity versus time. The examples on the left show the x-t and v-t graphsthat might be produced by a car starting from a traffic light, speeding up,cruising for a while at constant speed, and finally slowing down for a stopsign. If you have an air freshener hanging from your rear-view mirror, thenyou will see an effect on the air freshener during the beginning and endingperiods when the velocity is changing, but it will not be tilted during theperiod of constant velocity represented by the flat plateau in the middle ofthe v-t graph.

Students often mix up the things being represented on these two typesof graphs. For instance, many students looking at the top graph say thatthe car is speeding up the whole time, since “the graph is becoming greater.”What is getting greater throughout the graph is x, not v.

Similarly, many students would look at the bottom graph and think itshowed the car backing up, because “it’s going backwards at the end.” Butwhat is decreasing at the end is v, not x. Having both the x-t and v-t graphsin front of you like this is often convenient, because one graph may beeasier to interpret than the other for a particular purpose. Stacking themlike this means that corresponding points on the two graphs’ time axes arelined up with each other vertically. However, one thing that is a littlecounterintuitive about the arrangement is that in a situation like this oneinvolving a car, one is tempted to visualize the landscape stretching alongthe horizontal axis of one of the graphs. The horizontal axes, however,represent time, not position. The correct way to visualize the landscape isby mentally rotating the horizon 90 degrees counterclockwise and imagin-ing it stretching along the upright axis of the x-t graph, which is the onlyaxis that represents different positions in space.

2.7 ∫ Applications of CalculusThe integral symbol, ∫, in the heading for this section indicates that it is

meant to be read by students in calculus-based physics. Students in analgebra-based physics course should skip these sections. The calculus-relatedsections in this book are meant to be usable by students who are takingcalculus concurrently, so at this early point in the physics course I do notassume you know any calculus yet. This section is therefore not much morethan a quick preview of calculus, to help you relate what you’re learning inthe two courses.

Newton was the first person to figure out the tangent-line definition ofvelocity for cases where the x-t graph is nonlinear. Before Newton, nobodyhad conceptualized the description of motion in terms of x-t and v-t graphs.In addition to the graphical techniques discussed in this chapter, Newtonalso invented a set of symbolic techniques called calculus. If you have anequation for x in terms of t, calculus allows you, for instance, to find anequation for v in terms of t. In calculus terms, we say that the function v(t)

2

0 4 8t (s)

20

x(m)

v(m/s)

Section 2.6 Graphs of Velocity Versus Time

68

is the derivative of the function x(t). In other words, the derivative of afunction is a new function that tells how rapidly the original function waschanging. We now use neither Newton’s name for his technique (he called it“the method of fluxions”) nor his notation. The more commonly usednotation is due to Newton’s German contemporary Leibnitz, whom theEnglish accused of plagiarizing the calculus from Newton. In the Leibnitznotation, we write

v = dxdt

to indicate that the function v(t) equals the slope of the tangent line of thegraph of x(t) at every time t. The Leibnitz notation is meant to evoke thedelta notation, but with a very small time interval. Because the dx and dt arethought of as very small ∆x’s and ∆t’s, i.e. very small differences, the part ofcalculus that has to do with derivatives is called differential calculus.

Differential calculus consists of three things:

• The concept and definition of the derivative, which is covered inthis book, but which will be discussed more formally in your mathcourse.

• The Leibnitz notation described above, which you’ll need to getmore comfortable with in your math course.

• A set of rules for that allows you to find an equation for thederivative of a given function. For instance, if you happened tohave a situation where the position of an object was given by theequation x=2t7, you would be able to use those rules to find dx/dt=14t6. This bag of tricks is covered in your math course.

Chapter 2 Velocity and Relative Motion

69

SummarySelected Vocabulary

center of mass .................... the balance point of an objectvelocity .............................. the rate of change of position; the slope of the tangent line on an x-t

graph.Notation

x ........................................ a point in spacet ........................................ a point in time, a clock reading∆ ....................................... “change in;” the value of a variable afterwards minus its value before∆x ..................................... a distance, or more precisely a change in x, which may be less than the

distance traveled; its plus or minus sign indicates direction∆t ...................................... a duration of timev ........................................ velocityvAB .................................................... the velocity of object A relative to object B

Standard Terminology Avoided in This Bookdisplacement ..................... a name for the symbol ∆x.speed ................................. the absolute value of the velocity, i.e. the velocity stripped of any informa-

tion about its directionSummary

An object’s center of mass is the point at which it can be balanced. For the time being, we are studying themathematical description only of the motion of an object’s center of mass in cases restricted to one dimension.The motion of an object’s center of mass is usually far simpler than the motion of any of its other parts.

It is important to distinguish location, x, from distance, ∆x, and clock reading, t, from time interval ∆t. Whenan object’s x-t graph is linear, we define its velocity as the slope of the line, ∆x/∆t. When the graph is curved,we generalize the definition so that the velocity is the slope of the tangent line at a given point on the graph.

Galileo’s principle of inertia states that no force is required to maintain motion with constant velocity in astraight line, and absolute motion does not cause any observable physical effects. Things typically tend toreduce their velocity relative to the surface of our planet only because they are physically rubbing against theplanet (or something attached to the planet), not because there is anything special about being at rest withrespect to the earth’s surface. When it seems, for instance, that a force is required to keep a book slidingacross a table, in fact the force is only serving to cancel the contrary force of friction.

Absolute motion is not a well-defined concept, and if two observers are not at rest relative to one anotherthey will disagree about the absolute velocities of objects. They will, however, agree about relative velocities. Ifobject A is in motion relative to object B, and B is in motion relative to C, then A’s velocity relative to C is givenby vAC=vAB+vBC. Positive and negative signs are used to indicate the direction of an object’s motion.

Summary

70

Homework Problems1 . The graph shows the motion of a car stuck in stop-and-go freewaytraffic. (a) If you only knew how far the car had gone during this entiretime period, what would you think its velocity was? (b) What is the car’smaximum velocity?

2. (a) Let θ be the latitude of a point on the Earth's surface. Derive analgebra equation for the distance, L, traveled by that point during onerotation of the Earth about its axis, i.e. over one day, expressed in terms ofL, θ, and R, the radius of the earth. Check: Your equation should give L=0for the North Pole.

(b) At what speed is Fullerton, at latitude θ=34°, moving with therotation of the Earth about its axis? Give your answer in units of mi/h. [Seethe inside front cover of the book for the relevant data.]

3«. A person is parachute jumping. During the time between when sheleaps out of the plane and when she opens her chute, her altitude is given bythe equation

y=(10000 m) - (50 m/s)[t+(5.0 s)e-t/5.0 s] .

Find her velocity at t=7.0 s. (This can be done on a calculator, withoutknowing calculus.) Because of air resistance, her velocity does not increaseat a steady rate as it would for an object falling in vacuum.

4 S. A light-year is a unit of distance used in astronomy, and defined as thedistance light travels in one year. The speed of light is 3.0x108 m/s. Findhow many meters there are in one light-year, expressing your answer inscientific notation.

5 S. You’re standing in a freight train, and have no way to see out. If youhave to lean to stay on your feet, what, if anything, does that tell you aboutthe train’s velocity? Its acceleration? Explain.

6 ∫. A honeybee’s position as a function of time is given by x=10t-t3, where tis in seconds and x in meters. What is its velocity at t=3.0 s?

0 4 8 12time (s)

10

20

30

40

50

60

70

80

90

distance(m)

Problem 1.

S A solution is given in the back of the book. « A difficult problem. A computerized answer check is available. ∫ A problem that requires calculus.

Chapter 2 Velocity and Relative Motion

71© 1998 Benjamin Crowell

3 Acceleration andFree Fall

3.1 The Motion of Falling ObjectsThe motion of falling objects is the simplest and most common ex-

ample of motion with changing velocity. The early pioneers of physics had acorrect intuition that the way things drop was a message directly fromNature herself about how the universe worked. Other examples seem lesslikely to have deep significance. A walking person who speeds up is makinga conscious choice. If one stretch of a river flows more rapidly than another,it may be only because the channel is narrower there, which is just anaccident of the local geography. But there is something impressively consis-tent, universal, and inexorable about the way things fall.

Stand up now and simultaneously drop a coin and a bit of paper side byside. The paper takes much longer to hit the ground. That’s why Aristotlewrote that heavy objects fell more rapidly. Europeans believed him for twothousand years.

Now repeat the experiment, but make it into a race between the coinand your shoe. My own shoe is about 50 times heavier than the nickel I hadhandy, but it looks to me like they hit the ground at exactly the samemoment. So much for Aristotle! Galileo, who had a flair for the theatrical,did the experiment by dropping a bullet and a heavy cannonball from a talltower. Aristotle’s observations had been incomplete, his interpretation a vastoversimplification.

It is inconceivable that Galileo was the first person to observe a discrep-ancy with Aristotle’s predictions. Galileo was the one who changed thecourse of history because he was able to assemble the observations into acoherent pattern, and also because he carried out systematic quantitative(numerical) measurements rather than just describing things qualitatively.

Why is it that some objects, like the coin and the shoe, have similarmotion, but others, like a feather or a bit of paper, are different? Galileo

Galileo dropped a cannonball and amusketball simultaneously from atower, and observed that they both hitthe ground at the same time. Thiscontradicted Aristotle’s long-acceptedidea that heavier objects fell faster.

Galileo and the ChurchGalileo’s contradiction of Aristotle had serious consequences. He was interrogated by the Church authorities and

convicted of teaching that the earth went around the sun as a matter of fact and not, as he had promised previously,as a mere mathematical hypothesis. He was placed under permanent house arrest, and forbidden to write about orteach his theories. Immediately after being forced to recant his claim that the earth revolved around the sun, the oldman is said to have muttered defiantly “and yet it does move.”

The story is dramatic, but there are some omissions in the commonly taught heroic version. There was a rumorthat the Simplicio character represented the Pope. Also, some of the ideas Galileo advocated had controversial religiousovertones. He believed in the existence of atoms, and atomism was thought by some people to contradict the Church’sdoctrine of transubstantiation, which said that in the Catholic mass, the blessing of the bread and wine literallytransformed them into the flesh and blood of Christ. His support for a cosmology in which the earth circled the sunwas also disreputable because one of its supporters, Giordano Bruno, also proposed a bizarre synthesis of Christianitywith the ancient Egyptian religion.

72

speculated that in addition to the force that always pulls objects down, therewas an upward force exerted by the air. Anyone can speculate, but Galileowent beyond speculation and came up with two clever experiments to probethe issue. First, he experimented with objects falling in water, which probedthe same issues but made the motion slow enough that he could take timemeasurements with a primitive pendulum clock. With this technique, heestablished the following facts:

• All heavy, streamlined objects (for example a steel rod droppedpoint-down) reach the bottom of the tank in about the sameamount of time, only slightly longer than the time they would taketo fall the same distance in air.• Objects that are lighter or less streamlined take a longer time toreach the bottom.

This supported his hypothesis about two contrary forces. He imaginedan idealized situation in which the falling object did not have to push itsway through any substance at all. Falling in air would be more like this idealcase than falling in water, but even a thin, sparse medium like air would besufficient to cause obvious effects on feathers and other light objects thatwere not streamlined. Today, we have vacuum pumps that allow us to sucknearly all the air out of a chamber, and if we drop a feather and a rock sideby side in a vacuum, the feather does not lag behind the rock at all.

How the speed of a falling object increases with timeGalileo’s second stroke of genius was to find a way to make quantitative

measurements of how the speed of a falling object increased as it wentalong. Again it was problematic to make sufficiently accurate time measure-ments with primitive clocks, and again he found a tricky way to slow thingsdown while preserving the essential physical phenomena: he let a ball rolldown a slope instead of dropping it vertically. The steeper the incline, themore rapidly the ball would gain speed. Without a modern video camera,Galileo had invented a way to make a slow-motion version of falling.

Velocity increases more gradually onthe gentle slope, but the motion isotherwise the same as the motion ofa falling object.

The v-t graph of a falling object is aline.

v

t

Although Galileo’s clocks were only good enough to do accurateexperiments at the smaller angles, he was confident after making a system-atic study at a variety of small angles that his basic conclusions were gener-ally valid. Stated in modern language, what he found was that the velocity-versus-time graph was a line. In the language of algebra, we know that a linehas an equation of the form y=ax+b, but our variables are v and t, so itwould be v=at+b. (The constant b can be interpreted simply as the initialvelocity of the object, i.e. its velocity at the time when we started our clock,

which we conventionally write as v o .)

Chapter 3 Acceleration and Free Fall

73

Self-Check

An object is rolling down an incline. After it has been rolling for a short time, itis found to travel 13 cm during a certain one-second interval. During thesecond after that, if goes 16 cm. How many cm will it travel in the second afterthat?

A contradiction in Aristotle’s reasoningGalileo’s inclined-plane experiment disproved the long-accepted claim

by Aristotle that a falling object had a definite “natural falling speed”proportional to its weight. Galileo had found that the speed just kept onincreasing, and weight was irrelevant as long as air friction was negligible.Not only did Galileo prove experimentally that Aristotle had been wrong,but he also pointed out a logical contradiction in Aristotle’s own reasoning.Simplicio, the stupid character, mouths the accepted Aristotelian wisdom:

SIMPLICIO: There can be no doubt but that a particular body ... has afixed velocity which is determined by nature...

SALVIATI: If then we take two bodies whose natural speeds aredifferent, it is clear that, [according to Aristotle], on uniting the two, themore rapid one will be partly held back by the slower, and the slowerwill be somewhat hastened by the swifter. Do you not agree with mein this opinion?

SIMPLICIO: You are unquestionably right.

SALVIATI: But if this is true, and if a large stone moves with a speed of,say, eight [unspecified units] while a smaller moves with a speed offour, then when they are united, the system will move with a speedless than eight; but the two stones when tied together make a stonelarger than that which before moved with a speed of eight. Hence theheavier body moves with less speed than the lighter; an effect whichis contrary to your supposition. Thus you see how, from yourassumption that the heavier body moves more rapidly than the lighterone, I infer that the heavier body moves more slowly.

[tr. Crew and De Salvio]

What is gravity?The physicist Richard Feynman liked to tell a story about how when he

was a little kid, he asked his father, “Why do things fall?” As an adult, hepraised his father for answering, “Nobody knows why things fall. It’s a deepmystery, and the smartest people in the world don’t know the basic reasonfor it.” Contrast that with the average person’s off-the-cuff answer, “Oh, it’sbecause of gravity.” Feynman liked his father’s answer, because his fatherrealized that simply giving a name to something didn’t mean that youunderstood it. The radical thing about Galileo’s and Newton’s approach toscience was that they concentrated first on describing mathematically whatreally did happen, rather than spending a lot of time on untestable specula-tion such as Aristotle’s statement that “Things fall because they are trying toreach their natural place in contact with the earth.” That doesn’t mean thatscience can never answer the “why” questions. Over the next month or twoas you delve deeper into physics, you will learn that there are more funda-mental reasons why all falling objects have v-t graphs with the same slope,regardless of their mass. Nevertheless, the methods of science always imposelimits on how deep our explanation can go.

(a) Galileo’s experiments show that allfalling objects have the same motionif air resistance is negligible.

Aristotle said that heavier objects fellfaster than lighter ones. If two rocksare tied together, that makes an extra-heavy rock, (b), which should fallfaster. But Aristotle’s theory would alsopredict that the light rock would holdback the heavy rock, resulting in aslower fall, (c).

Its speed increases at a steady rate, so in the next second it will travel 19 cm.

(b) (c)

Section 3.1 The Motion of Falling Objects

74

3.2 AccelerationDefinition of acceleration for linear v-t graphs

Galileo’s experiment with dropping heavy and light objects from atower showed that all falling objects have the same motion, and his in-clined-plane experiments showed that the motion was described by v=ax+vo.The initial velocity vo depends on whether you drop the object from rest orthrow it down, but even if you throw it down, you cannot change the slope,a, of the v-t graph.

Since these experiments show that all falling objects have linear v-tgraphs with the same slope, the slope of such a graph is apparently animportant and useful quantity. We use the word acceleration, and thesymbol a, for the slope of such a graph. In symbols, a=∆v/∆t. The accelera-tion can be interpreted as the amount of speed gained in every second, andit has units of velocity divided by time, i.e. “meters per second per second,”or m/s/s. Continuing to treat units as if they were algebra symbols, wesimplify “m/s/s” to read “m/s2.” Acceleration can be a useful quantity fordescribing other types of motion besides falling, and the word and thesymbol “a” can be used in a more general context. We reserve the morespecialized symbol “g” for the acceleration of falling objects, which on thesurface of our planet equals 9.8 m/s2. Often when doing approximatecalculations or merely illustrative numerical examples it is good enough touse g=10 m/s2, which is off by only 2%.

ExampleQuestion : A despondent physics student jumps off a bridge, andfalls for three seconds before hitting the water. How fast is hegoing when he hits the water?Solution : Approximating g as 10 m/s2, he will gain 10 m/s ofspeed each second. After one second, his velocity is 10 m/s,after two seconds it is 20 m/s, and on impact, after falling forthree seconds, he is moving at 30 m/s.

Example: extracting acceleration from a graphQuestion : The x-t and v-t graphs show the motion of a carstarting from a stop sign. What is the car’s acceleration?Solution : Acceleration is defined as the slope of the v-t graph.The graph rises by 3 m/s during a time interval of 3 s, so theacceleration is (3 m/s)/(3 s)=1 m/s2.Incorrect solution #1 : The final velocity is 3 m/s, andacceleration is velocity divided by time, so the acceleration is (3m/s)/(10 s)=0.3 m/s2. The solution is incorrect because you can’t find the slope of agraph from one point. This person was just using the point at theright end of the v-t graph to try to find the slope of the curve.Incorrect solution #2 : Velocity is distance divided by time sov=(4.5 m)/(3 s)=1.5 m/s. Acceleration is velocity divided by time,so a=(1.5 m/s)/(3 s)=0.5 m/s2. The solution is incorrect because velocity is the slope of thetangent line. In a case like this where the velocity is changing,you can’t just pick two points on the x-t graph and use them tofind the velocity.

v(m/s)

t (s)0 1 2 3

10

20

30

2

4

x (m

)

1

2

3

7 8 9 10t (s)

v (m

/s)

Chapter 3 Acceleration and Free Fall

75

Example: converting g to different unitsQuestion : What is g in units of cm/s2?Solution : The answer is going to be how many cm/s of speed afalling object gains in one second. If it gains 9.8 m/s in onesecond, then it gains 980 cm/s in one second, so g=980 cm/s2.Alternatively, we can use the method of fractions that equal one:

9.8 m/s2

× 100 cm

1 m/=

980 cm

s2

Question : What is g in units of miles/hour2?Solution :

9.8 ms2

× 1 mile1600 m

× 3600 s1 hour

2

= 7.9×104 mile / hour2

This large number can be interpreted as the speed, in miles perhour, you would gain by falling for one hour. Note that we had tosquare the conversion factor of 3600 s/hour in order to cancelout the units of seconds squared in the denominator.

Question : What is g in units of miles/hour/s?Solution :

9.8 ms2

× 1 mile1600 m

× 3600 s1 hour

= 22 mile/hour/s

This is a figure that Americans will have an intuitive feel for. Ifyour car has a forward acceleration equal to the acceleration of afalling object, then you will gain 22 miles per hour of speed everysecond. However, using mixed time units of hours and secondslike this is usually inconvenient for problem-solving. It would belike using units of foot-inches for area instead of ft2 or in2.

The acceleration of gravity is different in different locations.Everyone knows that gravity is weaker on the moon, but actually it is

not even the same everywhere on Earth, as shown by the sampling ofnumerical data in the following table.

location latitudeelevation(m)

g(m/s2)

north pole 90° N 0 9.8322

Reykjavik, Iceland 64° N 0 9.8225

Fullerton, California 34° N 0 9.7957

Guayaquil, Ecuador 2° S 0 9.7806

Mt. Cotopaxi, Ecuador 1° S 5896 9.7624

Mt. Everest 28° N 8848 9.7643

The main variables that relate to the value of g on Earth are latitude andelevation. Although you have not yet learned how g would be calculatedbased on any deeper theory of gravity, it is not too hard to guess why gdepends on elevation. Gravity is an attraction between things that have

Section 3.1 The Motion of Falling Objects

76

mass, and the attraction gets weaker with increasing distance. As you ascendfrom the seaport of Guayaquil to the nearby top of Mt. Cotopaxi, you aredistancing yourself from the mass of the planet. The dependence on latitudeoccurs because we are measuring the acceleration of gravity relative to theearth’s surface, but the earth’s rotation causes the earth’s surface to fall outfrom under you. (We will discuss both gravity and rotation in more detaillater in the course.)

Much more spectacular differences in the strength of gravity can ofcourse be observed away from the Earth’s surface:

location g (m/s2)

asteroid Vesta (surface) 0.3

Earth's moon (surface) 1.6

Mars (surface) 3.7

Earth (surface) 9.8

Jupiter (cloud-tops) 26

Sun (visible surface) 270

typical neutron star (surface) 1012

black hole (center)

infinite according tosome theories, on theorder of 1052

according to others

A typical neutron star is not so different in size from a large asteroid, but isorders of magnitude more massive, so the mass of a body definitely corre-lates with the g it creates. On the other hand, a neutron star has about thesame mass as our Sun, so why is its g billions of times greater? If you had themisfortune of being on the surface of a neutron star, you’d be within a fewthousand miles of all its mass, whereas on the surface of the Sun, you’d stillbe millions of miles from most if its mass.

This false-color map shows variationsin the strength of the earth’s gravity.Purple areas have the strongestgravity, yellow the weakest. The over-all trend toward weaker gravity at theequator and stronger gravity at thepoles has been artificially removed toallow the weaker local variations toshow up. The map covers only theoceans because of the technique usedto make it: satellites look for bulgesand depressions in the surface of theocean. A very slight bulge will occurover an undersea mountain, forinstance, because the mountain’sgravitational attraction pulls watertoward it. The US governmentoriginally began collecting data likethese for military use, to correct for thedeviations in the paths of missiles. Thedata have recently been released forscientific and commercial use (e.g.searching for sites for off-shore oilwells).

Chapter 3 Acceleration and Free Fall

77

Discussion questionsA. What is wrong with the following definitions of g?

(a) “g is gravity.”(b) “g is the speed of a falling object.”(c) “g is how hard gravity pulls on things.”

B. When advertisers specify how much acceleration a car is capable of, theydo not give an acceleration as defined in physics. Instead, they usually specifyhow many seconds are required for the car to go from rest to 60 miles/hour.Suppose we use the notation “a” for the acceleration as defined in physics,and “acar ad” for the quantity used in advertisem*nts for cars. In the US’s non-metric system of units, what would be the units of a and acar ad? How would theuse and interpretation of large and small, positive and negative values bedifferent for a as opposed to acar ad?C. Two people stand on the edge of a cliff. As they lean over the edge, oneperson throws a rock down, while the other throws one straight up with anexactly opposite initial velocity. Compare the speeds of the rocks on impact atthe bottom of the cliff.

3.3 Positive and Negative AccelerationGravity always pulls down, but that does not mean it always speeds

things up. If you throw a ball straight up, gravity will first slow it down tov=0 and then begin increasing its speed. When I took physics in highschool, I got the impression that positive signs of acceleration indicatedspeeding up, while negative accelerations represented slowing down, i.e.deceleration. Such a definition would be inconvenient, however, because wewould then have to say that the same downward tug of gravity couldproduce either a positive or a negative acceleration. As we will see in thefollowing example, such a definition also would not be the same as the slopeof the v-t graph

Let’s study the example of the rising and falling ball. In the example ofthe person falling from a bridge, I assumed positive velocity values withoutcalling attention to it, which meant I was assuming a coordinate systemwhose x axis pointed down. In this example, where the ball is reversingdirection, it is not possible to avoid negative velocities by a tricky choice ofaxis, so let’s make the more natural choice of an axis pointing up. The ball’svelocity will initially be a positive number, because it is heading up, in thesame direction as the x axis, but on the way back down, it will be a negativenumber. As shown in the figure, the v-t graph does not do anything specialat the top of the ball’s flight, where v equals 0. Its slope is always negative.In the left half of the graph, the negative slope indicates a positive velocitythat is getting closer to zero. On the right side, the negative slope is due to anegative velocity that is getting farther from zero, so we say that the ball isspeeding up, but its velocity is decreasing!

To summarize, what makes the most sense is to stick with the originaldefinition of acceleration as the slope of the v-t graph, ∆v/∆t. By thisdefinition, it just isn’t necessarily true that things speeding up have positiveacceleration while things slowing down have negative acceleration. Theword “deceleration” is not used much by physicists, and the word “accelera-tion” is used unblushingly to refer to slowing down as well as speeding up:“There was a red light, and we accelerated to a stop.”

-5

5

0.5 1 1.5t (s)

v (m

/s)

1

2

x (m

)

a =−10 m/s2

a =−10 m/s2

a =−10 m/s2

Section 3.3 Positive and negative acceleration

78

ExampleQuestion : In the example above, what happens if you calculatethe acceleration between t=1.0 and 1.5 s?Answer : Reading from the graph, it looks like the velocity isabout -1 m/s at t=1.0 s, and around -6 m/s at t=1.5 s. Theacceleration, figured between these two points, is

a = ∆v∆t

=( – 6 m / s) – ( – 1 m / s)

(1.5 s) – (1.0 s)= – 10 m / s2 .

Even though the ball is speeding up, it has a negativeacceleration.

Another way of convincing you that this way of handling the plus andminus signs makes sense is to think of a device that measures acceleration.After all, physics is supposed to use operational definitions, ones that relateto the results you get with actual measuring devices. Consider an airfreshener hanging from the rear-view mirror of your car. When you speedup, the air freshener swings backward. Suppose we define this as a positivereading. When you slow down, the air freshener swings forward, so we’llcall this a negative reading on our accelerometer. But what if you put the carin reverse and start speeding up backwards? Even though you’re speedingup, the accelerometer responds in the same way as it did when you weregoing forward and slowing down. There are four possible cases:

motion of car

accelerom-eterswings

slope ofv-tgraph

direction offorce actingon car

forward, speeding up backward + forward

forward, slowing down forward - backward

backward, speeding up forward - backward

backward, slowing down backward + forward

Note the consistency of the three right-hand columns — nature is

Chapter 3 Acceleration and Free Fall

79

trying to tell us that this is the right system of classification, not the left-hand column.

Because the positive and negative signs of acceleration depend on thechoice of a coordinate system, the acceleration of an object under theinfluence of gravity can be either positive or negative. Rather than having towrite things like “g=9.8 m/s2 or -9.8 m/s2” every time we want to discuss g’snumerical value, we simply define g as the absolute value of the accelerationof objects moving under the influence of gravity. We consistently let g=9.8m/s2, but we may have either a=g or a=-g, depending on our choice of acoordinate system.

ExampleQuestion : A person kicks a ball, which rolls up a sloping street,comes to a halt, and rolls back down again. The ball hasconstant acceleration. The ball is initially moving at a velocity of4.0 m/s, and after 10.0 s it has returned to where it started. At theend, it has sped back up to the same speed it had initially, but inthe opposite direction. What was its acceleration?Solution : By giving a positive number for the initial velocity, thestatement of the question implies a coordinate axis that points upthe slope of the hill. The “same” speed in the opposite directionshould therefore be represented by a negative number, -4.0 m/s.The acceleration is a=∆v/∆t=(vafter-vbefore)/10.0 s=[(-4.0 m/s)-(4.0m/s)]/10.0 s=-0.80 m/s2. The acceleration was no different duringthe upward part of the roll than on the downward part of the roll.Incorrect solution : Acceleration is ∆v/∆t, and at the end it’s notmoving any faster or slower than when it started, so ∆v=0 anda=0. The velocity does change, from a positive number to anegative number.

Discussion questionsA. A child repeatedly jumps up and down on a trampoline. Discuss the signand magnitude of his acceleration, including both the time when he is in the airand the time when his feet are in contact with the trampoline.B. Sally is on an amusem*nt park ride which begins with her chair beinghoisted straight up a tower at a constant speed of 60 miles/hour. Despite sternwarnings from her father that he’ll take her home the next time shemisbehaves, she decides that as a scientific experiment she really needs torelease her corndog over the side as she’s on the way up. She does not throwit. She simply sticks it out of the car, lets it go, and watches it against thebackground of the sky, with no trees or buildings as reference points. Whatdoes the corndog’s motion look like as observed by Sally? Does its speed everappear to her to be zero? What acceleration does she observe it to have: is itever positive? negative? zero? What would her enraged father answer if askedfor a similar description of its motion as it appears to him, standing on theground?C. Can an object maintain a constant acceleration, but meanwhile reverse thedirection of its velocity?D. Can an object have a velocity that is positive and increasing at the sametime that its acceleration is decreasing?Discussion question B.

Section 3.3 Positive and negative acceleration

80

200

400

600

10

20

30

40

50

0 2 4 6 8 10 12 14t (s)

x (m

)v

(m/s

)E. The four figures show a refugee from a Picasso painting blowing on a rollingwater bottle. In some cases the person’s blowing is speeding the bottle up, butin others it is slowing it down. The arrow inside the bottle shows whichdirection it is going, and a coordinate system is shown at the bottom of eachfigure. In each case, figure out the plus or minus signs of the velocity andacceleration. It may be helpful to draw a v-t graph in each case.

x

x

x

x

(a) (b)

(c) (d)

3.4 Varying AccelerationSo far we have only been discussing examples of motion for which the

v-t graph is linear. If we wish to generalize our definition to v-t graphs thatare more complex curves, the best way to proceed is similar to how wedefined velocity for curved x-t graphs:

definition of accelerationThe acceleration of an object at any instant is the slope of the tangentline passing through its v-versus-t graph at the relevant point.

Example: a skydiverQuestion : The graphs show the results of a fairly realisticcomputer simulation of the motion of a skydiver, including theeffects of air friction. The x axis has been chosen pointing down,so x is increasing as she falls. Find (a) the skydiver’sacceleration at t=3.0 s, and also (b) at t=7.0 s.

Chapter 3 Acceleration and Free Fall

81

Solution : I’ve added tangent lines at the two points in question.(a) To find the slope of the tangent line, I pick two points on theline (not necessarily on the actual curve): (3.0 s, 28 m/s) and (5.0s, 42 m/s). The slope of the tangent line is (42 m/s-28 m/s)/(5.0 s- 3.0 s)=7.0 m/s2.(b) Two points on this tangent line are (7.0 s, 47 m/s) and (9.0 s,52 m/s). The slope of the tangent line is (52 m/s-47 m/s)/(9.0 s -7.0 s)=2.5 m/s2.Physically, what’s happening is that at t=3.0 s, the skydiver is notyet going very fast, so air friction is not yet very strong. Shetherefore has an acceleration almost as great as g. At t=7.0 s,she is moving almost twice as fast (about 100 miles per hour),and air friction is extremely strong, resulting in a significantdeparture from the idealized case of no air friction.

In the above example, the x-t graph was not even used in the solution ofthe problem, since the definition of acceleration refers to the slope of the v-tgraph. It is possible, however, to interpret an x-t graph to find out some-thing about the acceleration. An object with zero acceleration, i.e. constantvelocity, has an x-t graph that is a straight line. A straight line has nocurvature. A change in velocity requires a change in the slope of the x-tgraph, which means that it is a curve rather than a line. Thus accelerationrelates to the curvature of the x-t graph. Figure (c) shows some examples.

In the skydiver example, the x-t graph was more strongly curved at thebeginning, and became nearly straight at the end. If the x-t graph is nearlystraight, then its slope, the velocity, is nearly constant, and the accelerationis therefore small. We can thus interpret the acceleration as representing thecurvature of the x-t graph. If the “cup” of the curve points up, the accelera-tion is positive, and if it points down, the acceleration is negative.

Section 3.4 Varying Acceleration

10

20

30

40

50

0 2 4 6 8 10 12 14

t (s)

v (m

/s)

(3.0 s, 28 m/s)

(5.0 s, 42 m/s)

(7.0 s, 47 m/s)(9.0 s, 52 m/s)

x

t

large,positivea

x

t

a=0

x

t

a>0

a<0

x

t

small,positivea

x

t

large,negativea

x

t

a=0

82

Since the relationship between a and v is analogous to the relationshipbetween v and x, we can also make graphs of acceleration as a function oftime, as shown in figures (a) and (b) above.

Figure (c) summarizes the relationships among the three types ofgraphs.

Discussion questionsA. Describe in words how the changes in the a-t graph for the skydiver relateto the behavior of the v-t graph.B. Explain how each set of graphs contains inconsistencies.

Chapter 3 Acceleration and Free Fall

t

v

x

a

t

v

x

a

t

v

x

a

1 2 3

Discussion question B.

position

velocity

acceleration

slope oftangent line

slope oftangent line

curvature

=rate of change of position

=rate of change of velocity

5

10

01234

1

0 1 2 3 4 5t (s)

10203040

200

400

600

5

0 2 4 6 8 101214t (s)

x (m

)v

(m/s

)a

(m/s

2 )

x (m

)v

(m/s

)a

(m/s

2 )

(a) (b)

50

(c)

83

3.5 The Area Under the Velocity-Time GraphA natural question to ask about falling objects is how fast they fall, but

Galileo showed that the question has no answer. The physical law that hediscovered connects a cause (the attraction of the planet Earth’s mass) to aneffect, but the effect is predicted in terms of an acceleration rather than avelocity. In fact, no physical law predicts a definite velocity as a result of aspecific phenomenon, because velocity cannot be measured in absoluteterms, and only changes in velocity relate directly to physical phenomena.

The unfortunate thing about this situation is that the definitions ofvelocity and acceleration are stated in terms of the tangent-line technique,which lets you go from x to v to a, but not the other way around. Without atechnique to go backwards from a to v to x, we cannot say anything quanti-tative, for instance, about the x-t graph of a falling object. Such a techniquedoes exist, and I used it to make the x-t graphs in all the examples above.

First let’s concentrate on how to get x information out of a v-t graph. Inexample (a), an object moves at a speed of 20 m/s for a period of 4.0 s. Thedistance covered is ∆x=v∆t=(20 m/s)x(4.0 s)=80 m. Notice that the quanti-ties being multiplied are the width and the height of the shaded rectangle— or, strictly speaking, the time represented by its width and the velocityrepresented by its height. The distance of ∆x=80 m thus corresponds to thearea of the shaded part of the graph.

The next step in sophistication is an example like (b), where the objectmoves at a constant speed of 10 m/s for two seconds, then for two secondsat a different constant speed of 20 m/s. The shaded region can be split intoa small rectangle on the left, with an area representing ∆x=20 m, and a tallerone on the right, corresponding to another 40 m of motion. The totaldistance is thus 60 m, which corresponds to the total area under the graph.

An example like (c) is now just a trivial generalization; there is simply alarge number of skinny rectangular areas to add up. But notice that graph(c) is quite a good approximation to the smooth curve (d). Even though wehave no formula for the area of a funny shape like (d), we can approximateits area by dividing it up into smaller areas like rectangles, whose area iseasier to calculate. If someone hands you a graph like (d) and asks you tofind the area under it, the simplest approach is just to count up the littlerectangles on the underlying graph paper, making rough estimates offractional rectangles as you go along.

10

20

0 2 4 6 8t (s)

(c)

v (m

/s)

10

20

0 2 4 6 8t (s)

(d)

v (m

/s)

10

20

0 2 4 6 8t (s)

(a)

v (m

/s)

10

20

0 2 4 6 8t (s)

(b)

v (m

/s)

Section 3.5 The Area Under the Velocity-Time Graph

84

That’s what I’ve done above. Each rectangle on the graph paper is 1.0 swide and 2 m/s tall, so it represents 2 m. Adding up all the numbers gives∆x=41 m. If you needed better accuracy, you could use graph paper withsmaller rectangles.

It’s important to realize that this technique gives you ∆x, not x. The v-tgraph has no information about where the object was when it started.

The following are important points to keep in mind when applying thistechnique:

• If the range of v values on your graph does not extend down tozero, then you will get the wrong answer unless you compensate byadding in the area that is not shown.

• As in the example, one rectangle on the graph paper does notnecessarily correspond to one meter of distance.

• Negative velocity values represent motion in the opposite direction,so area under the t axis should be subtracted, i.e. counted as“negative area.”

• Since the result is a ∆x value, it only tells you xafter-xbefore, whichmay be less than the actual distance traveled. For instance, theobject could come back to its original position at the end, whichwould correspond to ∆x=0, even though it had actually moved anonzero distance.

Finally, note that one can find ∆v from an a-t graph using an entirelyanalogous method. Each rectangle on the a-t graph represents a certainamount of velocity change.

Discussion question

Roughly what would a pendulum’s v-t graph look like? What would happenwhen you applied the area-under-the-curve technique to find the pendulum’s∆x for a time period covering many swings?

10

20

0 2 4 6 8t (s)

v (m

/s)

2 m

1 m

2 m

2 m

2 m

0.5 m

2 m

2 m

2 m

2 m

2 m

2 m

2 m

1.5 m

1 m

1 m

1.5 m

1.5 m

1.5 m

1.5 m

1.5 m

1.5 m

1.5 m

1.5 m

1.5 m

1.5 m

0.5 m

Chapter 3 Acceleration and Free Fall

85

3.6 Algebraic Results for Constant AccelerationAlthough the area-under-the-curve technique can be applied to any

graph, no matter how complicated, it may be laborious to carry out, and iffractions of rectangles must be estimated the result will only be approxi-mate. In the special case of motion with constant acceleration, it is possibleto find a convenient shortcut which produces exact results. When theacceleration is constant, the v-t graph is a straight line, as shown in thefigure. The area under the curve can be divided into a triangle plus arectangle, both of whose areas can be calculated exactly: A=bh for a rect-

angle and A= 1 21 2 bh for a triangle. The height of the rectangle is the initialvelocity, vo, and the height of the triangle is the change in velocity frombeginning to end, ∆v. The object’s ∆x is therefore given by the equation

∆x = v o∆t + 12

∆v∆t . This can be simplified a little by using the definition of

acceleration, a=∆v/∆t to eliminate ∆v, giving

∆x = v o∆t + 12a∆t 2

[motion with constant acceleration] .

Since this is a second-order polynomial in ∆t, the graph of ∆x versus ∆tis a parabola, and the same is true of a graph of x versus t — the two graphsdiffer only by shifting along the two axes. Although I have derived theequation using a figure that shows a positive vo, positive a, and so on, it stillturns out to be true regardless of what plus and minus signs are involved.

Another useful equation can be derived if one wants to relate the changein velocity to the distance traveled. This is useful, for instance, for findingthe distance needed by a car to come to a stop. For simplicity, we start byderiving the equation for the special case of vo=0, in which the final velocityvf is a synonym for ∆v. Since velocity and distance are the variables of

interest, not time, we take the equation ∆x = 1 21 2 a∆t2 and use ∆t=∆v/a to

eliminate ∆t. This gives ∆x = 1 21 2 (∆v)2/a, which can be rewritten as

v f2 = 2a∆x [motion with constant acceleration, v o = 0] .

For the more general case where v o ≠ 0 , we skip the tedious algebraleading to the more general equation,

v f2 = v o

2+2a∆x [motion with constant acceleration] .

To help get this all organized in your head, first let’s categorize thevariables as follows:

Variables that change during motion with constant acceleration:

x, v, t

Variable that doesn’t change:

a

t

v

∆v

vo

∆t

Section 3.6 Algebraic Results for Constant Acceleration

86

If you know one of the changing variables and want to find another,there is always an equation that relates those two:

vx

t

a=∆v

∆t∆x = vo∆t + a∆t212

vf2 = vo

2 + 2a∆x

The symmetry among the three variables is imperfect only because theequation relating x and t includes the initial velocity.

There are two main difficulties encountered by students in applyingthese equations:

• The equations apply only to motion with constant acceleration. Youcan’t apply them if the acceleration is changing.

• Students are often unsure of which equation to use, or may causethemselves unnecessary work by taking the longer path around thetriangle in the chart above. Organize your thoughts by listing thevariables you are given, the ones you want to find, and the ones youaren’t given and don’t care about.

ExampleQuestion : You are trying to pull an old lady out of the way of anoncoming truck. You are able to give her an acceleration of 20 m/s2. Starting from rest, how much time is required in order to moveher 2 m?Solution : First we organize our thoughts:

Variables given: ∆x, a, voVariables desired: ∆tIrrelevant variables: vf

Consulting the triangular chart above, the equation we need is

clearly ∆x = v o∆t + 12a∆t 2

, since it has the four variables of

interest and omits the irrelevant one. Eliminating the vo term and

solving for ∆t gives ∆t = 2∆xa =0.4 s.

Discussion questionsA Check that the units make sense in the three equations derived in thissection.B. In chapter 1, I gave examples of correct and incorrect reasoning aboutproportionality, using questions about the scaling of area and volume. Try totranslate the incorrect modes of reasoning shown there into mistakes about thefollowing question: If the acceleration of gravity on Mars is 1/3 that on Earth,how many times longer does it take for a rock to drop the same distance onMars?

Chapter 3 Acceleration and Free Fall

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3.7* Biological Effects of WeightlessnessThe usefulness of outer space was brought home to North Americans in

1998 by the unexpected failure of the communications satellite that hadbeen handling almost all of the continent’s cellular phone traffic. Comparedto the massive economic and scientific payoffs of satellites and space probes,human space travel has little to boast about after four decades. Sendingpeople into orbit has just been too expensive to be an effective scientific orcommercial activity. The 1986 Challenger disaster dealt a blow to NASA’sconfidence, and with the end of the cold war, U.S. prestige as a superpowerwas no longer a compelling reason to send Americans into space. All thatmay change soon, with a new generation of much cheaper reusable space-ships. (The space shuttle is not truly reusable. Retrieving the boosters out ofthe ocean is no cheaper than building new ones, but NASA brings themback and uses them over for public relations, to show how frugal they are.)Space tourism is even beginning to make economic sense! No fewer thanthree private companies are now willing to take your money for a reserva-tion on a two-to-four minute trip into space, although none of them has afirm date on which to begin service. Within a decade, a space cruise may bethe new status symbol among those sufficiently rich and brave.

Space sicknessWell, rich, brave, and possessed of an iron stomach. Travel agents will

probably not emphasize the certainty of constant space-sickness. For usanimals evolved to function in g=9.8 m/s2, living in g=0 is extremelyunpleasant. The early space program focused obsessively on keeping theastronaut-trainees in perfect physical shape, but it soon became clear that abody like a Greek demigod’s was no defense against that horrible feelingthat your stomach was falling out from under you and you were never goingto catch up. Our inner ear, which normally tells us which way is down,tortures us when down is nowhere to be found. There is contradictoryinformation about whether anyone ever gets over it; the “right stuff ” culturecreates a strong incentive for astronauts to deny that they are sick.

Effects of long space missionsWorse than nausea are the health-threatening effects of prolonged

weightlessness. The Russians are the specialists in long-term missions, inwhich cosmonauts suffer harm to their blood, muscles, and, most impor-tantly, their bones.

The effects on the muscles and skeleton appear to be similar to thoseexperienced by old people and people confined to bed for a long time.Everyone knows that our muscles get stronger or weaker depending on theamount of exercise we get, but the bones are likewise adaptable. Normallyold bone mass is continually being broken down and replaced with newmaterial, but the balance between its loss and replacement is upset whenpeople do not get enough weight-bearing exercise. The main effect is on thebones of the lower body. More research is required to find out whetherastronauts’ loss of bone mass is due to faster breaking down of bone, slowerreplacement, or both. It is also not known whether the effect can be sup-pressed via diet or drugs.

The other set of harmful physiological effects appears to derive from theredistribution of fluids. Normally, the veins and arteries of the legs are

Artist’s conceptions of the X-33spaceship, a half-scale uncrewedversion of the planned VentureStarvehicle, which could cut the cost ofsending people into space by an orderof magnitde. The X-33 being built byLockheed-Martin, and is scheduled fora first test flight in December of 1999.Unlike the space shuttle, the X-33/VentureStar is a single unit — when ittakes off, it does it by itself, withoutany expensive extra boosters. Theaerospace industry is pushing for aSouthern California launch site for theprogram.Courtesy of NASA.

Section 3.7* Biological Effects of Weightlessness

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tightly constricted to keep gravity from making blood collect there. It isuncomfortable for adults to stand on their heads for very long, because thehead’s blood vessels are not able to constrict as effectively. Weightlessastronauts’ blood tends to be expelled by the constricted blood vessels of thelower body, and pools around their hearts, in their thoraxes, and in theirheads. The only immediate result is an uncomfortable feeling of bloatednessin the upper body, but in the long term, a harmful chain of events is set inmotion. The body’s attempts to maintain the correct blood volume are mostsensitive to the level of fluid in the head. Since astronauts have extra fluid intheir heads, the body thinks that the over-all blood volume has become toogreat. It responds by decreasing blood volume below normal levels. Thisincreases the concentration of red blood cells, so the body then decides thatthe blood has become too thick, and reduces the number of blood cells. Inmissions lasting up to a year or so, this is not as harmful as the musculo-skeletal effects, but it is not known whether longer period in space wouldbring the red blood cell count down to harmful levels.

Reproduction in spaceFor those enthralled by the romance of actual human colonization of

space, human reproduction in weightlessness becomes an issue. An already-pregnant Russian cosmonaut did spend some time in orbit in the 1960’s,and later gave birth to a normal child on the ground. Recently, one ofNASA’s public relations concerns about the space shuttle program has beento discourage speculation about space sex, for fear of a potential taxpayers’backlash against the space program as an expensive form of exotic pleasure.

Scientific work has been concentrated on studying plant and animalreproduction in space. Green plants, fungi, insects, fish, and amphibianshave all gone through at least one generation in zero-gravity experimentswithout any serious problems. In many cases, animal embryos conceived inorbit begin by developing abnormally, but later in development they seemto correct themselves. However, chicken embryos fertilized on earth lessthan 24 hours before going into orbit have failed to survive. Since chickensare the organisms most similar to humans among the species investigated sofar, it is not at all certain that humans could reproduce successfully in azero-gravity space colony.

Simulated gravityIf humans are ever to live and work in space for more than a year or so,

the only solution is probably to build spinning space stations to provide theillusion of weight, as discussed in section 9.2. Normal gravity could besimulated, but tourists would probably enjoy g=2 m/s2 or 5 m/s2. Spaceenthusiasts have proposed entire orbiting cities built on the rotating cylin-der plan. Although science fiction has focused on human colonization ofrelatively earthlike bodies such as our moon, Mars, and Jupiter’s icy moonEuropa, there would probably be no practical way to build large spinningstructures on their surfaces. If the biological effects of their 2-3 m/s2

gravitational accelerations are as harmful as the effect of g=0, then we maybe left with the surprising result that interplanetary space is more hospitableto our species than the moons and planets.

Weightless U.S. and Russianastronauts aboard the Mir spacestation.

Spacewalking astronauts beginconstruction of the InternationalSpace Station, December 1998.The space station will not rotate toprovide simulated gravity.

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3.8 ∫ Applications of CalculusIn the Applications of Calculus section at the end of the previous

chapter, I discussed how the slope-of-the-tangent-line idea related to thecalculus concept of a derivative, and the branch of calculus known asdifferential calculus. The other main branch of calculus, integral calculus,has to do with the area-under-the-curve concept discussed in section 3.5 ofthis chapter. Again there is a concept, a notation, and a bag of tricks fordoing things symbolically rather than graphically. In calculus, the areaunder the v-t graph between t=t1 and t=t2 is notated like this:

area under the curve = ∆x = v dt

t 1

t 2

The expression on the right is called an integral, and the s-shaped symbol,the integral sign, is read as “integral of....”

Integral calculus and differential calculus are closely related. For in-stance, if you take the derivative of the function x(t), you get the functionv(t), and if you integrate the function v(t), you get x(t) back again. In otherwords, integration and differentiation are inverse operations. This is knownas the fundamental theorem of calculus.

On an unrelated topic, there is a special notation for taking the deriva-tive of a function twice. The acceleration, for instance, is the second (i.e.double) derivative of the position, because differentiating x once gives v, andthen differentiating v gives a. This is written as

a = d 2x

dt 2 .

The seemingly inconsistent placement of the twos on the top and bottomconfuses all beginning calculus students. The motivation for this funnynotation is that acceleration has units of m/s2, and the notation correctlysuggests that: the top looks like it has units of meters, the bottom seconds2.The notation is not meant, however, to suggest that t is really squared.

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SummarySelected Vocabulary

gravity ............................... A general term for the phenomenon of attraction between things havingmass. The attraction between our planet and a human-sized object causesthe object to fall.

acceleration ....................... The rate of change of velocity; the slope of the tangent line on a v-t graph.Notation

a ........................................ accelerationg ........................................ the acceleration of objects in free fall

SummaryGalileo showed that when air resistance is negligible all falling bodies have the same motion regardless of

mass. Moreover, their v-t graphs are straight lines. We therefore define a quantity called acceleration as theslope, ∆v/∆t, of an object’s v-t graph. In cases other than free fall, the v-t graph may be curved, in which casethe definition is generalized as the slope of a tangent line on the v-t graph. The acceleration of objects in freefall varies slightly across the surface of the earth, and greatly on other planets.

Positive and negative signs of acceleration are defined according to whether the v-t graph slopes up ordown. This definition has the advantage that a force in a given direction always produces the same sign ofacceleration.

The area under the v-t graph gives ∆x, and analogously the area under the a-t graph gives ∆v.

For motion with constant acceleration, the following three equations hold:

∆x = vo∆t + 12a∆t 2

v f2 = vo

2 + 2a∆x

a = ∆v∆t

They are not valid if the acceleration is changing.

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Homework Problems1 . The graph represents the velocity of a bee along a straight line. At t=0,the bee is at the hive. (a) When is the bee farthest from the hive? (b) Howfar is the bee at its farthest point from the hive? (c) At t=13 s, how far is thebee from the hive?

10 2 3 4 5 6 7 8 9 10 11 12 13time (s)

-1

-2

-3

-4

1

2

3

4

5

velocity(m/s)

2. A rock is dropped in a pond. Draw plots of its position versus time,velocity versus time, and acceleration versus time. Include its whole motion,from the moment it is dropped to the moment it comes to rest on thebottom of the pond.

3. In an 18th-century naval battle, a cannon ball is shot horizontally, passesthrough the side of an enemy ship's hull, flies across the galley, and lodgesin a bulkhead. Draw plots of its horizontal position, velocity, and accelera-tion as functions of time, starting while it is inside the cannon and has notyet been fired, and ending when it comes to rest.

4. Draw graphs of position, velocity, and acceleration as functions of timefor a person bunjee jumping. (In bunjee jumping, a person has a stretchyelastic cord tied to his/her ankles, and jumps off of a high platform. At thebottom of the fall, the cord brings the person up short. Presumably theperson bounces up a little.)

5. A ball rolls down the ramp shown in the figure below, consisting of acircular knee, a straight slope, and a circular bottom. For each part of theramp, tell whether the ball’s velocity is increasing, decreasing, or constant,and also whether the ball’s acceleration is increasing, decreasing, or con-stant. Explain your answers. Assume there is no air friction or rollingresistance.

6. At the end of its arc, the velocity of a pendulum is zero. Is its acceleration

Problem 3.

Section Homework Problems

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also zero at this point? Explain.

7. What is the acceleration of a car that moves at a steady velocity of 100km/h for 100 seconds? Explain your answer.

8. A physics homework question asks, "If you start from rest and accelerateat 1.54 m/s2 for 3.29 s, how far do you travel by the end of that time?" Astudent answers as follows:

1.54 x 3.29 = 5.07 m

His Aunt Wanda is good with numbers, but has never taken physics. Shedoesn't know the formula for the distance traveled under constant accelera-tion over a given amount of time, but she tells her nephew his answercannot be right. How does she know?

9 . You are looking into a deep well. It is dark, and you cannot see thebottom. You want to find out how deep it is, so you drop a rock in, and youhear a splash 3 seconds later. Approximately how deep is the well?

10«. You take a trip in your spaceship to another star. Setting off, youincrease your speed at a constant acceleration. Once you get half-way there,you start decelerating, at the same rate, so that by the time you get there,you have slowed down to zero speed. You see the tourist attractions, andthen head home by the same method.

(a) Find a formula for the time, T, required for the round trip, in terms of d,the distance from our sun to the star, and a, the magnitude of the accelera-tion. Note that the acceleration is not constant over the whole trip, but thetrip can be broken up into constant-acceleration parts.

(b) The nearest star to the Earth (other than our own sun) is ProximaCentauri, at a distance of d=4x1016 m. Suppose you use an acceleration ofa=10 m/s2, just enough to compensate for the lack of true gravity and makeyou feel comfortable. How long does the round trip take, in years?

(c) Using the same numbers for d and a, find your maximum speed.Compare this to the speed of light, which is 3.0x108 m/s. (Later in thiscourse, you will learn that there are some new things going on in physicswhen one gets close to the speed of light, and that it is impossible to exceedthe speed of light. For now, though, just use the simpler ideas we've learnedso far.)

11. You climb half-way up a tree, and drop a rock. Then you climb to thetop, and drop another rock. How many times greater is the velocity of thesecond rock on impact? Explain. (The answer is not two times greater.)

12. Two children stand atop a tall building. One drops a rock over the edge,while simultaneously the second throws a rock downward so that it has an

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initial speed of 10 m/s. Compare the accelerations of the two objects whilein flight.

13 ∫. A person is parachute jumping. During the time between when sheleaps out of the plane and when she opens her chute, her altitude is given byan equation of the form

y = b – c t + ke – t / k ,

where e is the base of natural logarithms, and b, c, and k are constants.Because of air resistance, her velocity does not increase at a steady rate as itwould for an object falling in vacuum.

(a) What units would b, c, and k have to have for the equation to makesense?

(b) Find the person's velocity, v, as a function of time. [You will need to usethe chain rule, and the fact that d(ex)/dx=ex.]

(c) Use your answer from part (b) to get an interpretation of the constant c.[Hint: e –x approaches zero for large values of x.]

(d) Find the person's acceleration, a, as a function of time.

(e) Use your answer from part (b) to show that if she waits long enough toopen her chute, her acceleration will become very small.

14 S. The top part of the figure shows the position-versus-time graph for anobject moving in one dimension. On the bottom part of the figure, sketchthe corresponding v-versus-t graph.

15 S. On New Year's Eve, a stupid person fires a pistol straight up. Thebullet leaves the gun at a speed of 100 m/s. How long does it take beforethe bullet hits the ground?

16 S. If the acceleration of gravity on Mars is 1/3 that on Earth, how manytimes longer does it take for a rock to drop the same distance on Mars?Ignore air resistance.

17 S∫. A honeybee’s position as a function of time is given by x=10t-t3,where t is in seconds and x in meters. What is its acceleration at t=3.0 s?

18 S. In July 1999, Popular Mechanics carried out tests to find which carsold by a major auto maker could cover a quarter mile (402 meters) in theshortest time, starting from rest. Because the distance is so short, this typeof test is designed mainly to favor the car with the greatest acceleration, notthe greatest maximum speed (which is irrelevant to the average person). Thewinner was the Dodge Viper, with a time of 12.08 s. The car’s top (andpresumably final) speed was 118.51 miles per hour (52.98 m/s). (a) If a car,starting from rest and moving with constant acceleration, covers a quartermile in this time interval, what is its acceleration? (b) What would be thefinal speed of a car that covered a quarter mile with the constant accelera-tion you found in part a? (c) Based on the discrepancy between your answerin part b and the actual final speed of the Viper, what do you concludeabout how its acceleration changed over time?

Problem 14.

x

v

t

t

Homework Problems

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95© 1998 Benjamin Crowell

4 Force and MotionIf I have seen farther than others, it is because I have stood on the shoul-ders of giants.

Newton, referring to Galileo

4.1 ForceWe need only explain changes in motion, not motion itself

So far you’ve studied the measurement of motion in some detail, butnot the reasons why a certain object would move in a certain way. Thischapter deals with the “why” questions. Aristotle’s ideas about the causes ofmotion were completely wrong, just like all his other ideas about physicalscience, but it will be instructive to start with them, because they amount toa road map of modern students’ incorrect preconceptions.

Aristotle thought he needed to explain both why motion occurs andwhy motion might change. Newton inherited from Galileo the importantcounter-Aristotelian idea that motion needs no explanation, that it is onlychanges in motion that require a physical cause.

Aristotle gave three reasons for motion:

• Natural motion, such as falling, came from the tendency of objectsto go to their “natural” place, on the ground, and come to rest.

• Voluntary motion was the type of motion exhibited by animals,which moved because they chose to.

• Forced motion occurred when an object was acted on by someother object that made it move.

Even as great and skeptical a genius as Galileo was unable tomake much progress on the causes of motion. It was not until ageneration later that Isaac Newton (1642-1727) was able to attackthe problem successfully. In many ways, Newton’s personality wasthe opposite of Galileo’s. Where Galileo agressively publicized hisideas, Newton had to be coaxed by his friends into publishing a bookon his physical discoveries. Where Galileo’s writing had been popularand dramatic, Newton originated the stilted, impersonal style that mostpeople think is standard for scientific writing. (Scientific journals todayencourage a less ponderous style, and papers are often written in thefirst person.) Galileo’s talent for arousing animosity among the richand powerful was matched by Newton’s skill at making himself apopular visitor at court. Galileo narrowly escaped being burned at thestake, while Newton had the good fortune of being on the winningside of the revolution that replaced King James II with William andMary of Orange, leading to a lucrative post running the English royalmint.

Newton discovered the relationship between force and motion,and revolutionized our view of the universe by showing that the samephysical laws applied to all matter, whether living or nonliving, on oroff of our planet’s surface. His book on force and motion, theMathematical Principles of Natural Philosophy , was uncontradictedby experiment for 200 years, but his other main work, Optics , was onthe wrong track due to his conviction that light was composed ofparticles rather than waves. Newton was also an avid alchemist andan astrologer, an embarrassing fact that modern scientists would liketo forget.

Isaac Newton

Aristotle said motion had to be causedby a force. To explain why an arrowkept flying after the bowstring was nolonger pushing on it, he said the airrushed around behind the arrow andpushed it forward. We know this iswrong, because an arrow shot in avacuum chamber does not instantlydrop to the floor as it leaves the bow.Galileo and Newton realized that aforce would only be needed to changethe arrow’s motion, not to make itsmotion continue.

96

Motion changes due to an interaction between two objectsIn the Aristotelian theory, natural motion and voluntary motion are

one-sided phenomena: the object causes its own motion. Forced motion issupposed to be a two-sided phenomenon, because one object imposes its“commands” on another. Where Aristotle conceived of some of the phe-nomena of motion as one-sided and others as two-sided, Newton realizedthat a change in motion was always a two-sided relationship of a forceacting between two physical objects.

The one-sided “natural motion” description of falling makes a crucialomission. The acceleration of a falling object is not caused by its own“natural” tendencies but by an attractive force between it and the planetEarth. Moon rocks brought back to our planet do not “want” to fly back upto the moon because the moon is their “natural” place. They fall to the floorwhen you drop them, just like our homegrown rocks. As we’ll discuss inmore detail later in this course, gravitational forces are simply an attractionthat occurs between any two physical objects. Minute gravitational forcescan even be measured between human-scale objects in the laboratory.

The idea of natural motion also explains incorrectly why things come torest. A basketball rolling across a beach slows to a stop because it is interact-ing with the sand via a frictional force, not because of its own desire to be atrest. If it was on a frictionless surface, it would never slow down. Many ofAristotle’s mistakes stemmed from his failure to recognize friction as a force.

The concept of voluntary motion is equally flawed. You may have beena little uneasy about it from the start, because it assumes a clear distinctionbetween living and nonliving things. Today, however, we are used to havingthe human body likened to a complex machine. In the modern world-view,the border between the living and the inanimate is a fuzzy no-man’s landinhabited by viruses, prions, and silicon chips. Furthermore, Aristotle’sstatement that you can take a step forward “because you choose to” inap-propriately mixes two levels of explanation. At the physical level of explana-tion, the reason your body steps forward is because of a frictional forceacting between your foot and the floor. If the floor was covered with apuddle of oil, no amount of “choosing to” would enable you to take agraceful stride forward.

Forces can all be measured on the same numerical scaleIn the Aristotelian-scholastic tradition, the description of motion as

natural, voluntary, or forced was only the broadest level of classification, likesplitting animals into birds, reptiles, mammals, and amphibians. Theremight be thousands of types of motion, each of which would follow its ownrules. Newton’s realization that all changes in motion were caused by two-sided interactions made it seem that the phenomena might have more incommon than had been apparent. In the Newtonian description, there isonly one cause for a change in motion, which we call force. Forces may beof different types, but they all produce changes in motion according to thesame rules. Any acceleration that can be produced by a magnetic force canequally well be produced by an appropriately controlled stream of water. Wecan speak of two forces as being equal if they produce the same change inmotion when applied in the same situation, which means that they pushedor pulled equally hard in the same direction.

“Our eyes receive blue light reflectedfrom this painting because Monetwanted to represent water with thecolor blue.” This is a valid statementat one level of explanation, but physicsworks at the physical level ofexplanation, in which blue light getsto your eyes because it is reflected byblue pigments in the paint.

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The idea of a numerical scale of force and the newton unit were intro-duced in chapter 0. To recapitulate briefly, a force is when a pair of objectspush or pull on each other, and one newton is the force required to acceler-ate a 1-kg object from rest to a speed of 1 m/s in 1 second.

More than one force on an objectAs if we hadn’t kicked poor Aristotle around sufficiently, his theory has

another important flaw, which is important to discuss because it corre-sponds to an extremely common student misconception. Aristotle con-ceived of forced motion as a relationship in which one object was the bossand the other “followed orders.” It therefore would only make sense for anobject to experience one force at a time, because an object couldn’t followorders from two sources at once. In the Newtonian theory, forces arenumbers, not orders, and if more than one force acts on an object at once,the result is found by adding up all the forces. It is unfortunate that the usethe English word “force” has become standard, because to many people itsuggests that you are “forcing” an object to do something. The force of theearth’s gravity cannot “force” a boat to sink, because there are other forcesacting on the boat. Adding them up gives a total of zero, so the boataccelerates neither up nor down.

Objects can exert forces on each other at a distanceAristotle declared that forces could only act between objects that were

touching, probably because he wished to avoid the type of occult specula-tion that attributed physical phenomena to the influence of a distant andinvisible pantheon of gods. He was wrong, however, as you can observewhen a magnet leaps onto your refrigerator or when the planet earth exertsgravitational forces on objects that are in the air. Some types of forces, suchas friction, only operate between objects in contact, and are called contactforces. Magnetism, on the other hand, is an example of a noncontact force.Although the magnetic force gets stronger when the magnet is closer toyour refrigerator, touching is not required.

WeightIn physics, an object’s weight , FW, is defined as the earth’s gravitational

force on it. The SI unit of weight is therefore the Newton. People com-monly refer to the kilogram as a unit of weight, but the kilogram is a unit ofmass, not weight. Note that an object’s weight is not a fixed property of thatobject. Objects weigh more in some places than in others, depending on thelocal strength of gravity. It is their mass that always stays the same. Abaseball pitcher who can throw a 90-mile-per-hour fastball on earth wouldnot be able to throw any faster on the moon, because the ball’s inertiawould still be the same.

Positive and negative signs of forceWe’ll start by considering only cases of one-dimensional center-of-mass

motion in which all the forces are parallel to the direction of motion, i.e.either directly forward or backward. In one dimension, plus and minussigns can be used to indicate directions of forces, as shown in the figure. Wecan then refer generically to addition of forces, rather than having to speaksometimes of addition and sometimes of subtraction. We add the forcesshown in the figure and get 11 N. In general, we should choose a one-

In this example, positive signs havebeen used consistently for forces tothe right, and negative signs for forcesto the left. The numerical value of aforce carries no information about theplace on the saxophone where theforce is applied.

+8 N

+4 N

+2 N

-3 N

Section 4.1 Force

98

dimensional coordinate system with its x axis parallel the direction ofmotion. Forces that point along the positive x axis are positive, and forces inthe opposite direction are negative. Forces that are not directly along the xaxis cannot be immediately incorporated into this scheme, but that’s OK,because we’re avoiding those cases for now.

Discussion questionsIn chapter 0, I defined 1 N as the force that would accelerate a 1-kg mass fromrest to 1 m/s in 1 s. Anticipating the following section, you might guess that 2N could be defined as the force that would accelerate the same mass to twicethe speed, or twice the mass to the same speed. Is there an easier way todefine 2 N based on the definition of 1 N?

4.2 Newton’s First LawWe are now prepared to make a more powerful restatement of the

principle of inertia.

Newton's First LawIf the total force on an object is zero, its center of masscontinues in the same state of motion.

In other words, an object initially at rest is predicted to remain at rest if thetotal force on it is zero, and an object in motion remains in motion with thesame velocity in the same direction. The converse of Newton’s first law isalso true: if we observe an object moving with constant velocity along astraight line, then the total force on it must be zero.

In a future physics course or in another textbook, you may encounterthe term net force, which is simply a synonym for total force.

What happens if the total force on an object is not zero? It accelerates.Numerical prediction of the resulting acceleration is the topic of Newton’ssecond law, which we’ll discuss in the following section.

This is the first of Newton’s three laws of motion. It is not important tomemorize which of Newton’s three laws are numbers one, two, and three. Ifa future physics teacher asks you something like, “Which of Newton’s lawsare you thinking of,” a perfectly acceptable answer is “The one aboutconstant velocity when there’s zero total force.” The concepts are moreimportant than any specific formulation of them. Newton wrote in Latin,and I am not aware of any modern textbook that uses a verbatim translationof his statement of the laws of motion. Clear writing was not in vogue inNewton’s day, and he formulated his three laws in terms of a concept nowcalled momentum, only later relating it to the concept of force. Nearly allmodern texts, including this one, start with force and do momentum later.

Example: an elevatorQuestion : An elevator has a weight of 5000 N. Compare theforces that the cable must exert to raise it at constant velocity,lower it at constant velocity, and just keep it hanging.Answer : In all three cases the cable must pull up with a force ofexactly 5000 N. Most people think you’d need at least a littlemore than 5000 N to make it go up, and a little less than 5000 Nto let it down, but that’s incorrect. Extra force from the cable is

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only necessary for speeding the car up when it starts going up orslowing it down when it finishes going down. Decreased force isneeded to speed the car up when it gets going down and to slowit down when it finishes going up. But when the elevator iscruising at constant velocity, Newton’s first law says that you justneed to cancel the force of the earth’s gravity.

To many students, the statement in the example that the cable’s upwardforce “cancels” the earth’s downward gravitational force implies that therehas been a contest, and the cable’s force has won, vanquishing the earth’sgravitational force and making it disappear. That is incorrect. Both forcescontinue to exist, but because they add up numerically to zero, the elevatorhas no center-of-mass acceleration. We know that both forces continue toexist because they both have side-effects other than their effects on the car’scenter-of-mass motion. The force acting between the cable and the carcontinues to produce tension in the cable and keep the cable taut. Theearth’s gravitational force continues to keep the passengers (whom we areconsidering as part of the elevator-object) stuck to the floor and to produceinternal stresses in the walls of the car, which must hold up the floor.

Example: terminal velocity for falling objectsQuestion : An object like a feather that is not dense orstreamlined does not fall with constant acceleration, because airresistance is nonnegligible. In fact, its acceleration tapers off tonearly zero within a fraction of a second, and the feather finishesdropping at constant speed (known as its terminal velocity). Whydoes this happen?Answer : Newton’s first law tells us that the total force on thefeather must have been reduced to nearly zero after a short time.There are two forces acting on the feather: a downwardgravitational force from the planet earth, and an upward frictionalforce from the air. As the feather speeds up, the air frictionbecomes stronger and stronger, and eventually it cancels out theearth’s gravitational force, so the feather just continues withconstant velocity without speeding up any more.

The situation for a skydiver is exactly analogous. It’s just thatthe skydiver experiences perhaps a million times moregravitational force than the feather, and it is not until she is fallingvery fast that the force of air friction becomes as strong as thegravitational force. It takes her several seconds to reach terminalvelocity, which is on the order of a hundred miles per hour.

Section 4.2 Newton’s First Law

100

More general combinations of forcesIt is too constraining to restrict our attention to cases where all the

forces lie along the line of the center of mass’s motion. For one thing, wecan’t analyze any case of horizontal motion, since any object on earth will besubject to a vertical gravitational force! For instance, when you are drivingyour car down a straight road, there are both horizontal forces and verticalforces. However, the vertical forces have no effect on the center of massmotion, because the road’s upward force simply counteracts the earth’sdownward gravitational force and keeps the car from sinking into theground.

Later in the book we’ll deal with the most general case of many forcesacting on an object at any angles, using the mathematical technique ofvector addition, but the following slight generalization of Newton’s first lawallows us to analyze a great many cases of interest:

Suppose that an object has two sets of forces acting on it, one set alongthe line of the object’s initial motion and another set perpendicular tothe first set. If both sets of forces cancel, then the object’s center of masscontinues in the same state of motion.

Example: a car crashQuestion : If you drive your car into a brick wall, what is themysterious force that slams your face into the steering wheel?Answer : Your surgeon has taken physics, so she is not going tobelieve your claim that a mysterious force is to blame. Sheknows that your face was just following Newton’s first law.Immediately after your car hit the wall, the only forces acting onyour head were the same canceling-out forces that had existedpreviously: the earth’s downward gravitational force and theupward force from your neck. There were no forward orbackward forces on your head, but the car did experience abackward force from the wall, so the car slowed down and yourface caught up.

Example: a passenger riding the subwayQuestion : Describe the forces acting on a person standing in asubway train that is cruising at constant velocity.Answer : No force is necessary to keep the person movingrelative to the ground. He will not be swept to the back of thetrain if the floor is slippery. There are two vertical forces on him,the earth’s downward gravitational force and the floor’s upwardforce, which cancel. There are no horizontal forces on him at all,so of course the total horizontal force is zero.

Example: forces on a sailboatQuestion : If a sailboat is cruising at constant velocity with thewind coming from directly behind it, what must be true about theforces acting on it?Answer : The forces acting on the boat must be canceling eachother out. The boat is not sinking or leaping into the air, soevidently the vertical forces are canceling out. The vertical forcesare the downward gravitational force exerted by the planet earthand an upward force from the water.

The air is making a forward force on the sail, and if the boatis not accelerating horizontally then the water’s backward

water's frictionalforce on boat

air's forceon sail

water's bouyantforce on boat

earth's gravitationalforce on boat

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frictional force must be canceling it out.Contrary to Aristotle, more force is not needed in order to

maintain a higher speed. Zero total force is always needed tomaintain constant velocity. Consider the following made-upnumbers:

boat moving at alow, constant

velocity

boat moving ata high, constant

velocity

forward force of thewind on the sail...... 10,000 N 20,000 N

backward force ofthe water on thehull........................ -10,000 N -20,000 N

total force on theboat...................... 0 N 0 N

The faster boat still has zero total force on it. The forward forceon it is greater, and the backward force smaller (more negative),but that’s irrelevant because Newton’s first law has to do with thetotal force, not the individual forces.

This example is quite analogous to the one about terminalvelocity of falling objects, since there is a frictional force thatincreases with speed. After casting off from the dock and raisingthe sail, the boat will accelerate briefly, and then reach itsterminal velocity, at which the water’s frictional force has becomeas great as the wind’s force on the sail.

Discussion questionA. Newton said that objects continue moving if no forces are acting on them,but his predecessor Aristotle said that a force was necessary to keep an objectmoving. Why does Aristotle’s theory seem more plausible, even though wenow believe it to be wrong? What insight was Aristotle missing about thereason why things seem to slow down naturally?B. In the first figure, what would have to be true about the saxophone’s initialmotion if the forces shown were to result in continued one-dimensionalmotion?C. The second figure requires an ever further generalization of the precedingdiscussion. After studying the forces, what does your physical intuition tell youwill happen? Can you state in words how to generalize the conditions for one-dimensional motion to include situations like this one?

8 N

2 N 3 N

3 N

4 N

4 NDiscussion question B. Discussion question C.

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4.3 Newton’s Second LawWhat about cases where the total force on an object is not zero, so that

Newton’s first law doesn’t apply? The object will have an acceleration. Theway we’ve defined positive and negative signs of force and accelerationguarantees that positive forces produce positive accelerations, and likewisefor negative values. How much acceleration will it have? It will clearlydepend on both the object’s mass and on the amount of force.

Experiments with any particular object show that its acceleration isdirectly proportional to the total force applied to it. This may seem wrong,since we know of many cases where small amounts of force fail to move anobject at all, and larger forces get it going. This apparent failure of propor-tionality actually results from forgetting that there is a frictional force inaddition to the force we apply to move the object. The object’s accelerationis exactly proportional to the total force on it, not to any individual force onit. In the absence of friction, even a very tiny force can slowly change thevelocity of a very massive object.

Experiments also show that the acceleration is inversely proportional tothe object’s mass, and combining these two proportionalities gives thefollowing way of predicting the acceleration of any object:

Newton's Second Lawa = Ftotal/m ,

wherem is an object's mass,Ftotal is the sum of the forces acting on it, anda is the acceleration of the object's center of mass.

We are presently restricted to the case where the forces of interest areparallel to the direction of motion.

Example: an accelerating busQuestion : A VW bus with a mass of 2000 kg accelerates from 0to 25 m/s (freeway speed) in 34 s. Assuming the acceleration isconstant, what is the total force on the bus?Solution : We solve Newton’s second law for Ftotal=ma, andsubstitute ∆v/∆t for a, giving

Ftotal = m∆v/∆t= (2000 kg)(25 m/s - 0 m/s)/(34 s)= 1.5 kN .

A generalizationAs with the first law, the second law can be easily generalized to include

a much larger class of interesting situations:

Suppose an object is being acted on by two sets of forces, one setlying along the object’s initial direction of motion and another setacting along a perpendicular line. If the forces perpendicular to theinitial direction of motion cancel out, then the object acceleratesalong its original line of motion according to a=Ftotal/m.

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The relationship between mass and weightWe have already seen the experimental evidence that when weight (the

force of the earth’s gravity) is the only force acting on an object, its accelera-tion equals the constant g, and g depends on where you are on the surface ofthe earth, but not on the mass of the object. Applying Newton’s second lawthen allows us to calculate the magnitude of the gravitational force on anyobject in terms of its mass:

|FW| = mg .

(The equation only gives the magnitude, i.e. the absolute value, of FW,because we’re defining g as a positive number, so it equals the absolute valueof a falling object’s acceleration.)

Example: calculating terminal velocityQuestion : Experiments show that the force of air friction on afalling object such as a skydiver or a feather can beapproximated fairly well with the equation |Fair|=cρAv2, where c isa constant, ρ is the density of the air, A is the cross-sectionalarea of the object as seen from below, and v is the object’svelocity. Predict the object’s terminal velocity, i.e. the finalvelocity it reaches after a long time.Solution : As the object accelerates, its greater v causes theupward force of the air to increase until finally the gravitationalforce and the force of air friction cancel out, after which theobject continues at constant velocity. We choose a coordinatesystem in which positive is up, so that the gravitational force isnegative and the force of air friction is positive. We want to findthe velocity at which

Fair + FW = 0 , i.e.

cρAv 2 – mg= 0 .Solving for v gives

vterminal = m gc ρ A

Self-CheckIt is important to get into the habit of interpreting equations. These two self-check questions may be difficult for you, but eventually you will get used to thiskind of reasoning.

(a) Interpret the equation vterminal = m g / c ρ A in the case of ρ=0.(b) How would the terminal velocity of a 4-cm steel ball compare to that of a 1-cm ball?

A simple double-pan balance works bycomparing the weight forces exertedby the earth on the contents of the twopans. Since the two pans are at almostthe same location on the earth’ssurface, the value of g is essentiallythe same for each one, and equalityof weight therefore also impliesequality of mass.

(a) The case of ρ=0 represents an object falling in a vacuum, i.e. there is no density of air. The terminal velocitywould be infinite. Physically, we know that an object falling in a vacuum would never stop speeding up, since therewould be no force of air friction to cancel the force of gravity. (b) The 4-cm ball would have a mass that was greaterby a factor of 4x4x4, but its cross-sectional area would be greater by a factor of 4x4. Its terminal velocity would be

greater by a factor of 43 / 42=2.

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Discussion questions

A. Show that the Newton can be reexpressed in terms of the three basic mksunits as the combination kg.m/s2.B. What is wrong with the following statements?

1. “g is the force of gravity.”2. “Mass is a measure of how much space something takes up.”

C. Criticize the following incorrect statement:“If an object is at rest and the total force on it is zero, it stays at rest.There can also be cases where an object is moving and keeps on movingwithout having any total force on it, but that can only happen when there’sno friction, like in outer space.”

D. The table on the left gives laser timing data for Ben Johnson’s 100 m dashat the 1987 World Championship in Rome. (His world record was later revokedbecause he tested positive for steroids.) How does the total force on himchange over the duration of the race?

4.4 What Force Is NotViolin teachers have to endure their beginning students’ screeching. A

frown appears on the woodwind teacher’s face as she watches her studenttake a breath with an expansion of his ribcage but none in his belly. Whatmakes physics teachers cringe is their students’ verbal statements aboutforces. Below I have listed several dicta about what force is not.

Force is not a property of one object.A great many of students’ incorrect descriptions of forces could be cured

by keeping in mind that a force is an interaction of two objects, not aproperty of one object.

Incorrect statement: “That magnet has a lot of force.” If the magnet is one millimeter away from a steel ball bearing,they may exert a very strong attraction on each other, but if theywere a meter apart, the force would be virtually undetectable.The magnet’s strength can be rated using certain electrical units(ampere-meters2), but not in units of force.

Force is not a measure of an object’s motion.If force is not a property of a single object, then it cannot be used as a

measure of the object’s motion.

Incorrect statement: “The freight train rumbled down the trackswith awesome force.” Force is not a measure of motion. If the freight train collideswith a stalled cement truck, then some awesome forces willoccur, but if it hits a fly the force will be small.

Force is not energy.There are two main approaches to understanding the motion of objects,

one based on force and one on a different concept, called energy. The SIunit of energy is the Joule, but you are probably more familiar with thecalorie, used for measuring food’s energy, and the kilowatt-hour, the unitthe electric company uses for billing you. Physics students’ previous famil-iarity with calories and kilowatt-hours is matched by their universal unfa-miliarity with measuring forces in units of Newtons, but the precise opera-tional definitions of the energy concepts are more complex than those of the

Discussion question D.

x (m) t (s)10 1.8420 2.8630 3.8040 4.6750 5.5360 6.3870 7.2380 8.1090 8.96100 9.83

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force concepts, and textbooks, including this one, almost universally placethe force description of physics before the energy description. During thelong period after the introduction of force and before the careful definitionof energy, students are therefore vulnerable to situations in which, withoutrealizing it, they are imputing the properties of energy to phenomena offorce.

Incorrect statement: “How can my chair be making an upwardforce on my rear end? It has no power!” Power is a concept related to energy, e.g. 100-watt lightbulbuses up 100 joules per second of energy. When you sit in a chair,no energy is used up, so forces can exist between you and thechair without any need for a source of power.

Force is not stored or used up.Because energy can be stored and used up, people think force also can

be stored or used up.Incorrect statement: “If you don’t fill up your tank with gas, you’llrun out of force.” Energy is what you’ll run out of, not force.

Forces need not be exerted by living things or machines.Transforming energy from one form into another usually requires some

kind of living or mechanical mechanism. The concept is not applicable toforces, which are an interaction between objects, not a thing to be trans-ferred or transformed.

Incorrect statement: “How can a wooden bench be making anupward force on my rear end? It doesn’t have any springs oranything inside it.” No springs or other internal mechanisms are required. If thebench didn’t make any force on you, you would obey Newton’ssecond law and fall through it. Evidently it does make a force onyou!

A force is the direct cause of a change in motion.I can click a remote control to make my garage door change from being

at rest to being in motion. My finger’s force on the button, however, wasnot the force that acted on the door. When we speak of a force on an objectin physics, we are talking about a force that acts directly. Similarly, whenyou pull a reluctant dog along by its leash, the leash and the dog are makingforces on each other, not your hand and the dog. The dog is not eventouching your hand.

Self-CheckWhich of the following things can be correctly described in terms of force?

(a) A nuclear submarine is charging ahead at full steam.(b) A nuclear submarine’s propellers spin in the water.(c) A nuclear submarine needs to refuel its reactor periodically.

Discussion questionsA. Criticize the following incorrect statement: “If you shove a book across atable, friction takes away more and more of its force, until finally it stops.”B. You hit a tennis ball against a wall. Explain any and all incorrect ideas in thefollowing description of the physics involved: “The ball gets some force fromyou when you hit it, and when it hits the wall, it loses part of that force, so itdoesn’t bounce back as fast. The muscles in your arm are the only things thata force can come from.”

(a) This is motion, not force. (b) This is a description of how the sub is able to get the water to produce a forwardforce on it. (c) The sub runs out of energy, not force.

Section 4.4 What Force Is Not

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4.5 Inertial and Noninertial Frames of ReferenceOne day, you’re driving down the street in your pickup truck, on your

way to deliver a bowling ball. The ball is in the back of the truck, enjoyingits little jaunt and taking in the fresh air and sunshine. Then you have toslow down because a stop sign is coming up. As you brake, you glance inyour rearview mirror, and see your trusty companion accelerating towardyou. Did some mysterious force push it forward? No, it only seems that waybecause you and the car are slowing down. The ball is faithfully obeyingNewton’s first law, and as it continues at constant velocity it gets aheadrelative to the slowing truck. No forces are acting on it (other than the samecanceling-out vertical forces that were always acting on it). The ball onlyappeared to violate Newton’s first law because there was something wrongwith your frame of reference, which was based on the truck.

How, then, are we to tell in which frames of reference Newton’s laws arevalid? It’s no good to say that we should avoid moving frames of reference,because there is no such thing as absolute rest or absolute motion. Allframes can be considered as being either at rest or in motion. According to

(a) In a frame of reference that moves withthe truck, the bowling ball appears to violateNewton's first law by accelerating despitehaving no horizontal forces on it.

(b) In an inertial frame of reference, which the surface of the earthapproximately is, the bowling ball obeys Newton's first law. Itmoves equal distances in equal time intervals, i.e. maintainsconstant velocity. In this frame of reference, it is the truck thatappears to have a change in velocity, which makes sense, sincethe road is making a horizontal force on it.

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an observer in India, the strip mall that constituted the frame of reference inpanel (b) of the figure was moving along with the earth’s rotation at hun-dreds of miles per hour.

The reason why Newton’s laws fail in the truck’s frame of reference isnot because the truck is moving but because it is accelerating. (Recall thatphysicists use the word to refer either to speeding up or slowing down.)Newton’s laws were working just fine in the moving truck’s frame ofreference as long as the truck was moving at constant velocity. It was onlywhen its speed changed that there was a problem. How, then, are we to tellwhich frames are accelerating and which are not? What if you claim thatyour truck is not accelerating, and the sidewalk, the asphalt, and the BurgerKing are accelerating? The way to settle such a dispute is to examine themotion of some object, such as the bowling ball, which we know has zerototal force on it. Any frame of reference in which the ball appears to obeyNewton’s first law is then a valid frame of reference, and to an observer inthat frame, Mr. Newton assures us that all the other objects in the universewill obey his laws of motion, not just the ball.

Valid frames of reference, in which Newton’s laws are obeyed, are calledinertial frames of reference. Frames of reference that are not inertial are callednoninertial frames. In those frames, objects violate the principle of inertiaand Newton’s first law. While the truck was moving at constant velocity,both it and the sidewalk were valid inertial frames. The truck became aninvalid frame of reference when it began changing its velocity.

You usually assume the ground under your feet is a perfectly inertialframe of reference, and we made that assumption above. It isn’t perfectlyinertial, however. Its motion through space is quite complicated, beingcomposed of a part due to the earth’s daily rotation around its own axis, themonthly wobble of the planet caused by the moon’s gravity, and the rota-tion of the earth around the sun. Since the accelerations involved arenumerically small, the earth is approximately a valid inertial frame.

Noninertial frames are avoided whenever possible, and we will seldom,if ever, have occasion to use them in this course. Sometimes, however, anoninertial frame can be convenient. Naval gunners, for instance, get alltheir data from radars, human eyeballs, and other detection systems that aremoving along with the earth’s surface. Since their guns have ranges of manymiles, the small discrepancies between their shells’ actual accelerations andthe accelerations predicted by Newton’s second law can have effects thataccumulate and become significant. In order to kill the people they want tokill, they have to add small corrections onto the equation a=Ftotal/m. Doingtheir calculations in an inertial frame would allow them to use the usualform of Newton’s second law, but they would have to convert all their datainto a different frame of reference, which would require cumbersomecalculations.

Discussion questionIf an object has a linear x-t graph in a certain inertial frame, what is the effecton the graph if we change to a coordinate system with a different origin? Whatis the effect if we keep the same origin but reverse the positive direction of thex axis? How about an inertial frame moving alongside the object? What if wedescribe the object’s motion in a noninertial frame?

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SummarySelected Vocabulary

weight ............................... the force of gravity on an object, equal to mginertial frame ..................... a frame of reference that is not accelerating, one in which Newton’s first

law is truenoninertial frame ............... an accelerating frame of reference, in which Newton’s first law is violated

Terminology Used in Some Other Booksnet force ............................ another way of saying “total force”

NotationFW ..................................................... the weight force

SummaryNewton’s first law of motion states that if all the forces on an object cancel each other out, then the object

continues in the same state of motion. This is essentially a more refined version of Galileo’s principle ofinertia, which did not refer to a numerical scale of force.

Newton’s second law of motion allows the prediction of an object’s acceleration given its mass and thetotal force on it, a=Ftotal/m. This is only the one-dimensional version of the law; the full-three dimensionaltreatment will come in chapter 8, Vectors. Without the vector techniques, we can still say that the situationremains unchanged by including an additional set of vectors that cancel among themselves, even if they arenot in the direction of motion.

Newton’s laws of motion are only true in frames of reference that are not accelerating, known as inertialframes.

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Homework Problems1. An object is observed to be moving at constant speed along a line. Canyou conclude that no forces are acting on it? Explain. [Based on a problemby Serway and Faughn.]

2. A car is normally capable of an acceleration of 3 m/s2. If it is towing atrailer with half as much mass as the car itself, what acceleration can itachieve? [Based on a problem from PSSC Physics.]

3. (a) Let T be the maximum tension that the elevator's cable can withstandwithout breaking, i.e. the maximum force it can exert. If the motor isprogrammed to give the car an acceleration a, what is the maximum massthat the car can have, including passengers, if the cable is not to break?[Numerical check, not for credit: for T=1.0x104 N and a=3.0 m/s2, yourequation should give an answer of 780 kg.] (b) Interpret the equation youderived in the special cases of a=0 and of a downward acceleration ofmagnitude g.

4 . A helicopter of mass m is taking off vertically. The only forces acting onit are the earth's gravitational force and the force, Fair, of the air pushing upon the propeller blades. (a) If the helicopter lifts off at t=0, what is itsvertical speed at time t? (b) Plug numbers into your equation from part a,using m=2300 kg, Fair=27000 N, and t=4.0 s.

5«. In the 1964 Olympics in Tokyo, the best men's high jump was 2.18m. Four years later in Mexico City, the gold medal in the same event wasfor a jump of 2.24 m. Because of Mexico City's altitude (2400 m), theacceleration of gravity there is lower than that in Tokyo by about 0.01 m/s2.Suppose a high-jumper has a mass of 72 kg.

(a) Compare his mass and weight in the two locations.

(b) Assume that he is able to jump with the same initial vertical velocityin both locations, and that all other conditions are the same except forgravity. How much higher should he be able to jump in Mexico City?

(Actually, the reason for the big change between '64 and '68 was theintroduction of the "Fosbury flop.")

6 ∫. A blimp is initially at rest, hovering, when at t=0 the pilot turns on themotor of the propeller. The motor cannot instantly get the propeller going,but the propeller speeds up steadily. The steadily increasing force betweenthe air and the propeller is given by the equation F=kt, where k is a con-stant. If the mass of the blimp is m, find its position as a function of time.(Assume that during the period of time you're dealing with, the blimp isnot yet moving fast enough to cause a significant backward force due to airresistance.)

7 S. A car is accelerating forward along a straight road. If the force of theroad on the car's wheels, pushing it forward, is a constant 3.0 kN, and thecar's mass is 1000 kg, then how long will the car take to go from 20 m/s to50 m/s?

Problem 6.

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110

8. Some garden shears are like a pair of scissors: one sharp blade slices pastanother. In the “anvil” type, however, a sharp blade presses against a flat onerather than going past it. A gardening book says that for people who are notvery physically strong, the anvil type can make it easier to cut toughbranches, because it concentrates the force on one side. Evaluate this claimbased on Newton’s laws. [Hint: Consider the forces acting on the branch,and the motion of the branch.]

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5 Analysis of Forces5.1 Newton’s Third Law

Newton created the modern concept of force starting from his insightthat all the effects that govern motion are interactions between two objects:unlike the Aristotelian theory, Newtonian physics has no phenomena inwhich an object changes its own motion.

Is one object always the “order-giver” and the other the “order-fol-lower”? As an example, consider a batter hitting a baseball. The bat defi-nitely exerts a large force on the ball, because the ball accelerates drastically.But if you have ever hit a baseball, you also know that the ball makes a forceon the bat — often with painful results if your technique is as bad as mine!

How does the ball’s force on the bat compare with the bat’s force on theball? The bat’s acceleration is not as spectacular as the ball’s, but maybe weshouldn’t expect it to be, since the bat’s mass is much greater. In fact, carefulmeasurements of both objects’ masses and accelerations would show thatm

balla

ball is very nearly equal to –m

bata

bat, which suggests that the ball’s force

on the bat is of the same magnitude as the bat’s force on the ball, but in theopposite direction.

Rockets work by pushing exhaust gases outthe back. Newton’s third law says that if therocket exerts a backward force on the gases,the gases must make an equal forward forceon the rocket. Rocket engines can functionabove the atmosphere, unlike propellers andjets, which work by pushing against the sur-rounding air.

Section 5.1 Newton’s Third Law

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The figures show two somewhat more practical laboratory experimentsfor investigating this issue accurately and without too much interferencefrom extraneous forces.

In the first experiment, a large magnet and a small magnet are weighedseparately, and then one magnet is hung from the pan of the top balance sothat it is directly above the other magnet. There is an attraction between thetwo magnets, causing the reading on the top scale to increase and thereading on the bottom scale to decrease. The large magnet is more “power-ful” in the sense that it can pick up a heavier paperclip from the samedistance, so many people have a strong expectation that one scale’s readingwill change by a far different amount than the other. Instead, we find thatthe two changes are equal in magnitude but opposite in direction, so theupward force of the top magnet on the bottom magnet is of the samemagnitude as the downward force of the bottom magnet on the top mag-net.

In the second experiment, two people pull on two spring scales. Regard-less of who tries to pull harder, the two forces as measured on the springscales are equal. Interposing the two spring scales is necessary in order tomeasure the forces, but the outcome is not some artificial result of thescales’ interactions with each other. If one person slaps another hard on thehand, the slapper’s hand hurts just as much as the slappee’s, and it doesn’tmatter if the recipient of the slap tries to be inactive. (Punching someone inthe mouth causes just as much force on the fist as on the lips. It’s just thatthe lips are more delicate. The forces are equal, but not the levels of painand injury.)

Newton, after observing a series of results such as these, decided thatthere must be a fundamental law of nature at work:

Newton's Third LawForces occur in equal and opposite pairs: wheneverobject A exerts a force on object B, object B must alsobe exerting a force on object A. The two forces areequal in magnitude and opposite in direction.

In one-dimensional situations, we can use plus and minus signs to indicatethe directions of forces, and Newton’s third law can be written succinctly asF

A on B = –F

B on A.

There is no cause and effect relationship between the two forces. Thereis no “original” force, and neither one is a response to the other. The pair offorces is a relationship, like marriage, not a back-and-forth process like atennis match. Newton came up with the third law as a generalization aboutall the types of forces with which he was familiar, such as frictional andgravitational forces. When later physicists discovered a new type force, suchas the force that holds atomic nuclei together, they had to check whether itobeyed Newton’s third law. So far, no violation of the third law has everbeen discovered, whereas the first and second laws were shown to havelimitations by Einstein and the pioneers of atomic physics.

magnet

magnet

scale

scale

(a) Two magnets exert forces on eachother.

(b) Two people’s hands exert forceson each other.

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The English vocabulary for describing forces is unfortunately rooted inAristotelianism, and often implies incorrectly that forces are one-wayrelationships. It is unfortunate that a half-truth such as “the table exerts anupward force on the book” is so easily expressed, while a more complete andcorrect description ends up sounding awkward or strange: “the table and thebook interact via a force,” or “the table and book participate in a force.”

To students, it often sounds as though Newton’s third law impliesnothing could ever change its motion, since the two equal and oppositeforces would always cancel. The two forces, however, are always on twodifferent objects, so it doesn’t make sense to add them in the first place —we only add forces that are acting on the same object. If two objects areinteracting via a force and no other forces are involved, then both objectswill accelerate — in opposite directions!

Excuse me, ma'am, but itappears that the money in your

purse would exactly cancelout my bar tab.

It doesn’t make sense for the man totalk about the woman’s money can-celing out his bar tab, because thereis no good reason to combine hisdebts and her assets. Similarly, itdoesn’t make sense to refer to theequal and opposite forces of Newton’sthird law as canceling. It only makessense to add up forces that are actingon the same object, whereas twoforces related to each other byNewton’s third law are always actingon two different objects.

Newton’s third law does not mean thatforces always cancel out so that noth-ing can ever move. If these two figureskaters, initially at rest, push againsteach other, they will both move.

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A mnemonic for using Newton’s third law correctlyMnemonics are tricks for memorizing things. For instance, the musical

notes that lie between the lines on the treble clef spell the word FACE,which is easy to remember. Many people use the mnemonic“SOHCAHTOA” to remember the definitions of the sine, cosine, andtangent in trigonometry. I have my own modest offering, POFOSTITO,which I hope will make it into the mnemonics hall of fame. It’s a way toavoid some of the most common problems with applying Newton’s thirdlaw correctly:

Pair ofOppositeForcesOf theSameTypeInvolvingTwoObjects

ExampleQuestion : A book is lying on a table. What force is the Newton’s-third-law partner of the earth’s gravitational force on the book?Answer : Newton’s third law works like “B on A, A on B,” so thepartner must be the book’s gravitational force pulling upward onthe planet earth. Yes, there is such a force! No, it does not causethe earth to do anything noticeable.Incorrect answer : The table’s upward force on the book is theNewton’s-third-law partner of the earth’s gravitational force onthe book. This answer violates two out of three of the commandments ofPOFOSTITO. The forces are not of the same type, because thetable’s upward force on the book is not gravitational. Also, threeobjects are involved instead of two: the book, the table, and theplanet earth.

ExampleQuestion : A person is pushing a box up a hill. What force isrelated by Newton’s third law to the person’s force on the box?Answer : The box’s force on the person.Incorrect answer : The person’s force on the box is opposed byfriction, and also by gravity. This answer fails all three parts of the POFOSTITO test, themost obvious of which is that three forces are referred to insteadof a pair.

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Optional Topic: Newton’s third law and action at a distanceNewton’s third law is completely symmetric in the sense that neitherforce constitutes a delayed response to the other. Newton’s third lawdoes not even mention time, and the forces are supposed to agreeat any given instant. This creates an interesting situation when itcomes to noncontact forces. Suppose two people are holdingmagnets, and when one person waves or wiggles her magnet, theother person feels an effect on his. In this way they can send signalsto each other from opposite sides of a wall, and if Newton’s third lawis correct, it would seem that the signals are transmitted instantly,with no time lag. The signals are indeed transmitted quite quickly,but experiments with electronically controlled magnets show that thesignals do not leap the gap instantly: they travel at the same speedas light, which is an extremely high speed but not an infinite one.

Is this a contradiction to Newton’s third law? Not really. According tocurrent theories, there are no true noncontact forces. Action at adistance does not exist. Although it appears that the wiggling of onemagnet affects the other with no need for anything to be in contactwith anything, what really happens is that wiggling a magnet un-leashes a shower of tiny particles called photons. The magnetshoves the photons out with a kick, and receives a kick in return, instrict obedience to Newton’s third law. The photons fly out in alldirections, and the ones that hit the other magnet then interact withit, again obeying Newton’s third law.

Photons are nothing exotic, really. Light is made of photons, but oureyes receive such huge numbers of photons that we do not perceivethem individually. The photons you would make by wiggling amagnet with your hand would be of a “color” that you cannot see, faroff the red end of the rainbow.

Discussion questionsA. When you fire a gun, the exploding gases push outward in all directions,causing the bullet to accelerate down the barrel. What third-law pairs areinvolved? [Hint: Remember that the gases themselves are an object.]B. Tam Anh grabs Sarah by the hand and tries to pull her. She tries to remainstanding without moving. A student analyzes the situation as follows. “If TamAnh’s force on Sarah is greater than her force on him, he can get her to move.Otherwise, she’ll be able to stay where she is.” What’s wrong with this analy-sis?C. You hit a tennis ball against a wall. Explain any and all incorrect ideas in thefollowing description of the physics involved: “According to Newton’s third law,there has to be a force opposite to your force on the ball. The opposite force isthe ball’s mass, which resists acceleration, and also air resistance.”

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5.2 Classification and Behavior of ForcesOne of the most basic and important tasks of physics is to classify the

forces of nature. I have already referred informally to “types” of forces suchas friction, magnetism, gravitational forces, and so on. Classificationsystems are creations of the human mind, so there is always some degree ofarbitrariness in them. For one thing, the level of detail that is appropriatefor a classification system depends on what you’re trying to find out. Somelinguists, the “lumpers,” like to emphasize the similarities among languages,and a few extremists have even tried to find signs of similarities betweenwords in languages as different as English and Chinese, lumping the world’slanguages into only a few large groups. Other linguists, the “splitters,”might be more interested in studying the differences in pronunciationbetween English speakers in New York and Connecticut. The splitters callthe lumpers sloppy, but the lumpers say that science isn’t worthwhile unlessit can find broad, simple patterns within the seemingly complex universe.

Scientific classification systems are also usually compromises betweenpracticality and naturalness. An example is the question of how to classifyflowering plants. Most people think that biological classification is aboutdiscovering new species, naming them, and classifying them in the class-order-family-genus-species system according to guidelines set long ago. Inreality, the whole system is in a constant state of flux and controversy. Onevery practical way of classifying flowering plants is according to whethertheir petals are separate or joined into a tube or cone — the criterion is soclear that it can be applied to a plant seen from across the street. But herepracticality conflicts with naturalness. For instance, the begonia has separatepetals and the pumpkin has joined petals, but they are so similar in so manyother ways that they are usually placed within the same order. Some taxono-mists have come up with classification criteria that they claim correspondmore naturally to the apparent relationships among plants, without havingto make special exceptions, but these may be far less practical, requiring forinstance the examination of pollen grains under an electron microscope.

In physics, there are two main systems of classification for forces. At thispoint in the course, you are going to learn one that is very practical and easyto use, and that splits the forces up into a relatively large number of types:seven very common ones that we’ll discuss explicitly in this chapter, plusperhaps ten less important ones such as surface tension, which we will notbother with right now.

Professional physicists, however, are almost all obsessed with findingsimple patterns, so recognizing as many as fifteen or twenty types of forcesstrikes them as distasteful and overly complex. Since about the year 1900,physics has been on an aggressive program to discover ways in which thesemany seemingly different types of forces arise from a smaller number offundamental ones. For instance, when you press your hands together, theforce that keeps them from passing through each other may seem to havenothing to do with electricity, but at the atomic level, it actually does arisefrom electrical repulsion between atoms. By about 1950, all the forces ofnature had been explained as arising from four fundamental types of forcesat the atomic and nuclear level, and the lumping-together process didn’tstop there. By the 1960’s the length of the list had been reduced to three,

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and some theorists even believe that they may be able to reduce it to two orone. Although the unification of the forces of nature is one of the mostbeautiful and important achievements of physics, it makes much more senseto start this course with the more practical and easy system of classification.The unified system of four forces will be one of the highlights of the end ofyour introductory physics sequence.

The practical classification scheme which concerns us now can be laidout in the form of the tree shown below. The most specific types of forcesare shown at the tips of the branches, and it is these types of forces that arereferred to in the POFOSTITO mnemonic. For example, electrical andmagnetic forces belong to the same general group, but Newton’s third lawwould never relate an electrical force to a magnetic force.

The broadest distinction is that between contact and noncontact forces,which has been discussed in the previous chapter. Among the contact forces,we distinguish between those that involve solids only and those that have todo with fluids, a term used in physics to include both gases and liquids. Theterms “repulsive,” “attractive,” and “oblique” refer to the directions of theforces.

• Repulsive forces are those that tend to push the two participatingobjects away from each other. More specifically, a repulsive contactforce acts perpendicular to the surfaces at which the two objectstouch, and a repulsive noncontact force acts along the line betweenthe two objects.

• Attractive forces pull the two objects toward one another, i.e. they actalong the same line as repulsive forces, but in the opposite direction.

• Oblique forces are those that act at some other angle.

normalforce

staticfriction

kineticfriction

fluidfriction gravity

electricalforces

magneticforces

contactforces noncontact

forces

forcesbetweensolids

forces betweensolids andfluids or fluidsand fluids

repulsive

oblique

notslipping

slipping

oblique alwaysattractive,depends onthe objects'masses

depends onthe objects'charge, anelectricalproperty

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It should not be necessary to memorize this diagram by rote. It is betterto reinforce your memory of this system by calling to mind yourcommonsense knowledge of certain ordinary phenomena. For instance, weknow that the gravitational attraction between us and the planet earth willact even if our feet momentarily leave the ground, and that althoughmagnets have mass and are affected by gravity, most objects that have massare nonmagnetic.

This diagram is meant to be as simple as possible while including mostof the forces we deal with in everyday life. If you were an insect, you wouldbe much more interested in the force of surface tension, which allowed youto walk on water. I have not included the nuclear forces, which are respon-sible for holding the nuclei of atoms, because they are not evident ineveryday life.

You should not be afraid to invent your own names for types of forcesthat do not fit into the diagram. For instance, the force that holds a piece oftape to the wall has been left off of the tree, and if you were analyzing asituation involving scotch tape, you would be absolutely right to refer to itby some commonsense name such as “sticky force.”

On the other hand, if you are having trouble classifying a certain force,you should also consider whether it is a force at all. For instance, if someoneasks you to classify the force that the earth has because of its rotation, youwould have great difficulty creating a place for it on the diagram. That’sbecause it’s a type of motion, not a type of force!

The normal forceThe normal force, F

N, is the force that keeps one solid object from

passing through another. “Normal” is simply a fancy word for “perpendicu-lar,” meaning that the force is perpendicular to the surface of contact.Intuitively, it seems the normal force magically adjusts itself to providewhatever force is needed to keep the objects from occupying the same space.If your muscles press your hands together gently, there is a gentle normalforce. Press harder, and the normal force gets stronger. How does thenormal force know how strong to be? The answer is that the harder you jamyour hands together, the more compressed your flesh becomes. Your flesh isacting like a spring: more force is required to compress it more. The same istrue when you push on a wall. The wall flexes imperceptibly in proportionto your force on it. If you exerted enough force, would it be possible for twoobjects to pass through each other? No, typically the result is simply tostrain the objects so much that one of them breaks.

Gravitational forcesAs we’ll discuss in more detail later in the course, a gravitational force

exists between any two things that have mass. In everyday life, the gravita-tional force between two cars or two people is negligible, so the onlynoticeable gravitational forces are the ones between the earth and varioushuman-scale objects. We refer to these planet-earth-induced gravitationalforces as weight forces, and as we have already seen, their magnitude isgiven by |F

W|=mg.

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Static and kinetic frictionIf you have pushed a refrigerator across a kitchen floor, you have felt a

certain series of sensations. At first, you gradually increased your force onthe refrigerator, but it didn’t move. Finally, you supplied enough force tounstick the fridge, and there was a sudden jerk as the fridge started moving.Once the fridge is unstuck, you can reduce your force significantly and stillkeep it moving.

While you were gradually increasing your force, the floor’s frictionalforce on the fridge increased in response. The two forces on the fridgecanceled, and the fridge didn’t accelerate. How did the floor know how torespond with just the right amount of force? The figures on the left showone possible model of friction that explains this behavior. (A scientific modelis a description that we expect to be incomplete, approximate, or unrealisticin some ways, but that nevertheless succeeds in explaining a variety ofphenomena.) Figure (a) shows a microscopic view of the tiny bumps andholes in the surfaces of the floor and the refrigerator. The weight of thefridge presses the two surfaces together, and some of the bumps in onesurface will settle as deeply as possible into some of the holes in the othersurface. In figure (b), your leftward force on the fridge has caused it to rideup a little higher on the bump in the floor labeled with a small arrow. Stillmore force is needed to get the fridge over the bump and allow it to startmoving. Of course, this is occurring simultaneously at millions of places onthe two surfaces.

Once you had gotten the fridge moving at constant speed, you foundthat you needed to exert less force on it. Since zero total force is needed tomake an object move with constant velocity, the floor’s rightward frictionalforce on the fridge has apparently decreased somewhat, making it easier foryou to cancel it out. Our model also gives a plausible explanation for thisfact: as the surfaces slide past each other, they don’t have time to settle downand mesh with one another, so there is less friction.

Even though this model is intuitively appealing and fairly successful, itshould not be taken too seriously, and in some situations it is misleading.For instance, fancy racing bikes these days are made with smooth tires thathave no tread — contrary to what we’d expect from our model, this doesnot cause any decrease in friction. Machinists know that two very smoothand clean metal surfaces may stick to each other firmly and be very difficultto slide apart. This cannot be explained in our model, but makes more sensein terms of a model in which friction is described as arising from chemicalbonds between the atoms of the two surfaces at their points of contact: veryflat surfaces allow more atoms to come in contact.

A model that correctly explains manyproperties of friction. The microscopicbumps and holes in two surfaces diginto each other, causing a frictionalforce.

(a)

(b) force

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Since friction changes its behavior dramatically once the surfaces comeunstuck, we define two separate types of frictional forces. Static friction isfriction that occurs between surfaces that are not slipping over each other.Slipping surfaces experience kinetic friction. “Kinetic” means having to dowith motion. The forces of static and kinetic friction, notated F

s and F

k, are

always parallel to the surface of contact between the two objects.

Self-CheckDoes static friction create heat? Kinetic friction?

The maximum possible force of static friction depends on what kinds ofsurfaces they are, and also on how hard they are being pressed together. Theapproximate mathematical relationships can be expressed as follows:

Fs = –F

applied, when |F

applied| < µ

s|F

N| ,

where µs is a unitless number, called the coefficient of static friction, which

depends on what kinds of surfaces they are. The maximum force that staticfriction can supply, µ

s|F

N|, represents the boundary between static and

kinetic friction. It depends on the normal force, which is numerically equalto whatever force is pressing the two surfaces together. In terms of ourmodel, if the two surfaces are being pressed together more firmly, a greatersideways force will be required in order to make the irregularities in thesurfaces ride up and over each other.

Note that just because we use an adjective such as “applied” to refer to aforce, that doesn’t mean that there is some special type of force called the“applied force.” The applied force could be any type of force, or it could bethe sum of more than one force trying to make an object move.

The force of kinetic friction on each of the two objects is in the direc-tion that resists the slippage of the surfaces. Its magnitude is usually wellapproximated as

|Fk|=µ

k|F

N|

where µk is the coefficient of kinetic friction. Kinetic friction is usually more

or less independent of velocity.

We choose a coordinate system inwhich the applied force, i.e. the forcetrying to move the objects, is positive.The friction force is then negative,since it is in the opposite direction. Asyou increase the applied force, theforce of static friction increases tomatch it and cancel it out, until themaximum force of static friction is sur-passed. The surfaces then begin slip-ping past each other, and the frictionforce becomes smaller in absolutevalue.

appliedforce

frictionforce

static friction kinetic friction

Only kinetic friction creates heat, as when you rub your hands together. If you move your hands up and downtogether without sliding them across each other, no heat is produced by the static friction.

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Self-CheckCan a frictionless surface exert a normal force? Can a frictional force existwithout a normal force?

If you try to accelerate or decelerate your car too quickly, the forcesbetween your wheels and the road become too great, and they beginslipping. This is not good, because kinetic friction is weaker than staticfriction, resulting in less control. Also, if this occurs while you are turning,the car’s handling changes abruptly because the kinetic friction force is in adifferent direction than the static friction force had been: contrary to thecar’s direction of motion, rather than contrary to the forces applied to thetire.

Most people respond with disbelief when told of the experimentalevidence that both static and kinetic friction are approximately independentof the amount of surface area in contact. Even after doing a hands-onexercise with spring scales to show that it is true, many students are unwill-ing to believe their own observations, and insist that bigger tires “give moretraction.” In fact, the main reason why you would not want to put smalltires on a big heavy car is that the tires would burst!

Although many people expect that friction would be proportional tosurface area, such a proportionality would make predictions contrary tomany everyday observations about friction. A dog’s feet, for example, havevery little surface area in contact with the ground compared to a human’sfeet, and yet we know that a dog can win a tug-of-war with a child of thesame size.

The reason why a smaller surface area does not lead to less friction isthat the force between the two surfaces is more concentrated, causing theirbumps and holes to dig into each other more deeply.

Fluid frictionTry to drive a nail into a waterfall and you will be confronted with the

main difference between solid friction and fluid friction. Fluid friction ispurely kinetic; there is no static fluid friction. The nail in the waterfall maytend to get dragged along by the water flowing past it, but it does not stickin the water. The same is true for gases such as air: recall that we are usingthe word “fluid” to include both gases and liquids.

Unlike solid kinetic friction, the force of fluid friction increases rapidlywith velocity. In many cases, the force is approximately proportional to thesquare of the velocity,

Ffluid friction

∝ cρAv2 ,

where A is the cross-sectional area of the object, ρ is the density of the fluid,and c is a constant of proportionality that depends partly on the type offluid and partly on how streamlined the object is.

Frictionless ice can certainly make a normal force, since otherwise a hockey puck would sink into the ice. Friction isnot possible without a normal force, however: we can see this from the equation, or from common sense, e.g. whilesliding down a rope you do not get any friction unless you grip the rope.

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Discussion questionsA. A student states that when he tries to push his refrigerator, the reason itwon’t move is because Newton’s third law says there’s an equal and oppositefrictional force pushing back. After all, the static friction force is equal andopposite to the applied force. How would you convince him he is wrong?B. Kinetic friction is usually more or less independent of velocity. However,inexperienced drivers tend to produce a jerk at the last moment of decelerationwhen they stop at a stop light. What does this tell you about the kinetic frictionbetween the brake shoes and the brake drums?C. Some of the following are correct descriptions of types of forces that couldbe added on as new branches of the classification tree. Others are not reallytypes of forces, and still others are not force phenomena at all. In each case,decide what’s going on, and if appropriate, figure out how you would incorpo-rate them into the tree.

sticky force ........ makes tape stick to thingsopposite force ... the force that Newton’s third law says relates to every force

you makeflowing force ...... the force that water carries with it as it flows out of a hosesurface tension .. lets insects walk on waterhorizontal force . a force that is horizontalmotor force ........ the force that a motor makes on the thing it is turningcanceled force ... a force that is being canceled out by some other force

5.3 Analysis of ForcesNewton’s first and second laws deal with the total of all the forces

exerted on a specific object, so it is very important to be able to figure outwhat forces there are. Once you have focused your attention on one objectand listed the forces on it, it is also helpful to describe all the correspondingforces that must exist according to Newton’s third law. We refer to this as“analyzing the forces” in which the object participates.

ExampleA barge is being pulled along a canal by teams of horses on the shores. Analyze all the forces in which thebarge participates.

force acting on barge force related to it by Newton’s third lawropes’ forward normal forces on barge barge’s backward normal force on ropeswater’s backward fluid friction force on barge barge’s forward fluid friction force on waterplanet earth’s downward gravitational force on barge barge’s upward gravitational force on earthwater’s upward “floating” force on barge barge’s downward “floating” force on water

Here I’ve used the word “floating” force as an example of a sensible invented term for a type of force notclassified on the tree in the previous section. A more formal technical term would be “hydrostatic force.”Note how the pairs of forces are all structured as “A’s force on B, B’s force on A”: ropes on barge and bargeon ropes; water on barge and barge on water. Because all the forces in the left column are forces acting onthe barge, all the forces in the right column are forces being exerted by the barge, which is why each entry inthe column begins with “barge.”

Often you may be unsure whether you have forgotten one of the forces.One way to check is to see what physical result would come from the forcesyou’ve found so far. Suppose, for instance, that you’d forgotten the “float-ing” force on the barge in the example above. Looking at the forces you’dfound, you would have found that there was a downward gravitational forceon the barge which was not canceled by any upward force. The barge isn’tsupposed to sink, so you know you need to find a fourth, upward force.

Another technique for finding missing forces is simply to go throughthe list of all the common types of forces and see if any of them apply.

The following is another example in which we can profit by checkingagainst our physical intuition for what should be happening.

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ExampleAs shown in the figure, Cindy is rappelling down a cliff. Her downward motion is at constant speed, and shetakes little hops off of the cliff, as shown by the dashed line. Analyze the forces in which she participates at amoment when her feet are on the cliff and she is pushing off.

force acting on Cindy force related to it by Newton’s third lawplanet earth’s downward gravitational force on Cindy Cindy’s upward gravitational force on earthropes upward frictional force on Cindy (her hand) Cindy’s downward frictional force on the ropecliff’s rightward normal force on Cindy Cindy’s leftward normal force on the cliff

The two vertical forces cancel, which is what they should be doing if she is to go down at a constant rate. Theonly horizontal force on her is the cliff’s force, which is not canceled by any other force, and which thereforewill produce an acceleration of Cindy to the right. This makes sense, since she is hopping off. (This solution isa little oversimplified, because the rope is slanting, so it also applies a small leftward force to Cindy. As sheflies out to the right, the slant of the rope will increase, pulling her back in more strongly.)

I believe that constructing the type of table described in this section isthe best method for beginning students. Most textbooks, however, prescribea pictorial way of showing all the forces acting on an object. Such a pictureis called a free-body diagram. It should not be a big problem if a futurephysics professor expects you to be able to draw such diagrams, because theconceptual reasoning is the same. You simply draw a picture of the object,with arrows representing the forces that are acting on it. Arrows represent-ing contact forces are drawn from the point of contact, noncontact forcesfrom the center of mass. Free-body diagrams do not show the equal andopposite forces exerted by the object itself.

Discussion questionsA. In the example of the barge going down the canal, I referred to a “floating”or “hydrostatic” force that keeps the boat from sinking. If you were adding anew branch on the force-classification tree to represent this force, where wouldit go?B. A pool ball is rebounding from the side of the pool table. Analyze the forcesin which the ball participates during the short time when it is in contact with theside of the table.C. The earth’s gravitational force on you, i.e. your weight, is always equal tomg, where m is your mass. So why can you get a shovel to go deeper into theground by putting one foot on the shovel and jumping onto it? Just becauseyou’re jumping, that doesn’t mean your mass or weight is any greater, does it?

Discussion question C.

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5.4 Transmission of Forces by Low-Mass ObjectsYou’re walking your dog. The dog wants to go faster than you do, and

the leash is taut. Does Newton’s third law guarantee that your force on yourend of the leash is equal and opposite to the dog’s force on its end? If they’renot exactly equal, is there any reason why they should be approximatelyequal?

If there was no leash between you, and you were in direct contact withthe dog, then Newton’s third law would apply, but Newton’s third lawcannot relate your force on the leash to the dog’s force on the leash, becausethat would involve three separate objects. Newton’s third law only says thatyour force on the leash is equal and opposite to the leash’s force on you,

FyL

= – FLy

,

and that the dog’s force on the leash is equal and opposite to its force on thedog

FdL

= – FLd

.

Still, we have a strong intuitive expectation that whatever force we makeon our end of the leash is transmitted to the dog, and vice-versa. We cananalyze the situation by concentrating on the forces that act on the leash,F

dL and F

yL. According to Newton’s second law, these relate to the leash’s

mass and acceleration:

FdL

+ FyL

= mLa

L .

The leash is far less massive then any of the other objects involved, andif m

L is very small, then apparently the total force on the leash is also very

small, FdL

+ FyL

≈ 0, and therefore

FdL

≈ – FyL

.

Thus even though Newton’s third law does not apply directly to thesetwo forces, we can approximate the low-mass leash as if it was not interven-ing between you and the dog. It’s at least approximately as if you and thedog were acting directly on each other, in which case Newton’s third lawwould have applied.

In general, low-mass objects can be treated approximately as if theysimply transmitted forces from one object to another. This can be true forstrings, ropes, and cords, and also for rigid objects such as rods and sticks.

If you look at a piece of string under a magnifying glass as you pull onthe ends more and more strongly, you will see the fibers straightening andbecoming taut. Different parts of the string are apparently exerting forceson each other. For instance, if we think of the two halves of the string astwo objects, then each half is exerting a force on the other half. If weimagine the string as consisting of many small parts, then each segment istransmitting a force to the next segment, and if the string has very little

If we imagine dividing a taut rope up intosmall segments, then any segment hasforces pulling outward on it at each end.If the rope is of negligible mass, then allthe forces equal +T or -T, where T, thetension, is a single number.

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mass, then all the forces are equal in magnitude. We refer to the magnitudeof the forces as the tension in the string, T. Although the tension is mea-sured in units of Newtons, it is not itself a force. There are many forceswithin the string, some in one direction and some in the other direction,and their magnitudes are only approximately equal. The concept of tensiononly makes sense as a general, approximate statement of how big all theforces are.

If a rope goes over a pulley or around some other object, then thetension throughout the rope is approximately equal so long as there is nottoo much friction. A rod or stick can be treated in much the same way as astring, but it is possible to have either compression or tension.

Since tension is not a type of force, the force exerted by a rope on someother object must be of some definite type such as static friction, kineticfriction, or a normal force. If you hold your dog’s leash with your handthrough the loop, then the force exerted by the leash on your hand is anormal force: it is the force that keeps the leash from occupying the samespace as your hand. If you grasp a plain end of a rope, then the forcebetween the rope and your hand is a frictional force.

A more complex example of transmission of forces is the way a caraccelerates. Many people would describe the car’s engine as making theforce that accelerates the car, but the engine is part of the car, so that’simpossible: objects can’t make forces on themselves. What really happens isthat the engine’s force is transmitted through the transmission to the axles,then through the tires to the road. By Newton’s third law, there will thus bea forward force from the road on the tires, which accelerates the car.

Discussion questionsWhen you step on the gas pedal, is your foot’s force being transmitted in thesense of the word used in this section?

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126

5.5 Objects Under StrainA string lengthens slightly when you stretch it. Similarly, we have

already discussed how an apparently rigid object such as a wall is actuallyflexing when it participates in a normal force. In other cases, the effect ismore obvious. A spring or a rubber band visibly elongates when stretched.

Common to all these examples is a change in shape of some kind:lengthening, bending, compressing, etc. The change in shape can bemeasured by picking some part of the object and measuring its position, x.For concreteness, let’s imagine a spring with one end attached to a wall.When no force is exerted, the unfixed end of the spring is at some positionx

o. If a force acts at the unfixed end, its position will change to some new

value of x. The more force, the greater the departure of x from xo.

relaxedspring

force Fbeingapplied

xo

x

Back in Newton’s time, experiments like this were considered cutting-edge research, and his contemporary Hooke is remembered today for doingthem and for coming up with a simple mathematical generalization calledHooke’s law:

F ≈ k(x-xo) [force required to stretch a spring; valid for small

forces only] .

Here k is a constant, called the spring constant, that depends on how stiffthe object is. If too much force is applied, the spring exhibits more compli-cated behavior, so the equation is only a good approximation if the force issufficiently small. Usually when the force is so large that Hooke’s law is abad approximation, the force ends up permanently bending or breaking thespring.

Although Hooke’s law may seem like a piece of trivia about springs, it isactually far more important than that, because all solid objects exertHooke’s-law behavior over some range of sufficiently small forces. Forexample, if you push down on the hood of a car, it dips by an amount thatis directly proportional to the force. (But the car’s behavior would not be asmathematically simple if you dropped a boulder on the hood!)

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Discussion questionsA. A car is connected to its axles through big, stiff springs called shockabsorbers, or “shocks.” Although we’ve discussed Hooke’s law above only inthe case of stretching a spring, a car’s shocks are continually going throughboth stretching and compression. In this situation, how would you interpret thepositive and negative signs in Hooke’s law?

5.6 Simple Machines: The PulleyEven the most complex machines, such as cars or pianos, are built out

of certain basic units called simple machines. The following are some of themain functions of simple machines:

transmitting a force: The chain on a bicycle transmits a force from thecrank set to the rear wheel.

changing the direction of a force: If you push down on a seesaw, theother end goes up.

changing the speed and precision of motion: When you make the“come here” motion, your biceps only moves a couple of centimeterswhere it attaches to your forearm, but your arm moves much fartherand more rapidly.

changing the amount of force: A lever or pulley can be used to increaseor decrease the amount of force.

You are now prepared to understand one-dimensional simple machines,of which the pulley is the main example.

Example: a pulleyQuestion : Farmer Bill says this pulley arrangement doubles theforce of his tractor. Is he just a dumb hayseed, or does he knowwhat he’s doing?Solution : To use Newton’s first law, we need to pick an objectand consider the sum of the forces on it. Since our goal is torelate the tension in the part of the cable attached to the stump tothe tension in the part attached to the tractor, we should pick anobject to which both those cables are attached, i.e. the pulleyitself. As discussed in section 5.4, the tension in a string or cableremains approximately constant as it passes around a pulley,provided that there is not too much friction. There are thereforetwo leftward forces acting on the pulley, each equal to the forceexerted by the tractor. Since the acceleration of the pulley isessentially zero, the forces on it must be canceling out, so therightward force of the pulley-stump cable on the pulley must bedouble the force exerted by the tractor. Yes, Farmer Bill knowswhat he’s talking about.

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128

SummarySelected Vocabulary

repulsive ............................ describes a force that tends to push the two participating objects apartattractive ........................... describes a force that tends to pull the two participating objects togetheroblique .............................. describes a force that acts at some other angle, one that is not a direct

repulsion or attractionnormal force ...................... the force that keeps two objects from occupying the same spacestatic friction ..................... a friction force between surfaces that are not slipping past each otherkinetic friction ................... a friction force between surfaces that are slipping past each otherfluid .................................. a gas or a liquidfluid friction ...................... a friction force in which at least one of the object is is a fluidspring constant .................. the constant of proportionality between force and elongation of a spring

or other object under strainNotation

FN ..................................... a normal forceFs ....................................... a static frictional forceFk ...................................... a kinetic frictional forceµs ...................................... the coefficient of static friction; the constant of proportionality between

the maximum static frictional force and the normal force; depends onwhat types of surfaces are involved

µk ...................................... the coefficient of kinetic friction; the constant of proportionality betweenthe kinetic frictional force and the normal force; depends on what types ofsurfaces are involved

k ........................................ the spring constant; the constant of proportionality between the forceexerted on an object and the amount by which the object is lengthened orcompressed

SummaryNewton’s third law states that forces occur in equal and opposite pairs. If object A exerts a force on object

B, then object B must simultaneously be exerting an equal and opposite force on object A. Each instance ofNewton’s third law involves exactly two objects, and exactly two forces, which are of the same type.

There are two systems for classifying forces. We are presently using the more practical but lessfundamental one. In this system, forces are classified by whether they are repulsive, attractive, or oblique;whether they are contact or noncontact forces; and whether the two objects involved are solids or fluids.

Static friction adjusts itself to match the force that is trying to make the surfaces slide past each other, untilthe maximum value is reached,

|Fs|<µ

s|F

N| .

Once this force is exceeded, the surfaces slip past one another, and kinetic friction applies,

|Fk|=µ

k|F

N| .

Both types of frictional force are nearly independent of surface area, and kinetic friction is usuallyapproximately independent of the speed at which the surfaces are slipping.

A good first step in applying Newton’s laws of motion to any physical situation is to pick an object ofinterest, and then to list all the forces acting on that object. We classify each force by its type, and find itsNewton’s-third-law partner, which is exerted by the object on some other object.

When two objects are connected by a third low-mass object, their forces are transmitted to each othernearly unchanged.

Objects under strain always obey Hooke’s law to a good approximation, as long as the force is small.Hooke’s law states that the stretching or compression of the object is proportional to the force exerted on it,

F ≈ k(x-xo) .

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Homework Problems1. If a big truck and a VW bug collide head-on, which will be acted on bythe greater force? Which will have the greater acceleration?

2. The earth is attracted to an object with a force equal and opposite to theforce of the earth on the object. If this is true, why is it that when youdrop an object, the earth does not have an acceleration equal and oppositeto that of the object?

3. When you stand still, there are two forces acting on you, the force ofgravity (your weight) and the normal force of the floor pushing up onyour feet. Are these forces equal and opposite? Does Newton's third lawrelate them to each other? Explain.

4. A magnet is stuck underneath a car. Analyze the forces in which themagnet participates, using a table in the format shown in section 5.3.

5. Give two examples of objects at rest relative to the earth that are beingkept from falling by forces other than the normal force. Do not use objectsin outer space. In each case, analyze the forces.

6. A person is rowing a boat, with her legs extended and her feet braced.Analyze the forces in which she participates while she is doing the part ofthe stroke that propels the boat, with the ends of the oars in the water (notthe part where the oars are lifted out of the water).

7. A farmer is in a stall with a cow when the cow decides to press himagainst the wall, pinning him with his feet off the ground. Analyze theforces in which the farmer participates.

8. A propeller plane is cruising east at constant speed and altitude. Analyzethe forces in which the plane participates.

9 . Today’s tallest buildings are really not that much taller than the tallestbuildings of the 1940s. The main problem with making an even tallerskyscraper is that every elevator needs its own shaft running the wholeheight of the building. So many elevators are needed to serve the building’sthousands of occupants that the elevator shafts start taking up too much ofthe space within the building. An alternative is to have elevators that canmove both horizontally and vertically: with such a design, many elevatorcars can share a few shafts, and they don’t get in each other’s way too muchbecause they can detour around each other. In this design, it becomesimpossible to hang the cars from cables, so they would instead have to rideon rails which they grab onto with wheels. Friction would keep them fromslipping. The figure shows such a frictional elevator in its vertical travelmode. (The wheels on the bottom are for when it needs to switch tohorizontal motion.) (a) If the coefficient of static friction between rubberand steel is µ

s, and the maximum mass of the car plus its passengers is M,

how much force must there be pressing each wheel against the rail in orderto keep the car from slipping? (Assume the car is not accelerating.) (b)Show that your result has physically reasonable behavior with respect to µ

s.

In other words, if there was less friction, would the wheels need to bepressed more firmly or less firmly? Does your equation behave that way?

Problem 9.

steel rail

car

rubber wheel

Homework Problems

130

10. Unequal masses M and m are suspended from a pulley as shown in thefigure.

(a) Analyze the forces in which mass m participates, using a table theformat shown in section 5.3. [The forces in which the other massesparticipate will of course be similar, but not numerically the same.]

(b) Find the magnitude of the accelerations of the two masses. [Hints: (1)Pick a coordinate system, and use positive and negative signs consistentlyto indicate the directions of the forces and accelerations. (2) The twoaccelerations of the two masses have to be equal in magnitude but ofopposite signs, since one side eats up rope at the same rate at which theother side pays it out. (3) You need to apply Newton’s second law twice,once to each mass, and then solve the two equations for the unknowns:the acceleration, a, and the tension in the rope, T.]

(c) Many people expect that in the special case of M=m, the two masseswill naturally settle down to an equilibrium position side by side. Based onyour answer from part (b), is this correct?

11. A tugboat of mass m pulls a ship of mass M, accelerating it. Ignorefluid friction acting on their hulls, although there will of course need to befluid friction acting on the tug’s propellers.

(a) Analyze the forces in which the tugboat participates, using a table inthe format shown in section 5.3. Don’t worry about vertical forces.

(b) Do the same for the ship.

(c) Assume now that water friction on the two vessels’ hulls is negligible. Ifthe force acting on the tug’s propeller is F, what is the tension, T, in thecable connecting the two ships? [Hint: Write down two equations, one forNewton’s second law applied to each object. Solve these for the twounknowns T and a.]

(d) Interpret your answer in the special cases of M=0 and M=∞.

12. Explain why it wouldn't make sense to have kinetic friction be stron-ger than static friction.

Problem 10.

Chapter 5 Analysis of Forces

131

Motion in Three Dimensions

Photo by Clarence White, ca. 1903.

6 Newton’s Laws in ThreeDimensions

6.1 Forces Have No Perpendicular EffectsSuppose you could shoot a rifle and arrange for a second bullet to be

dropped from the same height at the exact moment when the first left thebarrel. Which would hit the ground first? Nearly everyone expects that thedropped bullet will reach the dirt first, and Aristotle would have agreed.Aristotle would have described it like this. The shot bullet receives someforced motion from the gun. It travels forward for a split second, slowing

Section 6.1 Forces Have No Perpendicular Effects

132

down rapidly because there is no longer any force to make it continue inmotion. Once it is done with its forced motion, it changes to naturalmotion, i.e. falling straight down. While the shot bullet is slowing down,the dropped bullet gets on with the business of falling, so according toAristotle it will hit the ground first.

Luckily, nature isn’t as complicated as Aristotle thought! To convinceyourself that Aristotle’s ideas were wrong and needlessly complex, stand upnow and try this experiment. Take your keys out of your pocket, and beginwalking briskly forward. Without speeding up or slowing down, releaseyour keys and let them fall while you continue walking at the same pace.

You have found that your keys hit the ground right next to your feet.Their horizontal motion never slowed down at all, and the whole time theywere dropping, they were right next to you. The horizontal motion and thevertical motion happen at the same time, and they are independent of eachother. Your experiment proves that the horizontal motion is unaffected bythe vertical motion, but it’s also true that the vertical motion is not changedin any way by the horizontal motion. The keys take exactly the sameamount of time to get to the ground as they would have if you simplydropped them, and the same is true of the bullets: both bullets hit theground simultaneously.

These have been our first examples of motion in more than one dimen-sion, and they illustrate the most important new idea that is required tounderstand the three-dimensional generalization of Newtonian physics:

Forces have no perpendicular effects.When a force acts on an object, it has no effect on thepart of the object's motion that is perpendicular to theforce.

In the examples above, the vertical force of gravity had no effect on thehorizontal motions of the objects. These were examples of projectilemotion, which interested people like Galileo because of its military applica-tions. The principle is more general than that, however. For instance, if a

(horizontal scale reduced)

Aristotle

Newton

Chapter 6 Newton’s Laws in Three Dimensions

133

rolling ball is initially heading straight for a wall, but a steady wind beginsblowing from the side, the ball does not take any longer to get to the wall.In the case of projectile motion, the force involved is gravity, so we can saymore specifically that the vertical acceleration is 9.8 m/s2, regardless of thehorizontal motion.

Relationship to relative motionThese concepts are directly related to the idea that motion is relative.

Galileo’s opponents argued that the earth could not possibly be rotating ashe claimed, because then if you jumped straight up in the air you wouldn’tbe able to come down in the same place. Their argument was based on theirincorrect Aristotelian assumption that once the force of gravity began to acton you and bring you back down, your horizontal motion would stop. Inthe correct Newtonian theory, the earth’s downward gravitational force isacting before, during, and after your jump, but has no effect on yourmotion in the perpendicular (horizontal) direction.

If Aristotle had been correct, then we would have a handy way todetermine absolute motion and absolute rest: jump straight up in the air,and if you land back where you started, the surface from which you jumpedmust have been in a state of rest. In reality, this test gives the same result aslong as the surface under you is an inertial frame. If you try this in a jetplane, you land back on the same spot on the deck from which you started,regardless of whether the plane is flying at 500 miles per hour or parked onthe runway. The method would in fact only be good for detecting whetherthe plane was accelerating.

Discussion QuestionsA. The following is an incorrect explanation of a true fact about target shooting:

“Shooting a high-powered rifle with a high muzzle velocity is different fromshooting a less powerful gun. With a less powerful gun, you have to aimquite a bit above your target, but with a more powerful one you don’t haveto aim so high because the bullet doesn’t drop as fast.”

What is the correct explanation?B. You have thrown a rock, and it is flying through the air in an arc. If theearth's gravitational force on it is always straight down, why doesn't it just gostraight down once it leaves your hand?C. Consider the example of the bullet that is dropped at the same momentanother bullet is fired from a gun. What would the motion of the two bullets looklike to a jet pilot flying along in the same direction as the shot bullet and at thesame horizontal speed?

Section 6.1 Forces Have No Perpendicular Effects

134

6.2 Coordinates and Components‘Cause we’re allBold as love,Just ask the axis.

-Jimi HendrixHow do we convert these ideas into mathematics? The figure below

shows a good way of connecting the intuitive ideas to the numbers. In onedimension, we impose a number line with an x coordinate on a certainstretch of space. In two dimensions, we imagine a grid of squares which welabel with x and y values.

This object experiences a force that pulls itdown toward the bottom of the page. In eachequal time interval, it moves three units tothe right. At the same time, its vertical mo-tion is making a simple pattern of +1, 0, -1, -2, -3, -4, ... units. Its motion can be describedby an x coordinate that has zero accelera-tion and a y coordinate with constant accel-eration. The arrows labeled x and y serve toexplain that we are defining increasing x tothe right and increasing y as upward.

But of course motion doesn’t really occur in a series of discrete hops likein chess or checkers. The figure on the left shows a way of conceptualizingthe smooth variation of the x and y coordinates. The ball’s shadow on thewall moves along a line, and we describe its position with a single coordi-nate, y, its height above the floor. The wall shadow has a constant accelera-tion of –9.8 m/s2. A shadow on the floor, made by a second light source,also moves along a line, and we describe its motion with an x coordinate,measured from the wall.

The velocity of the floor shadow is referred to as the x component of thevelocity, written v

x. Similarly we can notate the acceleration of the floor

shadow as ax. Since v

x is constant, a

x is zero.

Similarly, the velocity of the wall shadow is called vy, its acceleration a

y.

This example has ay=–9.8 m/s2.

Because the earth’s gravitational force on the ball is acting along the yaxis, we say that the force has a negative y component, F

y, but F

x=F

z=0.

The general idea is that we imagine two observers, each of whomperceives the entire universe as if it was flattened down to a single line. They-observer, for instance, perceives y, v

y, and a

y, and will infer that there is a

force, Fy, acting downward on the ball. That is, a y component means the

aspect of a physical phenomenon, such as velocity, acceleration, or force,that is observable to someone who can only see motion along the y axis.

All of this can easily be generalized to three dimensions. In the exampleabove, there could be a z-observer who only sees motion toward or awayfrom the back wall of the room.

x

y

x

y

Chapter 6 Newton’s Laws in Three Dimensions

135

Example: a car going over a cliffQuestion : The police find a car at a distance w=20 m from thebase of a cliff of height h=100 m. How fast was the car goingwhen it went over the edge? Solve the problem symbolically first,then plug in the numbers.Solution : Let’s choose y pointing up and x pointing away fromthe cliff. The car’s vertical motion was independent of its horizon-tal motion, so we know it had a constant vertical acceleration ofa=-g=-9.8 m/s2. The time it spent in the air is therefore related tothe vertical distance it fell by the constant-acceleration equation

∆y = 12

ay∆t 2 ,

or

– h = 12

( – g)∆t 2 .

Solving for ∆t gives

∆t = 2hg .

Since the vertical force had no effect on the car’s horizontalmotion, it had a

x=0, i.e. constant horizontal velocity. We can

apply the constant-velocity equation

vx = ∆x∆t ,

i.e.

vx = w∆t .

We now substitute for ∆t to find

vx = w / 2hg ,

which simplifies to

vx = wg

2h .

Plugging in numbers, we find that the car’s speed when it wentover the edge was 4 m/s, or about 10 mi/hr.

Projectiles move along parabolasWhat type of mathematical curve does a projectile follow through

space? To find out, we must relate x to y, eliminating t. The reasoning is verysimilar to that used in the example above. Arbitrarily choosing x=y=t=0 tobe at the top of the arc, we conveniently have x=∆x, y=∆y, and t=∆t, so

y = – 12a yt

2

x = v xt

We solve the second equation for t=x/vx and eliminate t in the first equa-

tion:

y = – 12a y

xv x

2 .

Since everything in this equation is a constant except for x and y, weconclude that y is proportional to the square of x. As you may or may notrecall from a math class, y∝x2 describes a parabola.Each water droplet follows a pa-

rabola. The faster water droplets’ pa-rabolas are bigger.

w

h

vx=?

x

y

Section 6.2 Coordinates and Components

A parabola can be defined as theshape made by cutting a cone paral-lel to its side. A parabola is also thegraph of an equation of the form

y∝ x 2.

136

Discussion QuestionA. At the beginning of this section I represented the motion of a projectile ongraph paper, breaking its motion into equal time intervals. Suppose insteadthat there is no force on the object at all. It obeys Newton’s first law andcontinues without changing its state of motion. What would the correspondinggraph-paper diagram look like? If the time interval represented by each arrowwas 1 second, how would you relate the graph-paper diagram to the velocitycomponents vx and vy?B. The figure shows two trajectories, made by splicing together lines andcircular arcs, which are unphysical for an object that is only being acted on bygravity. Prove that they are impossible based on Newton’s laws or on theprinciple that forces have no perpendicular effects.C. Make up several different coordinate systems oriented in different ways,and describe the ax and ay of a falling object in each one.

6.3 Newton’s Laws in Three DimensionsIt is now fairly straightforward to extend Newton’s laws to three dimen-

sions:

Newton’s First LawIf all three components of the total force on an object are zero, then itwill continue in the same state of motion.

Newton’s Second LawAn object’s acceleration components are predicted by the equations

ax = F

x,total/m ,

ay = F

y,total/m , and

az = F

z,total/m .

Newton’s Third LawIf two objects A and B interact via forces, then the components of theirforces on each other are equal and opposite:

FA on B,x

= –FB on A,x

,

FA on B,y

= –FB on A,y

, and

FA on B,z

= –FB on A,z

.

Chapter 6 Newton’s Laws in Three Dimensions

(1) (2)

Discussion question B.

137

Example: forces in perpendicular directions on the same objectQuestion : An object is initially at rest. Two constant forces beginacting on it, and continue acting on it for a while. As suggestedby the two arrows, the forces are perpendicular, and the right-ward force is stronger. What happens?Answer : Aristotle believed, and many students still do, that onlyone force can “give orders” to an object at one time. Theytherefore think that the object will begin speeding up and movingin the direction of the stronger force. In fact the object will movealong a diagonal. In the example shown in the figure, the objectwill respond to the large rightward force with a large accelerationcomponent to the right, and the small upward force will give it asmall acceleration component upward. The stronger force doesnot overwhelm the weaker force, or have any effect on theupward motion at all. The force components simply add together:

= 0

Fx,total = F1,x + F2,x

= 0

Fy,total = F1,y + F2,y

F1

F2

directionof motion

x

y

Section 6.3 Newton’s Laws in Three Dimensions

138

SummarySelected Vocabulary

component ........................ the part of a velocity, acceleration, or force that would be perceptible to anobserver who could only see the universe projected along a certain one-dimensional axis

parabola ............................ the mathematical curve whose graph has y proportional to x2

Notationx, y, z ................................. an object’s positions along the x, y, and z axesv

x, v

y, v

z.................................................... the x, y, and z components of an object’s velocity; the rates of change of

the object’s x, y, and z coordinatesa

x, a

y, a

z.................................................... the x, y, and z components of an object’s acceleration; the rates of change

of vx, v

y, and v

z

SummaryA force does not produce any effect on the motion of an object in a perpendicular direction. The most

important application of this principle is that the horizontal motion of a projectile has zero acceleration, whilethe vertical motion has an acceleration equal to g. That is, an object’s horizontal and vertical motions areindependent. The arc of a projectile is a parabola.

Motion in three dimensions is measured using three coordinates, x, y, and z . Each of these coordinateshas its own corresponding velocity and acceleration. We say that the velocity and acceleration both have x, y,and z components

Newton’s second law is readily extended to three dimensions by rewriting it as three equations predictingthe three components of the acceleration,

ax = F

x,total/m ,

ay = Fy,total/m ,

az = Fz,total/m ,

and likewise for the first and third laws.

Chapter 6 Newton’s Laws in Three Dimensions

139

Homework Problems1. (a) A ball is thrown straight up with velocity v. Find an equation for theheight to which it rises.(b) Generalize your equation for a ball thrown at an angle θ above hori-zontal, in which case its initial velocity components are v

x=v cos θ and v

y=v

sin θ.

2. At the Salinas Lettuce Festival Parade, Miss Lettuce of 1996 drops herbouquet while riding on a float. Compare the shape of its trajectory asseen by her to the shape seen by one of her admirers standing on thesidewalk.

3 . Two daredevils, Wendy and Bill, go over Niagara Falls. Wendy sits inan inner tube, and lets the 30 km/hr velocity of the river throw her outhorizontally over the falls. Bill paddles a kayak, adding an extra 10 km/hrto his velocity. They go over the edge of the falls at the same moment, sideby side. Ignore air friction. \Explain your reasoning.(a) Who hits the bottom first?(b) What is the horizontal component of Wendy's velocity on impact?(c) What is the horizontal component of Bill's velocity on impact?(d) Who is going faster on impact?

4. A baseball pitcher throws a pitch clocked at vx=73.3 mi/h. He throws

horizontally. By what amount, d, does the ball drop by the time it reacheshome plate, L=60.0 ft away? (a) First find a symbolic answer in terms of L,v

x, and g. (b) Plug in and find a numerical answer. Express your answer

in units of ft. [Note: 1 ft=12 in, 1 mi=5280 ft, and 1 in=2.54 cm]

L=60.0 ft

d=?

vx=73.3 mi/hr

5 S. A cannon standing on a flat field fires a cannonball with a muzzlevelocity v, at an angle θ above horizontal. The cannonball thus initially hasvelocity components v

x=v cos θ and v

y=v sin θ.

(a) Show that the cannon’s range (horizontal distance to where the can-

nonball falls) is given by the equation R = 2v 2 sin θ cos θg .

(b) Interpret your equation in the cases of θ=0 and θ=90°.

Homework Problems

140 Chapter 6 Newton’s Laws in Three Dimensions

6 ∫. Assuming the result of the previous problem for the range of a projec-

tile, R = 2v 2 sin θ cos θg , show that the maximum range is for θ=45°.

141

7 Vectors7.1 Vector Notation

The idea of components freed us from the confines of one-dimensionalphysics, but the component notation can be unwieldy, since every one-dimensional equation has to be written as a set of three separate equationsin the three-dimensional case. Newton was stuck with the componentnotation until the day he died, but eventually someone sufficiently lazy andclever figured out a way of abbreviating three equations as one.

Vectors are used in aerial navigation.

Section 7.1 Vector Notation

142

Example (a) shows both ways of writing Newton’s third law. Whichwould you rather write?

The idea is that each of the algebra symbols with an arrow written ontop, called a vector, is actually an abbreviation for three different numbers,the x, y, and z components. The three components are referred to as the

components of the vector, e.g. Fx is the x component of the vector F . The

notation with an arrow on top is good for handwritten equations, but isunattractive in a printed book, so books use boldface, F, to representvectors. After this point, I’ll use boldface for vectors throughout this book.

In general, the vector notation is useful for any quantity that has bothan amount and a direction in space. Even when you are not going to writeany actual vector notation, the concept itself is a useful one. We say thatforce and velocity, for example, are vectors. A quantity that has no directionin space, such as mass or time, is called a scalar. The amount of a vectorquantity is called its magnitude. The notation for the magnitude of a vectorA is |A|, like the absolute value sign used with scalars.

Often, as in example (b), we wish to use the vector notation to repre-sent adding up all the x components to get a total x component, etc. Theplus sign is used between two vectors to indicate this type of component-by-component addition. Of course, vectors are really triplets of numbers,not numbers, so this is not the same as the use of the plus sign with indi-vidual numbers. But since we don’t want to have to invent new words andsymbols for this operation on vectors, we use the same old plus sign, andthe same old addition-related words like “add,” “sum,” and “total.” Com-bining vectors this way is called vector addition.

Similarly, the minus sign in example (a) was used to indicate negatingeach of the vector’s three components individually. The equals sign is usedto mean that all three components of the vector on the left side of anequation are the same as the corresponding components on the right.

Example (c) shows how we abuse the division symbol in a similarmanner. When we write the vector ∆v divided by the scalar ∆t, we mean thenew vector formed by dividing each one of the velocity components by ∆t.

A vector has both adirection and an amount.

A scalar has only an amount.

(a) F A on B = –F B on A stands for

F A on B,x = – F B on A, x

F A on B,y = – F B on A, y

F A on B,z = – F B on A, z

(b) F total = F 1 + F 2 + ... stands for

F total,x = F 1,x + F 2,x + ...

F total,y = F 1,y + F 2,y + ...

F total,z = F 1,z + F 2,z + ...

(c) a = ∆v∆t

stands for

a x = ∆v x / ∆t

a y = ∆v y / ∆t

a z = ∆v z / ∆t

Chapter 7 Vectors

143

It’s not hard to imagine a variety of operations that would combinevectors with vectors or vectors with scalars, but only four of them areactually useful for physics:

operation definition

vector + vectorAdd component by component to makea new set of three numbers.

vector - vectorSubtract component by component tomake a new set of three numbers.

vector . scalarMultiply each component of the vectorby the scalar.

vector / scalar Divide each component of the vector bythe scalar.

As an example of an operation that is not useful for physics, there justaren’t any useful physics applications for dividing a vector by another vectorcomponent by component. In optional section 7.5, we discuss in moredetail the fundamental reasons why some vector operations are useful andothers useless.

We can do algebra with vectors, or with a mixture of vectors and scalarsin the same equation. Basically all the normal rules of algebra apply, but ifyou’re not sure if a certain step is valid, you should simply translate it intothree component-based equations and see if it works.

ExampleQuestion : If we are adding two force vectors, F+G, is it valid toassume as in ordinary algebra that F+G is the same as G+F?Answer : To tell if this algebra rule also applies to vectors, wesimply translate the vector notation into ordinary algebra nota-tion. In terms of ordinary numbers, the components of the vectorF+G would be F

x+G

x, F

y+G

y, and F

z+G

z, which are certainly the

same three numbers as Gx+F

x, G

y+F

y, and G

z+F

z. Yes, F+G is the

same as G+F.It is useful to define a symbol r for the vector whose components are x,

y, and z, and a symbol ∆r made out of ∆x, ∆y, and ∆z.

Although this may all seem a little formidable, keep in mind that itamounts to nothing more than a way of abbreviating equations! Also, tokeep things from getting too confusing the remainder of this chapterfocuses mainly on the ∆r vector, which is relatively easy to visualize.

Self-CheckTranslate the equations vx=∆x/∆t, vy=∆y/∆t, and vz=∆z/∆t for motion withconstant velocity into a single equation in vector notation.

v=∆r/∆t

Section 7.1 Vector Notation

144

Drawing vectors as arrowsA vector in two dimensions can be easily visualized by drawing an arrow

whose length represents its magnitude and whose direction represents itsdirection. The x component of a vector can then be visualized as the lengthof the shadow it would cast in a beam of light projected onto the x axis, andsimilarly for the y component. Shadows with arrowheads pointing backagainst the direction of the positive axis correspond to negative compo-nents.

In this type of diagram, the negative of a vector is the vector with thesame magnitude but in the opposite direction. Multiplying a vector by ascalar is represented by lengthening the arrow by that factor, and similarlyfor division.

Self-CheckGiven vector Q represented by an arrow below, draw arrows representing thevectors 1.5Q and -Q.

Q

Discussion QuestionsA. Would it make sense to define a zero vector?B. Would it make sense to define some vectors as positive and some asnegative?C. An object goes from one point in space to another. After it arrives at itsdestination, how does the magnitude of its ∆r vector compare with the distanceit traveled?

7.2 Calculations with Magnitude and DirectionIf you ask someone where Las Vegas is compared to Los Angeles, they

are unlikely to say that the ∆x is 290 km and the ∆y is 230 km, in a coordi-nate system where the positive x axis is east and the y axis points north.They will probably say instead that it’s 370 km to the northeast. If theywere being precise, they might specify the direction as 38° counterclockwisefrom east. In two dimensions, we can always specify a vector’s direction likethis, using a single angle. A magnitude plus an angle suffice to specifyeverything about the vector. The following two examples show how we usetrigonometry and the Pythagorean theorem to go back and forth betweenthe x-y and magnitude-angle descriptions of vectors.

x

y

x

y

x component(positive)

y component(negative)

1.5Q –Q

Chapter 7 Vectors

145

Example: finding the magnitude and angle from the componentsQuestion : Given that the ∆r vector from LA to Las Vegas has∆x=290 km and ∆y=230 km, how would we find the magnitudeand direction of ∆r?Solution : We find the magnitude of ∆r from the Pythagoreantheorem:

|∆r| = ∆x 2 + ∆y 2

= 370 kmWe know all three sides of the triangle, so the angle θ can befound using any of the inverse trig functions. For example, weknow the opposite and adjacent sides, so

θ = tan– 1∆y∆x

= 38° .

Example: finding the components from the magnitude and angleQuestion : Given that the straight-line distance from Los Angelesto Las Vegas is 370 km, and that the angle θ in the figure is 38°,how can the x and y components of the ∆r vector be found?Solution : The sine and cosine of θ relate the given information tothe information we wish to find:

cos θ = ∆x∆r

sin θ = ∆y∆r

Solving for the unknowns gives

∆x = ∆r cos θ= 290 km

∆y = ∆r sin θ= 230 km

LosAngeles

Las Vegas

∆x

∆y|∆r|

θ

Section 7.2 Calculations with Magnitude and Direction

146

The following example shows the correct handling of the plus andminus signs, which is usually the main cause of mistakes by students.

Example: negative componentsQuestion : San Diego is 120 km east and 150 km south of LosAngeles. An airplane pilot is setting course from San Diego toLos Angeles. At what angle should she set her course, measuredcounterclockwise from east, as shown in the figure?Solution : If we make the traditional choice of coordinate axes,with x pointing to the right and y pointing up on the map, then her∆x is negative, because her final x value is less than her initial xvalue. Her ∆y is positive, so we have

∆x = -120 km∆y = 150 km .

If we work by analogy with the previous example, we get

θ = tan– 1∆y∆x

= tan– 1 –1.25

= -51° .According to the usual way of defining angles in trigonometry, anegative result means an angle that lies clockwise from the xaxis, which would have her heading for the Baja California. Whatwent wrong? The answer is that when you ask your calculator totake the arctangent of a number, there are always two validpossibilities differing by 180°. That is, there are two possibleangles whose tangents equal -1.25:

tan 129° = -1.25tan -51° = -1.25

You calculator doesn’t know which is the correct one, so it justpicks one. In this case, the one it picked was the wrong one, andit was up to you to add 180° to it to find the right answer.

Discussion QuestionIn the example above, we dealt with components that were negative. Does itmake sense to talk about positive and negative vectors?

∆x

∆y

LosAngeles

|∆r|

θ

San Diego

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147

7.3 Techniques for Adding VectorsAddition of vectors given their components

The easiest type of vector addition is when you are in possession of thecomponents, and want to find the components of their sum.

ExampleQuestion : Given the ∆x and ∆y values from the previous ex-amples, find the ∆x and ∆y from San Diego to Las Vegas.Solution :

∆xtotal

= ∆x1 + ∆x

2

= -120 km + 290 km= 170 km

∆ytotal

= ∆y1 + ∆y

2

= 150 km + 230 km= 380

Addition of vectors given their magnitudes and directionsIn this case, you must first translate the magnitudes and directions into

components, and the add the components.

Graphical addition of vectorsOften the easiest way to add vectors is by making a scale drawing on a

piece of paper. This is known as graphical addition, as opposed to theanalytic techniques discussed previously.

ExampleQuestion : Given the magnitudes and angles of the ∆r vectorsfrom San Diego to Los Angeles and from Los Angeles to LasVegas, find the magnitude and angle of the ∆r vector from SanDiego to Las Vegas.Solution : Using a protractor and a ruler, we make a careful scaledrawing, as shown in the figure. A scale of 1 cm→10 km waschosen for this solution. With a ruler, we measure the distancefrom San Diego to Las Vegas to be 3.8 cm, which corresponds to380 km. With a protractor, we measure the angle θ to be 71°.

Even when we don’t intend to do an actual graphical calculation with aruler and protractor, it can be convenient to diagram the addition of vectorsin this way. With ∆r vectors, it intuitively makes sense to lay the vectors tip-to-tail and draw the sum vector from the tail of the first vector to the tip ofthe second vector. We can do the same when adding other vectors such asforce vectors.

Discussion QuestionsA. If you’re doing graphical addition of vectors, does it matter which vector youstart with and which vector you start from the other vector’s tip?B. If you add a vector with magnitude 1 to a vector of magnitude 2, whatmagnitudes are possible for the vector sum?C. Which of these examples of vector addition are correct, and which areincorrect?

LosAngeles

Las Vegas

San Diego

LosAngeles

Las Vegas

San Diego

190 km

370 km

141°

38° distance=?

θ=?

B

AA+B

A

B

Vectors can be added graphically byplacing them tip to tail, and thendrawing a vector from the tail of thefirst vector to the tip of the secondvector.

Section 7.3 Techniques for Adding Vectors

AB

A+BA

B

A+B A

BA+B

148

7.4* Unit Vector NotationWhen we want to specify a vector by its components, it can be cumber-

some to have to write the algebra symbol for each component:

∆x = 290 km, ∆y = 230 km

A more compact notation is to write

∆r = (290 km)x + (230 km)y ,

where the vectors x , y , and z , called the unit vectors, are defined as thevectors that have magnitude equal to 1 and directions lying along the x, y,and z axes. In speech, they are referred to as “x-hat” and so on.

A slightly different, and harder to remember, version of this notation is

unfortunately more prevalent. In this version, the unit vectors are called i ,

j , and k :

∆r = (290 km)i + (230 km)j .

7.5* Rotational InvarianceLet’s take a closer look at why certain vector operations are useful and

others are not. Consider the operation of multiplying two vectors compo-nent by component to produce a third vector:

Rx

= Px Q

x

Ry

= Py Q

y

Rz

= Pz Q

z

As a simple example, we choose vectors P and Q to have length 1, andmake them perpendicular to each other, as shown in figure (a). If wecompute the result of our new vector operation using the coordinate systemshown in (b), we find:

Rx

= 0R

y= 0

Rz

= 0The x component is zero because P

x =0, the y component is zero because

Qy=0, and the z component is of course zero because both vectors are in the

x-y plane. However, if we carry out the same operations in coordinatesystem (c), rotated 45 degrees with respect to the previous one, we find

Rx

= –1/2R

y= 1/2

Rz

= 0The operation’s result depends on what coordinate system we use, and sincethe two versions of R have different lengths (one being zero and the othernonzero), they don’t just represent the same answer expressed in twodifferent coordinate systems. Such an operation will never be useful inphysics, because experiments show physics works the same regardless ofwhich way we orient the laboratory building! The useful vector operations,such as addition and scalar multiplications, are rotationally invariant, i.e.come out the same regardless of the orientation of the coordinate system.

Chapter 7 Vectors

P

Q

x

y

xy

(a)

(b)

(c)

149

SummarySelected Vocabulary

vector ................................ a quantity that has both an amount (magnitude) and a direction in spacemagnitude ......................... the “amount” associated with a vectorscalar ................................. a quantity that has no direction in space, only an amount

NotationA....................................... vector with components A

x, A

y, and A

z

A ...................................... handwritten notation for a vector

|A| ..................................... the magnitude of vector Ar ........................................ the vector whose components are x, y, and z∆r ...................................... the vector whose components are ∆x, ∆y, and ∆z

x, y, z ............................... (optional topic) unit vectors; the vectors with magnitude 1 lying along thex, y, and z axes

i, j , k ............................... a harder to remember notation for the unit vectors

Standard Terminology Avoided in This Bookdisplacement vector ........... a name for the symbol ∆rspeed ................................. the magnitude of the velocity vector, i.e. the velocity stripped of any

information about its directionSummary

A vector is a quantity that has both a magnitude (amount) and a direction in space, as opposed to a scalar,which has no direction. The vector notation amounts simply to an abbreviation for writing the vector’s threecomponents.

In two dimensions, a vector can be represented either by its two components or by its magnitude anddirection. The two ways of describing a vector can be related by trigonometry.

The two main operations on vectors are addition of a vector to a vector, and multiplication of a vector by ascalar.

Vector addition means adding the components of two vectors to form the components of a new vector. Ingraphical terms, this corresponds to drawing the vectors as two arrows laid tip-to-tail and drawing the sumvector from the tail of the first vector to the tip of the second one. Vector subtraction is performed by negatingthe vector to be subtracted and then adding.

Multiplying a vector by a scalar means multiplying each of its components by the scalar to create a newvector. Division by a scalar is defined similarly.

Summary

150

Homework Problems1. Here are two vectors. Graphically calculate A+B, A-B, B-A, -2B, and A-2B. No numbers are involved.

2. Phnom Penh is 470 km east and 250 km south of Bangkok. Hanoi is60 km east and 1030 km north of Phnom Penh. (a) Choose a coordinatesystem, and translate these data into ∆x and ∆y values with the proper plusand minus signs. (b) Find the components of the ∆r vector pointingfrom Bangkok to Hanoi.

3 . If you walk 35 km at an angle 25° counterclockwise from east, andthen 22 km at 230° counterclockwise from east, find the distance anddirection from your starting point to your destination.

A

B

Problem 1.

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151

8 Vectors and MotionIn 1872, capitalist and former California governor Leland Stanford

asked photographer Eadweard Muybridge if he would work for him on aproject to settle a $25,000 bet (a princely sum at that time). Stanford’sfriends were convinced that a galloping horse always had at least one footon the ground, but Stanford claimed that there was a moment during eachcycle of the motion in which all four feet were in the air. The human eyewas simply not fast enough to settle the question. In 1878, Muybridgefinally succeeded in producing what amounted to a motion picture of thehorse, showing conclusively that all four feet did leave the ground at onepoint. (Muybridge was a colorful figure in San Francisco history, and hisacquittal for the murder of his wife’s lover was considered the trial of thecentury in California.)

The losers of the bet had probably been influenced by Aristotelianreasoning, for instance the expectation that a leaping horse would losehorizontal velocity while in the air with no force to push it forward, so thatit would be more efficient for the horse to run without leaping. But even forstudents who have converted wholeheartedly to Newtonianism, the rela-tionship between force and acceleration leads to some conceptual difficul-ties, the main one being a problem with the true but seemingly absurdstatement that an object can have an acceleration vector whose direction isnot the same as the direction of motion. The horse, for instance, has nearlyconstant horizontal velocity, so its a

x is zero. But as anyone can tell you who

has ridden a galloping horse, the horse accelerates up and down. The horse’s

Section

152

acceleration vector therefore changes back and forth between the up anddown directions, but is never in the same direction as the horse’s motion. Inthis chapter, we will examine more carefully the properties of the velocity,acceleration, and force vectors. No new principles are introduced, but anattempt is made to tie things together and show examples of the power ofthe vector formulation of Newton’s laws.

8.1 The Velocity VectorFor motion with constant velocity, the velocity vector is

v=∆r/∆t [ only for constant velocity ] .

The ∆r vector points in the direction of the motion, and dividing it by thescalar ∆t only changes its length, not its direction, so the velocity vectorpoints in the same direction as the motion. When the velocity is notconstant, i.e. when the x-t, y-t, and z-t graphs are not all linear, we use theslope-of-the-tangent-line approach to define the components v

x, v

y, and v

z,

from which we assemble the velocity vector. Even when the velocity vectoris not constant, it still points along the direction of motion.

Vector addition is the correct way to generalize the one-dimensionalconcept of adding velocities in relative motion, as shown in the followingexample:

Example: velocity vectors in relative motionQuestion : You wish to cross a river and arrive at a dock that isdirectly across from you, but the river’s current will tend to carryyou downstream. To compensate, you must steer the boat at anangle. Find the angle θ, given the magnitude, |v

WL|, of the water’s

velocity relative to the land, and the maximum speed, |vBW

|, ofwhich the boat is capable relative to the water.Solution : The boat’s velocity relative to the land equals thevector sum of its velocity with respect to the water and thewater’s velocity with respect to the land,

vBL

= vBW

+ vWL

.If the boat is to travel straight across the river, i.e. along the yaxis, then we need to have v

BL,x=0. This x component equals the

sum of the x components of the other two vectors,v

BL,x = v

BW,x + v

WL,x ,

or0 = -|v

BW| sin θ + |v

WL| .

Solving for θ, we findsin θ = |v

WL|/|v

BW| ,

θ = sin– 1 vWL

vBW .

vBW

vWL

θ

x

y

Chapter 8 Vectors and Motion

153

Discussion QuestionsA. Is it possible for an airplane to maintain a constant velocity vector but not aconstant |v|? How about the opposite -- a constant |v| but not a constantvelocity vector? Explain.B. New York and Rome are at about the same latitude, so the earth’s rotationcarries them both around nearly the same circle. Do the two cities have thesame velocity vector (relative to the center of the earth)?

8.2 The Acceleration VectorWhen all three acceleration components are constant, i.e. when the v

x-t,

vy-t, and v

z-t graphs are all linear, we can define the acceleration vector as

a=∆v/∆t [ only for constant acceleration] ,

which can be written in terms of initial and final velocities as

a=(vf-v

i)/∆t [ only for constant acceleration] .

If the acceleration is not constant, we define it as the vector made out of thea

x, a

y, and a

z components found by applying the slope-of-the-tangent-line

technique to the vx-t, v

y-t, and v

z-t graphs.

Now there are two ways in which we could have a nonzero acceleration.Either the magnitude or the direction of the velocity vector could change.This can be visualized with arrow diagrams as shown in the figure. Both themagnitude and direction can change simultaneously, as when a car acceler-ates while turning. Only when the magnitude of the velocity changes whileits direction stays constant do we have a ∆v vector and an accelerationvector in the same direction as the motion.

Self-CheckIn figure (a), is the object speeding up or slowing down? Describe how the ∆vvector is different depending on whether it is speeding up or slowing down.

If this all seems a little strange and abstract to you, you’re not alone. Itdoesn’t mean much to most physics students the first time someone tellsthem that acceleration is a vector, and that the acceleration vector does nothave to be in the same direction as the velocity vector. One way to under-stand those statements better is to imagine an object such as an air fresheneror a pair of fuzzy dice hanging from the rear-view mirror of a car. Such ahanging object, called a bob, constitutes an accelerometer. If you watch thebob as you accelerate from a stop light, you’ll see it swing backward. Thehorizontal direction in which the bob tilts is opposite to the direction of theacceleration. If you apply the brakes and the car’s acceleration vector pointsbackward, the bob tilts forward.

After accelerating and slowing down a few times, you think you’ve putyour accelerometer through its paces, but then you make a right turn.Surprise! Acceleration is a vector, and needn’t point in the same direction asthe velocity vector. As you make a right turn, the bob swings outward, toyour left. That means the car’s acceleration vector is to your right, perpen-

(a) A change in the magnitude of thevelocity vector implies an acceleration.

(b) A change in the direction of thevelocity vector also produces a non-zero ∆v vector, and thus a nonzeroacceleration vector, ∆v/∆t.

vivf

-vi

vf∆v

vivf

-vi

vf∆v

It is speeding up, because the final velocity vector has the greater magnitude. Speeding up produced a ∆v vector inthe same direction as the motion. Slowing down would have given a ∆v that bointed backward.

Section 8.2 The Acceleration Vector

154

dicular to your velocity vector. A useful definition of an acceleration vectorshould relate in a systematic way to the actual physical effects produced bythe acceleration, so a physically reasonable definition of the accelerationvector must allow for cases where it is not in the same direction as themotion.

Self-Check

In projectile motion, what direction does the acceleration vector have?

Discussion QuestionsA. When a car accelerates, why does a bob hanging from the rearview mirror

As we have already seen, the projectile has ax=0 and ay=-g,so the acceleration vector is pointing straight down.

a b c ed

∆vpointsdown

∆vpointsup

∆v≈0

a b c d e

The downward forceof gravity producesa downward accel-eration vector.

The upward force fromthe ground is greater thanthe downward force ofgravity. The total force onthe horse is upward, givingan upward acceleration.

This figure shows outlines traced fromthe first, third, fifth, seventh, and ninthframes in Muybridge’s series of pho-tographs of the galloping horse. Theestimated location of the horse’s cen-ter of mass is shown with a circle,which bobs above and below the hori-zontal dashed line.If we don’t care about calculating ve-locities and accelerations in any par-ticular system of units, then we canpretend that the time between framesis one unit. The horse’s velocity vec-tor as it moves from one point to thenext can then be found simply bydrawing an arrow to connect one po-sition of the center of mass to the next.This produces a series of velocity vec-tors which alternate between pointingabove and below horizontal.The ∆v vector is the vector which wewould have to add onto one velocityvector in order to get the next velocityvector in the series. The ∆v vector al-ternates between pointing down(around the time when the horse is inthe air, b) and up (around the timewhen the horse has two feet on theground, d).

The following are two examples of force, velocity, and accelerationvectors in complex motion.

Chapter 8 Vectors and Motion

155

swing toward the back of the car? Is it because a force throws it backward? Ifso, what force? Similarly, describe what happens in the other cases describedabove.B. The following is a question commonly asked by students:

“Why does the force vector always have to point in the same direction asthe acceleration vector? What if you suddenly decide to change your forceon an object, so that your force is no longer pointing the same direction thatthe object is accelerating?”

What misunderstanding is demonstrated by this question? Suppose, forexample, a spacecraft is blasting its rear main engines while moving forward,then suddenly begins firing its sideways-pointing maneuvering rocket as well.What does the student think Newton’s laws are predicting?

velocity acceleration force

In this example, the rappeller’s velocity has long periods of gradualchange interspersed with short periods of rapid change. Thesecorrespond to periods of small acceleration and force and periodsof large acceleration and force.

Section 8.2 The Acceleration Vector

156

8.3 The Force Vector and Simple MachinesForce is relatively easy to intuit as a vector. The force vector points in

the direction in which it is trying to accelerate the object it is acting on.

Since force vectors are so much easier to visualize than accelerationvectors, it is often helpful to first find the direction of the (total) forcevector acting on an object, and then use that information to determine thedirection of the acceleration vector. Newton’s second law, F

total=ma, tells us

that the two must be in the same direction.

An important application of force vectors is to analyze the forces actingin two-dimensional mechanical systems, as in the following example.

Example: pushing a block up a rampQuestion : Figure (a) shows a block being pushed up a friction-less ramp at constant speed by an applied force F

A. How much

force is required, in terms of the block’s mass, m, and the angleof the ramp, θ?Solution : Figure (b) shows the other two forces acting on theblock: a normal force, F

N, created by the ramp, and the weight

force, FW, created by the earth’s gravity. Because the block is

being pushed up at constant speed, it has zero acceleration, andthe total force on it must be zero. From figure (c), we find

|FA| = |F

W| sin θ

= mg sin θ .Since the sine is always less than one, the applied force is always less thanmg, i.e. pushing the block up the ramp is easier than lifting it straight up.This is presumably the principle on which the pyramids were constructed:the ancient Egyptians would have had a hard time applying the forces ofenough slaves to equal the full weight of the huge blocks of stone.

Essentially the same analysis applies to several other simple machines,such as the wedge and the screw.

Discussion QuestionsA. The figure shows a block being pressed diagonally upward against a wall,causing it to slide up the wall. Analyze the forces involved, including theirdirections.

Discussion question A.

(a) The applied force FA pushes theblock up the frictionless ramp.

(b) Three forces act on the block. Theirvector sum is zero.

(c) If the block is to move at constantvelocity, Newton’s first law says thatthe three force vectors acting on itmust add up to zero. To perform vec-tor addition, we put the vectors tip totail, and in this case we are addingthree vectors, so each one’s tail goesagainst the tip of the previous one.Since they are supposed to add up tozero, the third vector’s tip must comeback to touch the tail of the first vec-tor. They form a triangle, and since theapplied force is perpendicular to thenormal force, it is a right triangle.

FA

θ

FW

FNFA

θ

FA

FWFNθ

Chapter 8 Vectors and Motion

157

B. As discussed in chapter 5, the maximum force of static friction between twosurfaces is given by Fs<µsFN, where µs is a constant that depends on the typesof surfaces, and FN is the normal force, i.e. the force perpendicular to thesurfaces that keeps them from passing through each other. If a block of massm is resting on an inclined plane, what is the maximum angle of the incline forwhich static friction can prevent the block from sliding?C. The figure shows a roller coaster car rolling down and then up under theinfluence of gravity. Sketch the car’s velocity vectors and acceleration vectors.Pick an interesting point in the motion and sketch a set of force vectors actingon the car whose vector sum could have resulted in the right accelerationvector.

8.4 ∫ Calculus With VectorsThe definitions of the velocity and acceleration components given in

chapter 6 can be translated into calculus notation as

v = dxdt

x +dydt

y + dzdt

z

and

a =

dv x

dtx +

dv y

dty +

dv z

dtz .

To make the notation less cumbersome, we generalize the concept of thederivative to include derivatives of vectors, so that we can abbreviate theabove equations as

v = drdt

and

a = dvdt

.

In words, to take the derivative of a vector, you take the derivatives of itscomponents and make a new vector out of those. This definition meansthat the derivative of vector function has the familiar properties

d cf

dt= cd f

dt

and

d f + g

dt= d f

dt+

dgdt

.

The integral of a vector is likewise defined as integrating component bycomponent.

Section 8.4 ∫ Calculus With Vectors

Discussion question C.

158

Example: a fire-extinguisher stunt on iceQuestion : Prof. Puerile smuggles a fire extinguisher into askating rink. Climbing out onto the ice without any skates on, hesits down and pushes off from the wall with his feet, acquiring an

initial velocity voy . At t=0, he then discharges the fire extin-

guisher at a 45-degree angle so that it applies a force to him thatis backward and to the left, i.e. along the negative y axis and thepositive x axis. The fire extinguisher’s force is strong at first, butthen dies down according to the equation |F|=b–ct, where b andc are constants. Find the professor’s velocity as a function oftime.Solution : Measured counterclockwise from the x axis, the angleof the force vector becomes 315°. Breaking the force down into xand y components, we have

Fx

= |F| cos 315°=

12

(b–ct)

Fy

= |F| sin 315°=

12

(–b+ct) .

In unit vector notation, this is

F = 12

(b–ct) x + 12

(–b+ct)y .

Newton’s second law givesa = F/m

= b – ct2m x + –b + ct

2my .

To find the velocity vector as a function of time, we need tointegrate the acceleration vector with respect to time,

v = a dt

= b – ct

2mx + –b + ct

2my dt

= 12m

b – ct x + –b + ct y dt

A vector function can be integrated component by component, sothis can be broken down into two integrals,

v = x2m

b – ct dt + y2m

–b + ct dt

= bt – 1

2ct 2

2m+ const. #1 x

+ –bt + 1

2ct 2

2m+ const. #2 y

Here the physical significance of the two constants of integrationis that they give the initial velocity. Constant #1 is therefore zero,and constant #2 must equal v

o. The final result is

v = bt – 1

2ct 2

2mx +

–bt + 12ct 2

2m+ vo y .

Chapter 8 Vectors and Motion

159

SummaryThe velocity vector points in the direction of the object’s motion. Relative motion can be described by

vector addition of velocities.

The acceleration vector need not point in the same direction as the object’s motion. We use the word“acceleration” to describe any change in an object’s velocity vector, which can be either a change in itsmagnitude or a change in its direction.

An important application of the vector addition of forces is the use of Newton’s first law to analyze me-chanical systems.

Summary

160

Homework Problems1 . A dinosaur fossil is slowly moving down the slope of a glacier underthe influence of wind, rain and gravity. At the same time, the glacier ismoving relative to the continent underneath. The dashed lines representthe directions but not the magnitudes of the velocities. Pick a scale, anduse graphical addition of vectors to find the magnitude and the directionof the fossil's velocity relative to the continent. You will need a ruler andprotractor.

direction of motionof fossil relative toglacier, 2.3x10-7 m/s

direction of motionof glacier relativeto continent, 1.1x10-7 m/s

north

2. Is it possible for a helicopter to have an acceleration due east and avelocity due west? If so, what would be going on? If not, why not?

3 . A bird is initially flying horizontally east at 21.1 m/s, but one secondlater it has changed direction so that it is flying horizontally and 7° northof east, at the same speed. What are the magnitude and direction of itsacceleration vector during that one second time interval? (Assume itsacceleration was roughly constant.)

4. A person of mass M stands in the middle of a tightrope, which is fixedat the ends to two buildings separated by a horizontal distance L. The ropesags in the middle, stretching and lengthening the rope slightly. (a) If thetightrope walker wants the rope to sag vertically by no more than a heighth, find the minimum tension, T, that the rope must be able to withstandwithout breaking, in terms of h, g, M, and L. (b) Based on your equation,explain why it is not possible to get h=0, and give a physical interpretation.

L

h

5«. Your hand presses a block of mass m against a wall with a force FH

acting at an angle θ. Find the minimum and maximum possible values of|F

H| that can keep the block stationary, in terms of m, g, θ, and µ

s, the

coefficient of static friction between the block and the wall.θ

Problem 5.

Chapter 8 Vectors and Motion

161

θ

FliftFthrust

Problem 9.

Problem 10.

θ

ϕ

6. A skier of mass m is coasting down a slope inclined at an angle θcompared to horizontal. Assume for simplicity that the treatment ofkinetic friction given in chapter 5 is appropriate here, although a soft andwet surface actually behaves a little differently. The coefficient of kineticfriction acting between the skis and the snow is µ

k, and in addition the

skier experiences an air friction force of magnitude bv2, where b is aconstant. (a) Find the maximum speed that the skier will attain, in termsof the variables m, θ, µ

k, and b. (b) For angles below a certain minimum

angle θmin

, the equation gives a result that is not mathematically meaning-ful. Find an equation for θ

min, and give a physical explanation of what is

happening for θ<θmin

.

7 ∫. A gun is aimed horizontally to the west, and fired at t=0. The bullet's

position vector as a function of time is r = bx + cty + dtz , where b, c, andd are constants. (a) What units would b, c, and d need to have for theequation to make sense? (b) Find the bullet's velocity and acceleration as

functions of time. (c) Give physical interpretations of b, c, d, x , y , and z .

8 S. Annie Oakley, riding north on horseback at 30 mi/hr, shoots her rifle,aiming horizontally and to the northeast. The muzzle speed of the rifle is140 mi/hr. When the bullet hits a defenseless fuzzy animal, what is itsspeed of impact? Neglect air resistance, and ignore the vertical motion ofthe bullet.

9 S. A cargo plane has taken off from a tiny airstrip in the Andes, and isclimbing at constant speed, at an angle of θ=17° with respect to horizon-tal. Its engines supply a thrust of F

thrust=200 kN, and the lift from its

wings is Flift

=654 kN. Assume that air resistance (drag) is negligible, so theonly forces acting are thrust, lift, and weight. What is its mass, in kg?

10 S. A wagon is being pulled at constant speed up a slope θ by a ropethat makes an angle ϕ with the vertical. (a) Assuming negligible friction,show that the tension in the rope is given by the equation

F T = sin θsin θ + ϕ

F W ,

where FW

is the weight force acting on the wagon. (b) Interpret thisequation in the special cases of ϕ=0 and ϕ=180°-θ.

11 S. The angle of repose is the maximum slope on which an object willnot slide. On airless, geologically inert bodies like the moon or an asteroid,the only thing that determines whether dust or rubble will stay on a slopeis whether the slope is less steep than the angle of repose. (a) Find anequation for the angle of repose, deciding for yourself what are therelevant variables. (b) On an asteroid, where g can be thousands of timeslower than on Earth, would rubble be able to lie at a steeper angle ofrepose?

Homework Problems

162

163

9 Circular Motion9.1 Conceptual Framework for Circular Motion

I now live fifteen minutes from Disneyland, so my friends and family inmy native Northern California think it’s a little strange that I’ve nevervisited the Magic Kingdom again since a childhood trip to the south. Thetruth is that for me as a preschooler, Disneyland was not the Happiest Placeon Earth. My mother took me on a ride in which little cars shaped likerocket ships circled rapidly around a central pillar. I knew I was going todie. There was a force trying to throw me outward, and the safety featuresof the ride would surely have been inadequate if I hadn’t screamed thewhole time to make sure Mom would hold on to me. Afterward, sheseemed surprisingly indifferent to the extreme danger we had experienced.

Section 9.1 Conceptual Framework for Circular Motion

164

Circular motion does not produce an outward forceMy younger self ’s understanding of circular motion was partly right and

partly wrong. I was wrong in believing that there was a force pulling meoutward, away from the center of the circle. The easiest way to understandthis is to bring back the parable of the bowling ball in the pickup truckfrom chapter 4. As the truck makes a left turn, the driver looks in therearview mirror and thinks that some mysterious force is pulling the balloutward, but the truck is accelerating, so the driver’s frame of reference isnot an inertial frame. Newton’s laws are violated in a noninertial frame, sothe ball appears to accelerate without any actual force acting on it. Becausewe are used to inertial frames, in which accelerations are caused by forces,the ball’s acceleration creates a vivid illusion that there must be an outwardforce.

In an inertial frame everything makes more sense. The ball has no forceon it, and goes straight as required by Newton’s first law. The truck has aforce on it from the asphalt, and responds to it by accelerating (changingthe direction of its velocity vector) as Newton’s second law says it should.

(a) In the turning truck’s frame of reference, theball appears to violate Newton’s laws, display-ing a sideways acceleration that is not the re-sult of a force-interaction with any other object.

(b) In an inertial frame of reference, such as theframe fixed to the earth’s surface, the ball obeysNewton’s first law. No forces are acting on it, andit continues moving in a straight line. It is thetruck that is participating in an interaction withthe asphalt, the truck that accelerates as it shouldaccording to Newton’s second law.

Chapter 9 Circular Motion

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Circular motion does not persist without a forceI was correct about one thing, however. To make me curve around with

the car, I really did need some force such as a force from my mother,friction from the seat, or a normal force from the side of the car. (In fact, allthree forces were probably adding together.) One of the reasons why Galileofailed to refine the principle of inertia into a quantitative statement likeNewton’s first law is that he was not sure whether motion without a forcewould naturally be circular or linear. In fact, the most impressive exampleshe knew of the persistence of motion were mostly circular: the spinning of atop or the rotation of the earth, for example. Newton realized that inexamples such as these, there really were forces at work. Atoms on thesurface of the top are prevented from flying off straight by the ordinaryforce that keeps atoms stuck together in solid matter. The earth is nearly allliquid, but gravitational forces pull all its parts inward.

Uniform and nonuniform circular motionCircular motion always involves a change in the direction of the velocity

vector, but it is also possible for the magnitude of the velocity to change atthe same time. Circular motion is referred to as uniform if |v| is constant,and nonuniform if |v| is changing.

Your speedometer tells you the magnitude of your car’s velocity vector,so when you go around a curve while keeping your speedometer needlesteady, you are executing uniform circular motion. If your speedometerreading is changing as you turn, your circular motion is nonuniform.Uniform circular motion is simpler to analyze mathematically, so we willattack it first and then pass to the nonuniform case.

Self-CheckWhich of these are examples of uniform circular motion and which are nonuni-form?

(a) the clothes in a clothes dryer (assuming they remain against the inside ofthe drum, even at the top)

(b) a rock on the end of a string being whirled in a vertical circle

(a) An overhead view of a personswinging a rock on a rope. A force fromthe string is required to make the rock'svelocity vector keep changing direc-tion.

yesno

(b) If the string breaks, the rock willfollow Newton's first law and gostraight instead of continuing aroundthe circle.

(a) Uniform. They have the same motion as the drum itself, which is rotating as one solid piece. No part of thedrum can be rotating at a different speed from any other part. (b) Nonuniform. Gravity speeds it up on the waydown and slows it down on the way up.

Section 9.1 Conceptual Framework for Circular Motion

166

Only an inward force is required for uniform circular motion.The figures on the previous page showed the string pulling in straight

along a radius of the circle, but many people believe that when they aredoing this they must be “leading” the rock a little to keep it moving along.That is, they believe that the force required to produce uniform circularmotion is not directly inward but at a slight angle to the radius of the circle.This intuition is incorrect, which you can easily verify for yourself now ifyou have some string handy. It is only while you are getting the object goingthat your force needs to be at an angle to the radius. During this initialperiod of speeding up, the motion is not uniform. Once you settle downinto uniform circular motion, you only apply an inward force.

If you have not done the experiment for yourself, here is a theoreticalargument to convince you of this fact. We have discussed in chapter 6 theprinciple that forces have no perpendicular effects. To keep the rock fromspeeding up or slowing down, we only need to make sure that our force isperpendicular to its direction of motion. We are then guaranteed that itsforward motion will remain unaffected: our force can have no perpendiculareffect, and there is no other force acting on the rock which could slow itdown. The rock requires no forward force to maintain its forward motion,any more than a projectile needs a horizontal force to “help it over the top”of its arc.

Why, then, does a car driving in circles in a parking lot stop executinguniform circular motion if you take your foot off the gas? The source ofconfusion here is that Newton’s laws predict an object’s motion based on thetotal force acting on it. A car driving in circles has three forces on it

(1) an inward force from the asphalt, controlled with the steering wheel;(2) a forward force from the asphalt, controlled with the gas pedal; and(3) backward forces from air resistance and rolling resistance.

You need to make sure there is a forward force on the car so that thebackward forces will be exactly canceled out, creating a vector sum thatpoints directly inward.

In uniform circular motion, the acceleration vector is inwardSince experiments show that the force vector points directly inward,

Newton’s second law implies that the acceleration vector points inward aswell. This fact can also be proven on purely kinematical grounds, and wewill do so in the next section.

To make the brick go in a circle, I hadto exert an inward force on the rope.

A series of three hammer taps makesthe rolling ball trace a triangle, sevenhammers a heptagon. If the numberof hammers was large enough, the ballwould essentially be experiencing asteady inward force, and it would goin a circle. In no case is any forwardforce necessary.

Chapter 9 Circular Motion

When a car is going straight at con-stant speed, the forward and backwardforces on it are canceling out, produc-ing a total force of zero. When it movesin a circle at constant speed, there arethree forces on it, but the forward andbackward forces cancel out, so thevector sum is an inward force.

167

Discussion QuestionsA. In the game of crack the whip, a line of people stand holding hands, andthen they start sweeping out a circle. One person is at the center, and rotateswithout changing location. At the opposite end is the person who is runningthe fastest, in a wide circle. In this game, someone always ends up losing theirgrip and flying off. Suppose the person on the end loses her grip. What pathdoes she follow as she goes flying off? (Assume she is going so fast that sheis really just trying to put one foot in front of the other fast enough to keep fromfalling; she is not able to get any significant horizontal force between her feetand the ground.)B. Suppose the person on the outside is still holding on, but feels that she mayloose her grip at any moment. What force or forces are acting on her, and inwhat directions are they? (We are not interested in the vertical forces, whichare the earth's gravitational force pulling down, and the ground's normal forcepushing up.)C. Suppose the person on the outside is still holding on, but feels that she mayloose her grip at any moment. What is wrong with the following analysis of thesituation? "The person whose hand she's holding exerts an inward force onher, and because of Newton's third law, there's an equal and opposite forceacting outward. That outward force is the one she feels throwing her outward,and the outward force is what might make her go flying off, if it's strongenough."D. If the only force felt by the person on the outside is an inward force, whydoesn't she go straight in?E. In the amusem*nt park ride shown in the figure, the cylinder spins fasterand faster until the customer can pick her feet up off the floor without falling. Inthe old Coney Island version of the ride, the floor actually dropped out like atrap door, showing the ocean below. (There is also a version in which thewhole thing tilts up diagonally, but we’re discussing the version that stays flat.)If there is no outward force acting on her, why does she stick to the wall?Analyze all the forces on her.F. What is an example of circular motion where the inward force is a normalforce? What is an example of circular motion where the inward force isfriction? What is an example of circular motion where the inward force the sumof more than one force?G. Does the acceleration vector always change continuously in circularmotion? The velocity vector?

Discussion question E.

Discussion questions A-D.

Section 9.1 Conceptual Framework for Circular Motion

168

9.2 Uniform Circular MotionIn this section I derive a simple and very useful equation for the magni-

tude of the acceleration of an object undergoing constant acceleration. Thelaw of sines is involved, so I’ve recapped it on the left.

The derivation is brief, but the method requires some explanation andjustification. The idea is to calculate a ∆v vector describing the change inthe velocity vector as the object passes through an angle θ. We then calcu-late the acceleration, a=∆v/∆t. The astute reader will recall, however, thatthis equation is only valid for motion with constant acceleration. Althoughthe magnitude of the acceleration is constant for uniform circular motion,the acceleration vector changes its direction, so it is not a constant vector,and the equation a=∆v/∆t does not apply. The justification for using it isthat we will then examine its behavior when we make the time interval veryshort, which means making the angle θ very small. For smaller and smallertime intervals, the ∆v/∆t expression becomes a better and better approxima-tion, so that the final result of the derivation is exact.

In figure (a), the object sweeps out an angle θ. Its direction of motionalso twists around by an angle θ, from the vertical dashed line to the tiltedone. Figure (b) shows the initial and final velocity vectors, which have equalmagnitude, but directions differing by θ. In (c), the vectors have beenreassembled in the proper orientation for vector subtraction. They form anisosceles triangle with interior angles θ, η, and η. (Eta, η, is my favoriteGreek letter.) The law of sines gives

∆v

sin θ=

v

sin η .

This tells us the magnitude of ∆v, which is one of the two ingredients weneed for calculating the magnitude of a=∆v/∆t. The other ingredient is ∆t.The time required for the object to move through the angle θ is

∆t = length of arc

v .

Now if we measure our angles in radians we can use the definition of radianmeasure, which is (angle)=(length of arc)/(radius), giving ∆t=θr/|v|. Com-bining this with the first expression involving |∆v| gives

|a| = |∆v|/∆t

= v 2

r ⋅ sin θθ ⋅

1sin η .

When θ becomes very small, the small-angle approximation sin θ≈θobtains, and also η becomes close to 90°, so sin η≈1, and we have anequation for |a|:

|a| = v 2

r [uniform circular motion] .

θ

θ

(a)

A

B

Ca

bc

The law of sines:A/sin a = B/sin b = C/sin c

(b) (c)

vi

vf -vivf

∆v = vf + (-vi)

θ

ηη

Chapter 9 Circular Motion

169

Example: force required to turn on a bikeQuestion : A bicyclist is making a turn along an arc of a circlewith radius 20 m, at a speed of 5 m/s. If the combined mass ofthe cyclist plus the bike is 60 kg, how great a static friction forcemust the road be able to exert on the tires?Solution : Taking the magnitudes of both sides of Newton’ssecond law gives

|F| = |ma|= m|a| .

Substituting |a|=|v|2/r gives|F| =m|v|2/r

≈ 80 N(rounded off to one sig fig).

Example: Don’t hug the center line on a curve!Question : You’re driving on a mountain road with a steep dropon your right. When making a left turn, is it safer to hug thecenter line or to stay closer to the outside of the road?Solution : You want whichever choice involves the least accel-eration, because that will require the least force and entail theleast risk of exceeding the maximum force of static friction.Assuming the curve is an arc of a circle and your speed isconstant, your car is performing uniform circular motion, with|a|=|v|2/r. The dependence on the square of the speed showsthat driving slowly is the main safety measure you can take, butfor any given speed you also want to have the largest possiblevalue of r. Even though your instinct is to keep away from thatscary precipice, you are actually less likely to skid if you keeptoward the outside, because then you are describing a largercircle.

Example: acceleration related to radius and period of rotationQuestion : How can the equation for the acceleration in uniformcircular motion be rewritten in terms of the radius of the circleand the period, T, of the motion, i.e. the time required to goaround once?Solution : The period can be related to the speed as follows:

|v| = circumferenceT

= 2πr/T .Substituting into the equation|a|=|v|2/r gives

|a| = 4π2rT 2 .

Example: a clothes dryerQuestion : My clothes dryer has a drum with an inside radius of35 cm, and it spins at 48 revolutions per minute. What is theacceleration of the clothes inside?Solution : We can solve this by finding the period and plugging into the result of the previous example. If it makes 48 revolutions inone minute, then the period is 1/48 of a minute, or 1.25 s. To getan acceleration in mks units, we must convert the radius to 0.35m. Plugging in, the result is 8.8 m/s2.

Section 9.2 Uniform Circular Motion

170

Example: more about clothes dryers!Question : In a discussion question in the previous section, wemade the assumption that the clothes remain against the insideof the drum as they go over the top. In light of the previousexample, is this a correct assumption?Solution : No. We know that there must be some minimum speedat which the motor can run that will result in the clothes justbarely staying against the inside of the drum as they go over thetop. If the clothes dryer ran at just this minimum speed, thenthere would be no normal force on the clothes at the top: theywould be on the verge of losing contact. The only force acting onthem at the top would be the force of gravity, which would givethem an acceleration of g=9.8 m/s2. The actual dryer must berunning slower than this minimum speed, because it produces anacceleration of only 8.8 m/s2. My theory is that this is doneintentionally, to make the clothes mix and tumble.

Discussion QuestionA. A certain amount of force is needed to provide the acceleration of circularmotion. What if were are exerting a force perpendicular to the direction ofmotion in an attempt to make an object trace a circle of radius r, but the forceisn’t as big as m|v|2/r?B. Suppose a rotating space station is built that gives its occupants the illusionof ordinary gravity. What happens when a person in the station throws a ballstraight “up” in the air (i.e. towards the center)?

An artist’s conception of a rotating spacecolony in the form of a giant wheel. Aperson living in this noninertial frame ofreference has an illusion of a force pull-ing her outward, toward the deck, for thesame reason that a person in the pickuptruck has the illusion of a force pullingthe bowling ball. By adjusting the speedof rotation, the designers can make anacceleration |v|2/r equal to the usual ac-celeration of gravity on earth. On earth,your acceleration standing on the groundis zero, and a falling rock heads for yourfeet with an acceleration of 9.8 m/s2. Aperson standing on the deck of the spacecolony has an upward acceleration of 9.8m/s2, and when she lets go of a rock,her feet head up at the nonacceleratingrock. To her, it seems the same as truegravity.Art by NASA.

Chapter 9 Circular Motion

171

9.3 Nonuniform Circular MotionWhat about nonuniform circular motion? Although so far we have been

discussing components of vectors along fixed x and y axes, it now becomesconvenient to discuss components of the acceleration vector along the radialline (in-out) and the tangential line (along the direction of motion). Fornonuniform circular motion, the radial component of the accelerationobeys the same equation as for uniform circular motion,

ar

= |v|2/r ,

but the acceleration vector also has a tangential component,

at

= slope of the graph of |v| versus t .

The latter quantity has a simple interpretation. If you are going around acurve in your car, and the speedometer needle is moving, the tangentialcomponent of the acceleration vector is simply what you would havethought the acceleration was if you saw the speedometer and didn’t knowyou were going around a curve.

Example: Slow down before a turn, not during it.Question : When you’re making a turn in your car and you’reafraid you may skid, isn’t it a good idea to slow down?Solution : If the turn is an arc of a circle, and you’ve alreadycompleted part of the turn at constant speed without skidding,then the road and tires are apparently capable of enough staticfriction to supply an acceleration of |v|2/r. There is no reason whyyou would skid out now if you haven’t already. If you get nervousand brake, however, then you need to have a tangential accel-eration component in addition to the radial component you werealready able to produce successfully. This would require anacceleration vector with a greater magnitude, which in turn wouldrequire a larger force. Static friction might not be able to supplythat much force, and you might skid out. As in the previousexample on a similar topic, the safe thing to do is to approach theturn at a comfortably low speed.

ar

ata

ar

at

a

a

An object moving in a circle mayspeed up (top), keep the magnitudeof its velocity vector constant (middle),or slow down (bottom).

Section 9.3 Nonuniform Circular Motion

172

SummarySelected Vocabulary

uniform circular motion ......... circular motion in which the magnitude of the velocity vector remainsconstant

nonuniform circular motion ... circular motion in which the magnitude of the velocity vector changesradial ...................................... parallel to the radius of a circle; the in-out directiontangential ............................... tangent to the circle, perpendicular to the radial direction

Notationa

r................................................................... radial acceleration; the component of the acceleration vector along the in-

out directiona

t.................................................................... tangential acceleration; the component of the acceleration vector tangent

to the circleSummary

If an object is to have circular motion, a force must be exerted on it toward the center of the circle. There isno outward force on the object; the illusion of an outward force comes from our experiences in which our pointof view was rotating, so that we were viewing things in a noninertial frame.

An object undergoing uniform circular motion has an inward acceleration vector of magnitude

|a| = v 2

r .

In nonuniform circular motion, the radial and tangential components of the acceleration vector are

ar

= |v|2/r

at

= slope of the graph of |v| versus t .

Chapter 9 Circular Motion

173

Homework Problems1. When you’re done using an electric mixer, you can get most of thebatter off of the beaters by lifting them out of the batter with the motorrunning at a high enough speed. Let’s imagine, to make things easier tovisualize, that we instead have a piece of tape stuck to one of the beaters.(a) Explain why static friction has no effect on whether or not the tapeflies off. (b) Suppose you find that the tape doesn’t fly off when the motoris on a low speed, but speeding it up does cause it to fly off. Why wouldthe greater speed change things?

2. Show that the expression |v|2/r has the units of acceleration.

3 . A plane is flown in a loop-the-loop of radius 1.00 km. The planestarts out flying upside-down, straight and level, then begins curving upalong the circular loop, and is right-side up when it reaches the top . (Theplane may slow down somewhat on the way up.) How fast must the planebe going at the top if the pilot is to experience no force from the seat orthe seatbelt while at the top of the loop?

4 ∫. In this problem, you'll derive the equation |a|=|v|2/r using calculus.Instead of comparing velocities at two points in the particle's motion andthen taking a limit where the points are close together, you'll just take

derivatives. The particle's position vector is r=(r cos θ)x + (r sin θ)y ,

where x and y are the unit vectors along the x and y axes. By the defini-tion of radians, the distance traveled since t=0 is rθ, so if the particle istraveling at constant speed v=|v|, we have v=rθ/t. (a) Eliminate θ to get theparticle's position vector as a function of time. (b) Find the particle'sacceleration vector. (c) Show that the magnitude of the acceleration vectorequals v2/r.

5 S. Three cyclists in a race are rounding a semicircular curve. At themoment depicted, cyclist A is using her brakes to apply a force of 375 Nto her bike. Cyclist B is coasting. Cyclist C is pedaling, resulting in aforce of 375 N on her bike. Each cyclist, with her bike, has a mass of 75kg. At the instant shown, the instantaneous speed of all three cyclists is 10m/s. On the diagram, draw each cyclist's acceleration vector with its tailon top of her present position, indicating the directions and lengthsreasonably accurately. Indicate approximately the consistent scale you areusing for all three acceleration vectors. Extreme precision is not necessaryas long as the directions are approximately right, and lengths of vectorsthat should be equal appear roughly equal, etc. Assume all three cyclistsare traveling along the road all the time, not wandering across their lane orwiping out and going off the road.

Homework Problems

174

6 S«. The amusem*nt park ride shown in the figure consists of a cylindri-cal room that rotates about its vertical axis. When the rotation is fastenough, a person against the wall can pick his or her feet up off the floorand remain “stuck” to the wall without falling.(a) Suppose the rotation results in the person having a speed v. The radiusof the cylinder is r, the person’s mass is m, the downward acceleration ofgravity is g, and the coefficient of static friction between the person andthe wall is µ

s. Find an equation for the speed, v, required, in terms of the

other variables. (You will find that one of the variables cancels out.)(b) Now suppose two people are riding the ride. Huy is wearing denim,and Gina is wearing polyester, so Huy’s coefficient of static friction is threetimes greater. The ride starts from rest, and as it begins rotating faster andfaster, Gina must wait longer before being able to lift her feet withoutsliding to the floor. Based on your equation from part a, how many timesgreater must the speed be before Gina can lift her feet without slidingdown?

7 S. An engineer is designing a curved off-ramp for a freeway. Since theoff-ramp is curved, she wants to bank it to make it less likely that motor-ists going too fast will wipe out. If the radius of the curve is r, how greatshould the banking angle, θ, be so that for a car going at a speed v, nostatic friction force whatsoever is required to allow the car to make thecurve? State your answer in terms of v, r, and g, and show that the mass ofthe car is irrelevant.

8 . Lionel brand toy trains come with sections of track in standardlengths and shapes. For circular arcs, the most commonly used sectionshave diameters of 662 and 1067 mm at the inside of the outer rail. Themaximum speed at which a train can take the broader curve without flyingoff the tracks is 0.95 m/s. At what speed must the train be operated toavoid derailing on the tighter curve?

9. The figure shows a ball on the end of a string of length L attached to avertical rod which is spun about its vertical axis by a motor. The period(time for one rotation) is P.

(a) Analyze the forces in which the ball participates.

(b) Find how the angle θ depends on P, g, and L. [Hints: (1) Write downNewton’s second law for the vertical and horizontal components of forceand acceleration. This gives two equations, which can be solved for thetwo unknowns, θ and the tension in the string. (2) If you introducevariables like v and r, relate them to the variables your solution is supposedto contain, and eliminate them.]

(c) What happens mathematically to your solution if the motor is run veryslowly (very large values of P)? Physically, what do you think wouldactually happen in this case?

20 m

A

B

C

directionof travel

Problem 5.

v

r

Problem 6.

θ

Problem 7.

Chapter 9 Circular Motion

Problem 9.

θL

175

10. Psychology professor R.O. Dent requests funding for an experimenton compulsive thrill-seeking behavior in hamsters, in which the subject isto be attached to the end of a spring and whirled around in a horizontalcircle. The spring has equilibrium length b, and obeys Hooke’s law withspring constant k. It is stiff enough to keep from bending significantlyunder the hamster’s weight.

(a) Calculate the length of the spring when it is undergoing steady circularmotion in which one rotation takes a time T. Express your result in termsof k, b, and T.

(b) The ethics committee somehow fails to veto the experiment, but thesafety committee expresses concern. Why? Does your equation do any-thing unusual, or even spectacular, for any particular value of T? What doyou think is the physical significance of this mathematical behavior?

11«. The figure shows an old-fashioned device called a flyball governor,used for keeping an engine running at the correct speed. The whole thingrotates about the vertical shaft, and the mass M is free to slide up anddown. This mass would have a connection (not shown) to a valve thatcontrolled the engine. If, for instance, the engine ran too fast, the masswould rise, causing the engine to slow back down.

(a) Show that in the special case of a=0, the angle θ is given by

θ =

cos– 1 g(m + M)P 2

4π2mL ,

where P is the period of rotation (time required for one complete rota-tion).

(b) There is no closed-form solution for θ in the general case where a isnot zero. However, explain how the undesirable low-speed behavior of thea=0 device would be improved by making a nonzero.

[Based on an example by J.P. den Hartog.]

12. The figure shows two blocks of masses m1 and m

2 sliding in circles on

a frictionless table. Find the tension in the strings if the period of rotation(time required for one complete rotation) is P.

Problem 10.

Problem 11.

mm

M

L

a

θ

L1 L2m1 m2

Problem 12.

176

177

10 GravityCruise your radio dial today and try to find any popular song that

would have been imaginable without Louis Armstrong. By introducing soloimprovisation into jazz, Armstrong took apart the jigsaw puzzle of popularmusic and fit the pieces back together in a different way. In the same way,Newton reassembled our view of the universe. Consider the titles of somerecent physics books written for the general reader: The God Particle,Dreams of a Final Theory. When the subatomic particle called the neutrinowas recently proven for the first time to have mass, specialists in cosmologybegan discussing seriously what effect this would have on calculations of theultimate fate of the universe: would the neutrinos’ mass cause enough extragravitational attraction to make the universe eventually stop expanding andfall back together? Without Newton, such attempts at universal understand-ing would not merely have seemed a little pretentious, they simply wouldnot have occurred to anyone.

This chapter is about Newton’s theory of gravity, which he used toexplain the motion of the planets as they orbited the sun. Whereas thisbook has concentrated on Newton’s laws of motion, leaving gravity as adessert, Newton tosses off the laws of motion in the first 20 pages of thePrincipia Mathematica and then spends the next 130 discussing the motionof the planets. Clearly he saw this as the crucial scientific focus of his work.Why? Because in it he showed that the same laws of motion applied to theheavens as to the earth, and that the gravitational force that made an applefall was the same as the force that kept the earth’s motion from carrying itaway from the sun. What was radical about Newton was not his laws ofmotion but his concept of a universal science of physics.

Gravity is the only really important force on the cosmic scale. Left: a false-color image of saturn’s rings, composed of innu-merable tiny ice particles orbiting in circles under the influence of saturn’s gravity. Right: A stellar nursery, the Eagle Nebula.Each pillar of hydrogen gas is about as tall as the diameter of our entire solar system. The hydrogen molecules all attracteach other through gravitational forces, resulting in the formation of clumps that contract to form new stars.

178

10.1 Kepler’s LawsNewton wouldn’t have been able to figure out why the planets move the

way they do if it hadn’t been for the astronomer Tycho Brahe (1546-1601)and his protege Johannes Kepler (1571-1630), who together came up withthe first simple and accurate description of how the planets actually domove. The difficulty of their task is suggested by the figure below, whichshows how the relatively simple orbital motions of the earth and Marscombine so that as seen from earth Mars appears to be staggering in loopslike a drunken sailor.

sun

earth's orbitMars' orbit

Jan 1Feb 1Mar 1

Apr 1 May 1

Jun 1Jul 1Aug 1

As the earthand Mars

revolve aroundthe sun at different

rates, the combined effect of their motions

makes Mars appear to trace a strange, looped

path across the back-ground of the distant stars.

Brahe, the last of the great naked-eye astronomers, collected extensivedata on the motions of the planets over a period of many years, taking thegiant step from the previous observations’ accuracy of about 10 seconds ofarc (10/60 of a degree) to an unprecedented 1 second. The quality of hiswork is all the more remarkable considering that his observatory consistedof four giant brass protractors mounted upright in his castle in Denmark.Four different observers would simultaneously measure the position of aplanet in order to check for mistakes and reduce random errors.

With Brahe’s death, it fell to his former assistant Kepler to try to makesome sense out of the volumes of data. Kepler, in contradiction to his lateboss, had formed a prejudice, a correct one as it turned out, in favor of thetheory that the earth and planets revolved around the sun, rather than theearth staying fixed and everything rotating about it. Although motion isrelative, it is not just a matter of opinion what circles what. The earth’srotation and revolution about the sun make it a noninertial reference frame,which causes detectable violations of Newton’s laws when one attempts todescribe sufficiently precise experiments in the earth-fixed frame. Althoughsuch direct experiments were not carried out until the 19th century, what

Tycho Brahe made his name as anastronomer by showing that the brightnew star, today called a supernova,that appeared in the skies in 1572 wasfar beyond the Earth’s atmosphere.This, along with Galileo’s discovery ofsunspots, showed that contrary to Ar-istotle, the heavens were not perfectand unchanging. Brahe’s fame as anastronomer brought him patronagefrom King Frederick II, allowing him tocarry out his historic high-precisionmeasurements of the planets’ motions.A contradictory character, Brahe en-joyed lecturing other nobles about theevils of dueling, but had lost his ownnose in a youthful duel and had it re-placed with a prosthesis made of analloy of gold and silver. Willing to en-dure scandal in order to marry a peas-ant, he nevertheless used the feudalpowers given to him by the king toimpose harsh forced labor on the in-habitants of his parishes. The resultof their work, an Italian-style palacewith an observatory on top, surelyranks as one of the most luxuriousscience labs ever built. When the kingdied and his son reduced Brahe’s privi-leges, Brahe left in a huff for a newposition in Prague, taking his data withhim. He died of a ruptured bladder af-ter falling from a wagon on the wayhome from a party — at the time, itwas considered rude to leave the din-ner table to relieve oneself.

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convinced everyone of the sun-centered system in the 17th century was thatKepler was able to come up with a surprisingly simple set of mathematicaland geometrical rules for describing the planets’ motion using the sun-centered assumption. After 900 pages of calculations and many false startsand dead-end ideas, Kepler finally synthesized the data into the followingthree laws:

Kepler's elliptical orbit law: The planets orbit the sun in elliptical orbitswith the sun at one focus.

Kepler's equal-area law: The line connecting a planet to the sun sweepsout equal areas in equal amounts of time.

Kepler's law of periods: The time required for a planet to orbit the sun,called its period, is proportional to the long axis of the ellipse raised tothe 3/2 power. The constant of proportionality is the same for all theplanets.

Although the planets' orbits are ellipses rather than circles, most arevery close to being circular. The earth's orbit, for instance, is only flattenedby 1.7% relative to a circle. In the special case of a planet in a circularorbit, the two foci (plural of "focus") coincide at the center of the circle,and Kepler's elliptical orbit law thus says that the circle is centered on thesun. The equal-area law implies that a planet in a circular orbit movesaround the sun with constant speed. For a circular orbit, the law of periodsthen amounts to a statement that the time for one orbit is proportional tor3/2, where r is the radius. If all the planets were moving in their orbits at thesame speed, then the time for one orbit would simply depend on thecircumference of the circle, so it would only be proportional to r to the firstpower. The more drastic dependence on r3/2 means that the outer planetsmust be moving more slowly than the inner planets.

10.2 Newton’s Law of GravityThe sun's force on the planets obeys an inverse square law.

Kepler's laws were a beautifully simple explanation of what the planetsdid, but they didn't address why they moved as they did. Did the sun exerta force that pulled a planet toward the center of its orbit, or, as suggested byDescartes, were the planets circulating in a whirlpool of some unknownliquid? Kepler, working in the Aristotelian tradition, hypothesized not justan inward force exerted by the sun on the planet, but also a second force inthe direction of motion to keep the planet from slowing down. Somespeculated that the sun attracted the planets magnetically.

Once Newton had formulated his laws of motion and taught them tosome of his friends, they began trying to connect them to Kepler's laws. Itwas clear now that an inward force would be needed to bend the planets'paths. This force was presumably an attraction between the sun and each

An ellipse is a circle that has been dis-torted by shrinking and stretchingalong perpendicular axes.

An ellipse can be constructed by tyinga string to two pins and drawing likethis with the pencil stretching the stringtaut. Each pin constitutes one focusof the ellipse.

If the time interval taken by the planet to move from P to Q is equal to the timeinterval from R to S, then according to Kepler's equal-area law, the two shadedareas are equal. The planet is moving faster during interval RS than it did duringPQ, which Newton later determined was due to the sun's gravitational force accel-erating it. The equal-area law predicts exactly how much it will speed up.

sun P

QR

S

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planet. (Although the sun does accelerate in response to the attractions ofthe planets, its mass is so great that the effect had never been detected bythe prenewtonian astronomers.) Since the outer planets were moving slowlyalong more gently curving paths than the inner planets, their accelerationswere apparently less. This could be explained if the sun’s force was deter-mined by distance, becoming weaker for the farther planets. Physicists werealso familiar with the noncontact forces of electricity and magnetism, andknew that they fell off rapidly with distance, so this made sense.

In the approximation of a circular orbit, the magnitude of the sun’sforce on the planet would have to be

F = ma = mv2/r . (1)

Now although this equation has the magnitude, v, of the velocity vector init, what Newton expected was that there would be a more fundamentalunderlying equation for the force of the sun on a planet, and that thatequation would involve the distance, r, from the sun to the object, but notthe object’s speed, v.

Self-CheckIf eq. (1) really was generally applicable, what would happen to an objectreleased at rest in some empty region of the solar system?

Equation (1) was thus a useful piece of information which could berelated to the data on the planets simply because the planets happened to begoing in nearly circular orbits, but Newton wanted to combine it with otherequations and eliminate v algebraically in order to find a deeper truth.

To eliminate v, Newton used the equation

v = circumferenceT

= 2πr/T . (2)

Of course this equation would also only be valid for planets in nearlycircular orbits. Plugging this into eq. (1) to eliminate v gives

F = 4π2mr

T 2 . (3)

This unfortunately has the side-effect of bringing in the period, T, which weexpect on similar physical grounds will not occur in the final answer. That’swhere the circular-orbit case, T ∝ r3/2, of Kepler’s law of periods comes in.Using it to eliminate T gives a result that depends only on the mass of theplanet and its distance from the sun:

F ∝ m/r2 . [ force of the sun on a planet ofmass m at a distance r from the sun;same proportionality constant for allthe planets ]

(Since Kepler’s law of periods is only a proportionality, the final result is aproportionality rather than an equation, and there is this no point inhanging on to the factor of 4π2.)

It would just stay where it was. Plugging v=0 into eq. (1) would give F=0, so it would not accelerate from rest, andwould never fall into the sun. No astronomer had ever observed an object that did that!

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As an example, the "twin planets" Uranus and Neptune have nearly thesame mass, but Neptune is about twice as far from the sun as Uranus, so thesun's gravitational force on Neptune is about four times smaller.

The forces between heavenly bodies are the same type offorce as terrestrial gravity

OK, but what kind of force was it? It probably wasn't magnetic, sincemagnetic forces have nothing to do with mass. Then came Newton's greatinsight. Lying under an apple tree and looking up at the moon in the sky,he saw an apple fall. Might not the earth also attract the moon with thesame kind of gravitational force? The moon orbits the earth in the same waythat the planets orbit the sun, so maybe the earth's force on the fallingapple, the earth's force on the moon, and the sun's force on a planet wereall the same type of force.

There was an easy way to test this hypothesis numerically. If it was true,then we would expect the gravitational forces exerted by the earth to followthe same F∝m/r2 rule as the forces exerted by the sun, but with a differentconstant of proportionality appropriate to the earth's gravitational strength.The issue arises now of how to define the distance, r, between the earth andthe apple. An apple in England is closer to some parts of the earth than toothers, but suppose we take r to be the distance from the center of the earthto the apple, i.e. the radius of the earth. (The issue of how to measure r didnot arise in the analysis of the planets' motions because the sun and planetsare so small compared to the distances separating them.) Calling theproportionality constant k, we have

Fearth on apple

= k mapple

/ rearth

2

Fearth on moon

= k mmoon

/ dearth-moon

2 .

Newton's second law says a=F/m, so

aapple

= k / rearth

2

amoon

= k / dearth-moon

2 .

The Greek astronomer Hipparchus had already found 2000 years beforethat the distance from the earth to the moon was about 60 times the radiusof the earth, so if Newton's hypothesis was right, the acceleration of themoon would have to be 602=3600 times less than the acceleration of thefalling apple.

Applying a=v2/r to the acceleration of the moon yielded an accelerationthat was indeed 3600 times smaller than 9.8 m/s2, and Newton was con-vinced he had unlocked the secret of the mysterious force that kept themoon and planets in their orbits.

Newton's law of gravityThe proportionality F∝m/r2 for the gravitational force on an object of

mass m only has a consistent proportionality constant for various objects ifthey are being acted on by the gravity of the same object. Clearly the sun'sgravitational strength is far greater than the earth's, since the planets allorbit the sun and do not exhibit any very large accelerations caused by theearth (or by one another). What property of the sun gives it its greatgravitational strength? Its great volume? Its great mass? Its great tempera-ture? Newton reasoned that if the force was proportional to the mass of the

1

60

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object being acted on, then it would also make sense if the determiningfactor in the gravitational strength of the object exerting the force was itsown mass. Assuming there were no other factors affecting the gravitationalforce, then the only other thing needed to make quantitative predictions ofgravitational forces would be a proportionality constant. Newton called thatproportionality constant G, and the complete form of the law of gravity hehypothesized was

F = Gm1m

2/r2 . [ gravitational force between objects of mass

m1 and m

2, separated by a distance r; r is not

the radius of anything ]

Newton conceived of gravity as an attraction between any two masses in theuniverse. The constant G tells us the how many newtons the attractive forceis for two 1-kg masses separated by a distance of 1 m. The experimentaldetermination of G in ordinary units (as opposed to the special, nonmetric,units used in astronomy) is described in section 10.5. This difficult mea-surement was not accomplished until long after Newton’s death.

The proportionality to 1/r2 actually was not entirely unexpected.Proportionalities to 1/r2 are found in many other phenomena in whichsome effect spreads out from a point. For instance, the intensity of the lightfrom a candle is proportional to 1/r2, because at a distance r from thecandle, the light has to be spread out over the surface of an imaginarysphere of area 4πr2. The same is true for the intensity of sound from afirecracker, or the intensity of gamma radiation emitted by the Chernobylreactor. It's important, however, to realize that this is only an analogy. Forcedoes not travel through space as sound or light does, and force is not asubstance that can be spread thicker or thinner like butter on toast.

Although several of Newton's contemporaries had speculated that theforce of gravity might be proportional to 1/r2, none of them, even the oneswho had learned Newton's laws of motion, had had any luck proving thatthe resulting orbits would be ellipses, as Kepler had found empirically.Newton did succeed in proving that elliptical orbits would result from a 1/r2

force, but we postpone the proof until the end of the next volume of thetextbook because it can be accomplished much more easily using theconcepts of energy and angular momentum.

Newton also predicted that orbits in the shape of hyperbolas should bepossible, and he was right. Some comets, for instance, orbit the sun in veryelongated ellipses, but others pass through the solar system on hyperbolicpaths, never to return. Just as the trajectory of a faster baseball pitch isflatter than that of a more slowly thrown ball, so the curvature of a planet’sorbit depends on its speed. A spacecraft can be launched at relatively lowspeed, resulting in a circular orbit about the earth, or it can be launched at ahigher speed, giving a more gently curved ellipse that reaches farther fromthe earth, or it can be launched at a very high speed which puts it in aneven less curved hyperbolic orbit. As you go very far out on a hyperbola, itapproaches a straight line, i.e. its curvature eventually becomes nearly zero.

Newton also was able to prove that Kepler's second law (sweeping outequal areas in equal time intervals) was a logical consequence of his law ofgravity. Newton's version of the proof is moderately complicated, but theproof becomes trivial once you understand the concept of angular momen-

hyperbola

ellipse

circle

The conic sections are the curvesmade by cutting the surface of an infi-nite cone with a plane.

Chapter 10 Gravity

1 m

1 kg1 kg

6.67x10-11 N

The gravitational attraction betweentwo 1-kg masses separated by a dis-tance of 1 m is 6.67x10-11 N. Do notmemorize this number!

An imaginary cannon able to shootcannonballs at very high speeds isplaced on top of an imaginary, verytall mountain that reaches up abovethe atmosphere. Depending on thespeed at which the ball is fired, it mayend up in a tightly curved elliptical or-bit, a, a circular orbit, b, a bigger ellip-tical orbit, c, or a nearly straight hy-perbolic orbit, d.

a

b

d

c

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tum, which will be covered later in the course. There is therefore not muchpoint in giving a proof at this point.

Self-Check

Which of Kepler’s laws would it make sense to apply to hyperbolic orbits?

Discussion QuestionsA. How could Newton find the speed of the moon to plug in to a=v2/r?B. Two projectiles of different mass shot out of guns on the surface of the earthat the same speed and angle will follow the same trajectories, assuming thatair friction is negligible. (You can verify this by throwing two objects togetherfrom your hand and seeing if they separate or stay side by side.) Whatcorresponding fact would be true for satellites of the earth having differentmasses?C. What is wrong with the following statement? "A comet in an elliptical orbitspeeds up as it approaches the sun, because the sun's force on it is increas-ing."D. Why would it not make sense to expect the earth’s gravitational force on abowling ball to be inversely proportional to the square of the distance betweentheir surfaces rather than their centers?E. Does the earth accelerate as a result of the moon's gravitational force on it?Suppose two planets were bound to each other gravitationally the way theearth and moon are, but the two planets had equal masses. What would theirmotion be like?F. Spacecraft normally operate by firing their engines only for a few minutes ata time, and an interplanetary probe will spend months or years on its way to itsdestination without thrust. Suppose a spacecraft is in a circular orbit aroundMars, and it then briefly fires its engines in reverse, causing a sudden de-crease in speed. What will this do to its orbit? What about a forward thrust?

10.3 Apparent WeightlessnessIf you ask somebody at the bus stop why astronauts are weightless,

you’ll probably get one of the following two incorrect answers:

(1) They’re weightless because they’re so far from the earth.

(2) They’re weightless because they’re moving so fast.

The first answer is wrong, because the vast majority of astronauts never getmore than a thousand miles from the earth’s surface. The reduction ingravity caused by their altitude is significant, but not 100%. The secondanswer is wrong because Newton’s law of gravity only depends on distance,not speed.

The correct answer is that astronauts in orbit around the earth are notreally weightless at all. Their weightlessness is only apparent. If there was nogravitational force on the spaceship, it would obey Newton’s first law andmove off on a straight line, rather than orbiting the earth. Likewise, theastronauts inside the spaceship are in orbit just like the spaceship itself, withthe earth’s gravitational force continually twisting their velocity vectorsaround. The reason they appear to be weightless is that they are in the sameorbit as the spaceship, so although the earth’s gravity curves their trajectory

The equal-area law makes equally good sense in the case of a hyperbolic orbit (and observations verify it). Theelliptical orbit law had to be generalized by Newton to include hyperbolas. The law of periods doesn’t make sensein the case of a hyperbolic orbit, because a hyperbola never closes back on itself, so the motion never repeats.

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184

down toward the deck, the deck drops out from under them at the samerate.

Apparent weightlessness can also be experienced on earth. Any time youjump up in the air, you experience the same kind of apparent weightlessnessthat the astronauts do. While in the air, you can lift your arms more easilythan normal, because gravity does not make them fall any faster than therest of your body, which is falling out from under them. The Russian airforce now takes rich foreign tourists up in a big cargo plane and gives themthe feeling of weightlessness for a short period of time while the plane isnose-down and dropping like a rock.

10.4 Vector Addition of Gravitational ForcesPick a flower on earth and you move the farthest star.

Paul DiracWhen you stand on the ground, which part of the earth is pulling down

on you with its gravitational force? Most people are tempted to say that theeffect only comes from the part directly under you, since gravity alwayspulls straight down. Here are three observations that might help to changeyour mind:

• If you jump up in the air, gravity does not stop affecting you justbecause you are not touching the earth: gravity is a noncontact force.That means you are not immune from the gravity of distant parts ofour planet just because you are not touching them.

• Gravitational effects are not blocked by intervening matter. Forinstance, in an eclipse of the moon, the earth is lined up directlybetween the sun and the moon, but only the sun's light is blockedfrom reaching the moon, not its gravitational force -- if the sun'sgravitational force on the moon was blocked in this situation,astronomers would be able to tell because the moon's accelerationwould change suddenly. A more subtle but more easily observableexample is that the tides are caused by the moon's gravity, and tidaleffects can occur on the side of the earth facing away from the moon.Thus, far-off parts of the earth are not prevented from attracting youwith their gravity just because there is other stuff between you andthem.

• Prospectors sometimes search for underground deposits of denseminerals by measuring the direction of the local gravitational forces,i.e. the direction things fall or the direction a plumb bob hangs. Forinstance, the gravitational forces in the region to the west of such adeposit would point along a line slightly to the east of the earth'scenter. Just because the total gravitational force on you points down,that doesn't mean that only the parts of the earth directly below youare attracting you. It's just that the sideways components of all theforce vectors acting on you come very close to canceling out.

Chapter 10 Gravity

Gravity only appears to pull straightdown because the near perfect sym-metry of the earth makes the sidewayscomponents of the total force on anobject cancel almost exactly. If thesymmetry is broken, e.g. by a densemineral deposit, the total force is a littleoff to the side.

185

A cubic centimeter of lava in the earth's mantle, a grain of silica insideMt. Kilimanjaro , and a flea on a cat in Paris are all attracting you with theirgravity. What you feel is the vector sum of all the gravitational forcesexerted by all the atoms of our planet, and for that matter by all the atomsin the universe.

When Newton tested his theory of gravity by comparing the orbitalacceleration of the moon to the acceleration of a falling apple on earth, heassumed he could compute the earth's force on the apple using the distancefrom the apple to the earth's center. Was he wrong? After all, it isn't just theearth's center attracting the apple, it's the whole earth. A kilogram of dirt afew feet under his backyard in England would have a much greater force onthe apple than a kilogram of molten rock deep under Australia, thousandsof miles away. There's really no obvious reason why the force should comeout right if you just pretend that the earth's whole mass is concentrated atit* center. Also, we know that the earth has some parts that are more dense,and some parts that are less dense. The solid crust, on which we live, isconsiderably less dense that the molten rock underneath on which it floats.By all rights, the computation of the vector sum of all the forces exerted byall the earth's parts should be a horrendous mess.

Actually, Newton had sound mathematical reasons for treating theearth's mass as if it was concentrated at its center. First, although Newtonno doubt suspected the earth's density was nonuniform, he knew that thedirection of its total gravitational force was very nearly toward the earth'scenter. That was strong evidence that the distribution of mass was verysymmetric, so that we can think of the earth as being made of many layers,like an onion, with each layer having constant density throughout. (Todaythere is further evidence for symmetry based on measurements of how thevibrations from earthquakes and nuclear explosions travel through theearth.) Newton then concentrated on the gravitational forces exerted by asingle such thin shell, and proved the following mathematical theorem,known as the shell theorem:

If an object lies outside a thin, uniform shell of mass, then thevector sum of all the gravitational forces exerted by all the parts ofthe shell is the same as if all the shell's mass was concentrated at itscenter. If the object lies inside the shell, then all the gravitationalforces cancel out exactly.

For terrestrial gravity, each shell acts as though its mass was concen-trated at the earth's center, so the final result is the same as if the earth'swhole mass was concentrated at its center.

The second part of the shell theorem, about the gravitational forcescanceling inside the shell, is a little surprising. Obviously the forces wouldall cancel out if you were at the exact center of a shell, but why should theystill cancel out perfectly if you are inside the shell but off-center? Thewhole idea might seem academic, since we don't know of any hollowplanets in our solar system that astronauts could hope to visit, but actuallyit's a useful result for understanding gravity within the earth, which is animportant issue in geology. It doesn't matter that the earth is not actually

An object outside a spherical shell ofmass will feel gravitational forces fromevery part of the shell — strongerforces from the closer parts andweaker ones from the parts fartheraway. The shell theorem states thatthe vector sum of all the forces is thesame as if all the mass had been con-centrated at the center of the shell.

Section 10.4 Vector Addition of Gravitational Forces

186

hollow. In a mine shaft at a depth of, say, 2 km, we can use the shelltheorem to tell us that the outermost 2 km of the earth has no net gravita-tional effect, and the gravitational force is the same as what would beproduced if the remaining, deeper, parts of the earth were all concentratedat its center.

Discussion QuestionsA. If you hold an apple in your hand, does the apple exert a gravitational forceon the earth? Is it much weaker than the earth's gravitational force on theapple? Why doesn't the earth seem to accelerate upward when you drop theapple?B. When astronauts travel from the earth to the moon, how does the gravita-tional force on them change as they progress?C. How would the gravity in the first-floor lobby of a massive skyscrapercompare with the gravity in an open field outside of the city?

10.5 Weighing the EarthLet's look more closely at the application of Newton's law of gravity to

objects on the earth's surface. Since the earth's gravitational force is thesame as if its mass was all concentrated at its center, the force on a fallingobject of mass m is given by

F = G Mearth

m / rearth

2 .

The object's acceleration is F/m, so the object's mass cancels out and we geta constant acceleration for all falling objects, as we knew we should:

g = G Mearth

/ rearth

2 .

Newton knew neither the mass of the earth nor a numerical value forthe constant G. But if someone could measure G, then it would be possiblefor the first time in history to determine the mass of the earth! The onlyway to measure G is to measure the gravitational force between two objectsof known mass, but that's an exceedingly difficult task, because the forcebetween any two objects of ordinary size is extremely small. The Englishphysicist Henry Cavendish was the first to succeed, using the apparatusshown in the diagrams. The two larger balls were lead spheres 8 inches indiameter, and each one attracted the small ball near it. The two small ballshung from the ends of a horizontal rod, which itself hung by a thin thread.

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The frame from which the larger balls hung could be rotated by hand abouta vertical axis, so that for instance the large ball on the right would pull itsneighboring small ball toward us and while the small ball on the left wouldbe pulled away from us. The thread from which the small balls hung wouldthus be twisted through a small angle, and by calibrating the twist of thethread with known forces, the actual gravitational force could be deter-mined. Cavendish set up the whole apparatus in a room of his house,nailing all the doors shut to keep air currents from disturbing the delicateapparatus. The results had to be observed through telescopes stuck throughholes drilled in the walls. Cavendish's experiment provided the firstnumerical values for G and for the mass of the earth. The presently ac-cepted value of G is 6.67x10-11 N.m2/kg2.

The following page shows a modern-day Cavendish experiment con-structed by one of my students.

Cavendish’s apparatus viewed fromthe side, and a simplified versionviewed from above. The two large ballsare fixed in place, but the rod fromwhich the two small balls hang is freeto twist under the influence of thegravitational forces.

Section 10.5 Weighing the Earth

188 Chapter 10 Gravity

My student Narciso Guzman built this version of theCavendish experiment in his garage, from a descrip-tion on the Web at www.fourmilab.to. Two steel ballssit near the ends of a piece of styrofoam, which is sus-pending from a ladder by fishing line (not visible in thisphoto). To make vibrations die out more quickly, a smallpiece of metal from a soda can is attached underneaththe styrofoam arm, sticking down into a bowl of water.(The arm is not resting on the bowl.)

The sequence of four video frames on the right showsthe apparatus in action. Initially (top), lead bricks areinserted near the steel balls. They attract the balls, andthe arm begins to rotate counterclockwise.

The main difficulties in this experiment are isolatingthe apparatus from vibrations and air currents. Narcisohad to leave the room while the camcorder ran. Also, itis helpful if the apparatus can be far from walls orfurnitude that would create gravitational forces on it.

Another student is now working on turning this ex-periment into a numerical measurement of G, to be usedas a laboratory exercise at Fullerton College.

189

Knowing G not only allowed the determination of the earth’s mass butalso those of the sun and the other planets. For instance, by observing theacceleration of one of Jupiter’s moons, we can infer the mass of Jupiter. Thefollowing table gives the distances of the planets from the sun and themasses of the sun and planets.

average distance fromthe sun, in units ofthe earth's averagedistance from the sun

mass, in units of theearth's mass

sun — 330,000

mercury 0.38 .056

venus .72 .82

earth 1 1

mars 1.5 .11

jupiter 5.2 320

saturn 9.5 95

uranus 19 14

neptune 30 17

pluto 39 .002

Discussion QuestionsA. It would have been difficult for Cavendish to start designing an experimentwithout at least some idea of the order of magnitude of G. How could heestimate it in advance to within a factor of 10?B. Fill in the details of how one would determine Jupiter’s mass by observingthe acceleration of one of its moons. Why is it only necessary to know theacceleration of the moon, not the actual force acting on it? Why don’t we needto know the mass of the moon? What about a planet that has no moons, suchas Venus — how could its mass be found?C. The gravitational constant G is very difficult to measure accurately, and isthe least accurately known of all the fundamental numbers of physics such asthe speed of light, the mass of the electron, etc. But that's in the mks system,based on the meter as the unit of length, the kilogram as the unit of mass, andthe second as the unit of distance. Astronomers sometimes use a differentsystem of units, in which the unit of distance, called the astronomical unit ora.u., is the radius of the earth's orbit, the unit of mass is the mass of the sun,and the unit of time is the year (i.e. the time required for the earth to orbit thesun). In this system of units, G has a precise numerical value simply as amatter of definition. What is it?

Section 10.5 Weighing the Earth

190

SummarySelected Vocabulary

ellipse ................................ a flattened circle; one of the conic sectionsconic section...................... a curve formed by the intersection of a plane and an infinite conehyperbola .......................... another conic section; it does not close back on itselfperiod ................................ the time required for a planet to complete one orbit; more generally, the

time for one repetition of some repeating motionfocus .................................. one of two special points inside an ellipse: the ellipse consists of all points

such that the sum of the distances to the two foci equals a certain number;a hyperbola also has a focus

NotationG ....................................... the constant of proportionality in Newton’s law of gravity; the gravita-

tional force of attraction between two 1-kg spheres at a center-to-centerdistance of 1 m

SummaryKepler deduced three empirical laws from data on the motion of the planets:

Kepler's elliptical orbit law : The planets orbit the sun in elliptical orbits with the sun at one focus.

Kepler's equal-area law : The line connecting a planet to the sun sweeps out equal areas in equalamounts of time.

Kepler's law of periods : The time required for a planet to orbit the sun is proportional to the long axisof the ellipse raised to the 3/2 power. The constant of proportionality is the same for all the planets.

Newton was able to find a more fundamental explanation for these laws. Newton’s law of gravity statesthat the magnitude of the attractive force between any two objects in the universe is given by

F = Gm1m2/r 2 .

Weightlessness of objects in orbit around the earth is only apparent. An astronaut inside a spaceship issimply falling along with the spaceship. Since the spaceship is falling out from under the astronaut, it appearsas though there was no gravity accelerating the astronaut down toward the deck.

Gravitational forces, like all other forces, add like vectors. A gravitational force such as we ordinarily feel isthe vector sum of all the forces exerted by all the parts of the earth. As a consequence of this, Newton provedthe shell theorem for gravitational forces:

If an object lies outside a thin, uniform shell of mass, then the vector sum of all the gravitational forcesexerted by all the parts of the shell is the same as if all the shell's mass was concentrated at its center.If the object lies inside the shell, then all the gravitational forces cancel out exactly.

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191

Homework Problems1 . Roy has a mass of 60 kg. Laurie has a mass of 65 kg. They are 1.5 mapart.(a) What is the magnitude of the gravitational force of the earth on Roy?(b) What is the magnitude of Roy's gravitational force on the earth?(c) What is the magnitude of the gravitational force between Roy andLaurie?(d) What is the magnitude of the gravitational force between Laurie andthe sun?

2. During a solar eclipse, the moon, earth and sun all lie on the same line,with the moon between the earth and sun. Define your coordinates so thatthe earth and moon lie at greater x values than the sun. For each force,give the correct sign as well as the magnitude. (a) What force is exerted onthe moon by the sun? (b) On the moon by the earth? (c) On the earth bythe sun? (d) What total force is exerted on the sun? (e) On the moon? (f )On the earth?

3. Suppose that on a certain day there is a crescent moon, and you cantell by the shape of the crescent that the earth, sun and moon form atriangle with a 135° interior angle at the moon's corner. What is themagnitude of the total gravitational force of the earth and the sun on themoon?

moon

earth

sun

4. How high above the Earth's surface must a rocket be in order to have 1/100 the weight it would have at the surface? Express your answer in unitsof the radius of the Earth.

5. The star Lalande 21185 was found in 1996 to have two planets inroughly circular orbits, with periods of 6 and 30 years. What is the ratio ofthe two planets' orbital radii?

6. In a Star Trek episode, the Enterprise is in a circular orbit around aplanet when something happens to the engines. Spock then tells Kirk thatthe ship will spiral into the planet's surface unless they can fix the engines.Is this scientifically correct? Why?

7. (a) Suppose a rotating spherical body such as a planet has a radius r anda uniform density ρ, and the time required for one rotation is T. At thesurface of the planet, the apparent acceleration of a falling object isreduced by acceleration of the ground out from under it. Derive anequation for the apparent acceleration of gravity, g, at the equator in termsof r, ρ, T, and G.(b) Applying your equation from (a), by what fraction is your apparentweight reduced at the equator compared to the poles, due to the Earth'srotation?(c) Using your equation from (a), derive an equation giving the value of Tfor which the apparent acceleration of gravity becomes zero, i.e. objectscan spontaneously drift off the surface of the planet. Show that T only

Homework Problems

192

depends on ρ, and not on r.(d) Applying your equation from (c), how long would a day have to be inorder to reduce the apparent weight of objects at the equator of the Earthto zero? [Answer: 1.4 hours](e) Observational astronomers have recently found objects they calledpulsars, which emit bursts of radiation at regular intervals of less than asecond. If a pulsar is to be interpreted as a rotating sphere beaming out anatural "searchlight" that sweeps past the earth with each rotation, useyour equation from (c) to show that its density would have to be muchgreater than that of ordinary matter.(f ) Theoretical astronomers predicted decades ago that certain stars thatused up their sources of energy could collapse, forming a ball of neutronswith the fantastic density of ~1017 kg/m3. If this is what pulsars really are,use your equation from (c) to explain why no pulsar has ever been ob-served that flashes with a period of less than 1 ms or so.

8. You are considering going on a space voyage to Mars, in which yourroute would be half an ellipse, tangent to the Earth's orbit at one end andtangent to Mars' orbit at the other. Your spacecraft's engines will only beused at the beginning and end, not during the voyage. How long wouldthe outward leg of your trip last? (Assume the orbits of Earth and Mars arecircular.)

9.« (a) If the earth was of uniform density, would your weight be in-creased or decreased at the bottom of a mine shaft? Explain. (b) In real life,objects weight slightly more at the bottom of a mine shaft. What does thatallow us to infer about the Earth?

10 S. Ceres, the largest asteroid in our solar system, is a spherical bodywith a mass 6000 times less than the earth’s, and a radius which is 13times smaller. If an astronaut who weighs 400 N on earth is visiting thesurface of Ceres, what is her weight?

11 S. Prove, based on Newton’s laws of motion and Newton’s law ofgravity, that all falling objects have the same acceleration if they aredropped at the same location on the earth and if other forces such asfriction are unimportant. Do not just say, “g=9.8 m/s2 -- it’s constant.”You are supposed to be proving that g should be the same number for allobjects.

12 S. The figure shows an image from the Galileo space probe takenduring its August 1993 flyby of the asteroid Ida. Astronomers weresurprised when Galileo detected a smaller object orbiting Ida. This smallerobject, the only known satellite of an asteroid in our solar system, waschristened Dactyl, after the mythical creatures who lived on Mount Ida,and who protected the infant Zeus. For scale, Ida is about the size andshape of Orange County, and Dactyl the size of a college campus. Galileowas unfortunately unable to measure the time, T, required for Dactyl toorbit Ida. If it had, astronomers would have been able to make the firstaccurate determination of the mass and density of an asteroid. Find anequation for the density, ρ, of Ida in terms of Ida’s known volume, V, theknown radius, r, of Dactyl’s orbit, and the lamentably unknown variable T.(This is the same technique that was used successfully for determining themasses and densities of the planets that have moons.)

Chapter 10 Gravity

Problem 8.

Earth's orbit

Mars' orbit

your orbit

193

Problem 12.

13 ∫. If a bullet is shot straight up at a high enough velocity, it will neverreturn to the earth. This is known as the escape velocity. We will discussescape velocity using the concept of energy in the next book of the series,but it can also be gotten at using straightforward calculus. In this problem,you will analyze the motion of an object of mass m whose initial velocity isexactly equal to escape velocity. We assume that it is starting from thesurface of a spherically symmetric planet of mass M and radius b. The trickis to guess at the general form of the solution, and then determine thesolution in more detail. Assume (as is true) that the solution is of the formr = kt p, where r is the object’s distance from the center of the planet attime t, and k and p are constants. (a) Find the acceleration, and useNewton’s second law and Newton’s law of gravity to determine k and p.You should find that the result is independent of m. (b) What happens tothe velocity as t approaches infinity? (c) Determine escape velocity fromthe Earth’s surface.

14. Astronomers have recently observed stars orbiting at very high speedsaround an unknown object near the center of our galaxy. For stars orbitingat distances of about 1014 m from the object, the orbital velocities areabout 106 m/s. Assuming the orbits are circular, estimate the mass of theobject, in units of the mass of the sun, 2x1030 kg. If the object was atightly packed cluster of normal stars, it should be a very bright source oflight. Since no visible light is detected coming from it, it is instead be-lieved to be a supermassive black hole.

194

ExercisesExercise 0A: Models and Idealization

Equipment:coffee filtersramps (one per group)balls of various sizessticky tapevacuum pump and “guinea and feather” apparatus (one)

The motion of falling objects has been recognized since ancient times as an important piece of phys-ics, but the motion is inconveniently fast, so in our everyday experience it can be hard to tell exactlywhat objects are doing when they fall. In this exercise you will use several techniques to get aroundthis problem and study the motion. Your goal is to construct a scientific model of falling. A modelmeans an explanation that makes testable predictions. Often models contain simplifications or ideali-zations that make them easier to work with, even though they are not strictly realistic.

1. One method of making falling easier to observe is to use objects like feathers that we know fromeveryday experience will not fall as fast. You will use coffee filters, in stacks of various sizes, to test thefollowing two hypotheses and see which one is true, or whether neither is true:

Hypothesis 1A: When an object is dropped, it rapidly speeds up to a certain natural fallingspeed, and then continues to fall at that speed. The falling speed is proportional to the object’sweight. (A proportionality is not just a statement that if one thing gets bigger, the other does too.It says that if one becomes three times bigger, the other also gets three times bigger, etc.)

Hypothesis 1B: Different objects fall the same way, regardless of weight.

Test these hypotheses and discuss your results with your instructor.

2. A second way to slow down the action is to let a ball roll down a ramp. The steeper the ramp, thecloser to free fall. Based on your experience in part 1, write a hypothesis about what will happen whenyou race a heavier ball against a lighter ball down the same ramp, starting them both from rest.

Hypothesis:______________________________________________________________

Show your hypothesis to your instructor, and then test it.

You have probably found that falling was more complicated than you thought! Is there more than onefactor that affects the motion of a falling object? Can you imagine certain idealized situations that aresimpler?Try to agree verbally with your group on an informal model of falling that can make predictionsabout the experiments described in parts 3 and 4.

3. You have three balls: a standard “comparison ball” of medium weight, a light ball, and a heavy ball.Suppose you stand on a chair and (a) drop the light ball side by side with the comparison ball, then (b)drop the heavy ball side by side with the comparison ball, then (c) join the light and heavy ballstogether with sticky tape and drop them side by side with the comparison ball.

Use your model to make a prediction:____________________________________________________

Test your prediction.

4. Your instructor will pump nearly all the air out of a chamber containing a feather and a heavierobject, then let them fall side by side in the chamber.

Use your model to make a prediction:____________________________________________________

Exercise 1A: Scaling Applied to Leaves

Equipment:leaves of three sizes, having roughly similar proportions of length, width, and thickness

(example: blades of grass, large ficus leaves, and agave leaves)balance

1. Each group will have one leaf, and should measure its surface area and volume, and determine itssurface-to-volume ratio (surface area divided by volume). For consistency, every group should useunits of cm2 and cm3, and should only find the area of one side of the leaf. The area can be foundtracing the area of the leaf on graph paper and counting squares. The volume can be found by weigh-ing the leaf and assuming that its density is 1 g/cm3, which is nearly true since leaves are mostly water.Write your results on the board for comparison with the other groups’ numbers.2. Both the surface area and the volume are bigger for bigger leaves, but what about the surface tovolume ratios? What implications would this have for the plants’ abilities to survive in different environ-ments?

Exercise 2A: Changing Velocity

This exercise involves Michael Johnson’s world-record 200-meter sprint in the 1996 Olympics. Thetable gives the distance he has covered at various times. (The data are made up, except for his totaltime.) Each group is to find a value of ∆x/∆t between two specified instants, with the members of thegroup checking each other’s answers. We will then compare everyone’s results and discuss how thisrelates to velocity.

t (s) x (m)A 10.200 100.0000B 10.210 100.0990C 10.300 100.9912D 11.200 110.0168E 19.320 200.0000

group 1: Find ∆x/∆t using points A and B.group 2: Find ∆x/∆t using points A and C.group 3: Find ∆x/∆t using points A and D.group 4: Find ∆x/∆t using points A and E.

Exercise 3A: Reasoning with Ratios and Powers

Equipment:ping-pong balls and paddlestwo-meter sticks

You have probably bounced a ping pong ball straight up and down in the air. The time between hits isrelated to the height to which you hit the ball. If you take twice as much time between hits, how manytimes higher do you think you will have to hit the ball? Write down your hypoth-esis:__________________________

Your instructor will first beat out a tempo of 240 beats per minute (four beats per second), which youshould try to match with the ping-pong ball. Measure the height to which the ball rises:_______________

Now try it at 120 beats per minute:________________

Compare your hypothesis and your results with the rest of the class.

Exercise 4A: Force and Motion

Equipment:2-meter pieces of butcher paperwood blocks with hooksstringmasses to put on top of the blocks to increase frictionspring scales (preferably calibrated in Newtons)

Suppose a person pushes a crate, sliding it across the floor at a certain speed, and then repeats thesame thing but at a higher speed. This is essentially the situation you will act out in this exercise. Whatdo you think is different about her force on the crate in the two situations? Discuss this with your groupand write down your hypothesis:

_________________________________________________________________________

1. First you will measure the amount of friction between the wood block and the butcher paper whenthe wood and paper surfaces are slipping over each other. The idea is to attach a spring scale to theblock and then slide the butcher paper under the block while using the scale to keep the block frommoving with it. Depending on the amount of force your spring scale was designed to measure, youmay need to put an extra mass on top of the block in order to increase the amount of friction. It is agood idea to use long piece of string to attach the block to the spring scale, since otherwise one tendsto pull at an angle instead of directly horizontally.

First measure the amount of friction force when sliding the butcher paper as slowly as pos-sible:___________________

Now measure the amount of friction force at a significantly higher speed, say 1 meter per second. (Ifyou try to go too fast, the motion is jerky, and it is impossible to get an accurate reading.)___________________

Discuss your results. Why are we justified in assuming that the string’s force on the block (i.e. thescale reading) is the same amount as the paper’s frictional force on the block?

2. Now try the same thing but with the block moving and the paper standing still. Try two differentspeeds.

Do your results agree with your original hypothesis? If not, discuss what’s going on. How does theblock “know” how fast to go?

Exercise 4B: Interactions

Equipment:neodymium disc magnets (3/group)compasstriple-arm balance (2/group)clamp and 50-cm vertical rod for holding

balance upstringtapescissors

Your goal in this exercise is to compare the forcestwo magnets exert on each other, i.e. to comparemagnet A’s force on magnet B to magnet B’s forceon magnet A. Magnet B will be made out of two ofthe small disc magnets put together, so it is twiceas strong as magnet A.

1. Note that these magnets are extremely strong!Being careful not to pinch your skin, put two discmagnets together to make magnet B.

2. Familiarize yourself with how the magnets be-have. In addition to magnets A and B, there aretwo other magnets that can come into play. Thecompass needle itself is a magnet, and the planetearth is a magnet. Ordinarily the compass needletwists around under the influence of the earth, butthe disc magnets are very strong close up, so ifyou bring them within a few cm of the compass,the compass is essentially just responding to them.Investigate how different parts of magnets A andB interact with the compass, and label them ap-propriately. Investigate how magnets A and B canattract or repel one another.

3. You are ready to form a hypothesis about thefollowing situation. Suppose we set up two bal-ances as shown in the figure. The magnets arenot touching. The top magnet is hanging from ahook underneath the pan, giving the same resultas if it was on top of the pan. Make sure it is hang-ing under the center of the pan. You will want tomake sure the magnets are pulling on each other,not pushing each other away, so that the top mag-net will stay in one place.

The balances will not show the magnets’ trueweights, because the magnets are exerting forceson each other. The top balance will read a highernumber than it would without any magnetic forces,and the bottom balance will have a lower thannormal reading. The difference between eachmagnet’s true weight and the reading on the bal-

ance gives a measure of how strongly the magnetis being pushed or pulled by the other magnet.

How do you think the amount of pushing or pull-ing experienced by the two magnets will compare?In other words, which reading will change more,or will they change by the same amount? Writedown a hypothesis:________________________

Before going on to part 4, discuss your hypoth-esis with your instructor.

4. Now set up the experiment described abovewith two balances. Since we are interested in thechangse in the scale readings caused by the mag-netic forces, you will need to take a total of fourscale readings: one pair with the balances sepa-rated and one pair with the magnets close togetheras shown in the figure above.

When the balances are together and the magneticforces are acting, it is not possible to get both bal-ances to reach equilibrium at the same time, be-cause sliding the weights on one balance cancause its magnet to move up or down, tipping theother balance. Therefore, while you take a read-ing from one balance, you need to immobilize theother in the horizontal position by taping its tip soit points exactly at the zero mark.

You will also probably find that as you slide theweights, the pointer swings suddenly to the oppo-site side, but you can never get it to be stable inthe middle (zero) position. Try bringing the pointermanually to the zero position and then releasingit. If it swings up, you’re too low, and if it swingsdown, you’re too high. Search for the dividing linebetween the too-low region and the too-high re-gion.

If the changes in the scale readings are very small(say a few grams or less), you need to get themagnets closer together. It should be possible toget the scale readings to change by large amounts(up to 10 or 20 g).

magnet A taped to pencilmagnet B

pencil

Exercise 5A: Friction

Equipment:2-meter pieces of butcher paperwood blocks with hooksstringmasses to put on top of the blocks to increase frictionspring scales (preferably calibrated in Newtons)

1. Using the same equipment as in exercise 4A, test the statement that kinetic friction is approximatelyindependent of velocity.

2. Test the statement that kinetic friction is independent of surface area.

203

Solutions to SelectedProblems

Chapter 0

6. 134 mg × 10 – 3 g

1 mg × 10 – 3 kg1 g = 1.34 × 10 – 4 kg

Chapter 1

10. 1 mm2 × 1 cm10 mm

2= 10 – 2 cm2

Chapter 2

4. 1 light-year = v ∆t

= 3.0x108 m/s 1 year

× 365 days

1 year24 hours

1 day3600 s1 hour

= 9.5x1015 m

5. Velocity is relative, so having to lean tells younothing about the train’s velocity. Fullerton ismoving at a huge speed relative to Beijing, but thatdoesn’t produce any noticeable effect in either city.The fact that you have to lean tells you that the trainis accelerating.

Chapter 3

14.

x

v

t

t

15. Taking g to be 10 m/s, the bullet loses 10 m/s ofspeed every second, so it will take 10 s to come to astop, and then another 10 s to come back down, fora total of 20 s.

16. ∆x = 12at 2 , so for a fixed value of ∆x, we have

t ∝ 1/ a . Decreasing a by a factor of 3 means that t

will increase by a factor of 3 .

17. v = dxdt

= 10 – 3t 2

a = dvdt

= —6t

= —18 m/s2

18. (a) Solving ∆x = 12at 2 for a, we find a=2∆x/t2=5.51

m/s2. (b) v= 2a∆x =66.6 m/s. (c) The actual car’sfinal velocity is less than that of the idealized con-stant-acceleration car. If the real car and the idealizedcar covered the quarter mile in the same time but thereal car was moving more slowly at the end than theidealized one, the real car must have been goingfaster than the idealized car at the beginning of therace. The real car apparently has a greater accelera-tion at the beginning, and less acceleration at theend. This make sense, because every car has somemaximum speed, which is the speed beyond which itcannot accelerate.

Chapter 4

7. a= ∆v∆t , and also a= F

m , so

∆t = ∆va

= m∆vF

= (1000 kg)(50 m/s – 20 m/s)3000 N

= 10 s

Solutions to Selected Problems

204

Chapter 6

5. (a) The easiest strategy is to find the time spentaloft, and then find the range. The vertical motion andthe horizontal motion are independent. The verticalmotion has acceleration —g, and the cannonballspends enough time in the air to reverse its verticalvelocity component completely, so we have

∆vy

= vyf—v

yi

= —2v sin θ .

The time spent aloft is therefore

∆t = ∆vy / a

y

= 2v sin θ / g .

During this time, the horizontal distance traveled is

R = vx∆t

= 2 v 2 sin θ cos θ / g .

(b) The range becomes zero at both θ=0 and atθ=90°. The θ=0 case gives zero range because theball hits the ground as soon as it leaves the mouth ofthe cannon. A 90 degree angle gives zero rangebecause the cannonball has no horizontal motion.

Chapter 8

8. We want to find out about the velocity vector vBG

ofthe bullet relative to the ground, so we need to addAnnie’s velocity relative to the ground v

AG to the

bullet’s velocity vector vBA

relative to her. Letting thepositive x axis be east and y north, we have

vBA,x

= (140 mi/hr) cos 45°

= 100 mi/hr

vBA,y

= (140 mi/hr) sin 45°

= 100 mi/hr

and

vAG,x

= 0

vAG,y

= 30 mi/hr .

The bullet’s velocity relative to the ground thereforehas components

vBG,x

= 100 mi/hr and

vBG,y

= 130 mi/hr .

Its speed on impact with the animal is the magnitudeof this vector

|vBG

| = (100 mi/hr)2 + (130 mi/hr)2

= 160 mi/hr

(rounded off to 2 significant figures).

9. Since its velocity vector is constant, it has zeroacceleration, and the sum of the force vectors actingon it must be zero. There are three forces acting onthe plane: thrust, lift, and gravity. We are given thefirst two, and if we can find the third we can infer itsmass. The sum of the y components of the forces iszero, so

0 = Fthrust,y +Flift,y +FW,y

= |Fthrust

| sin θ + |Flift

| cos θ — mg .

The mass is

m = (|Fthrust

| sin θ + |Flift

| cos θ) / g

= 6.9x104 kg

10. (a) Since the wagon has no acceleration, the totalforces in both the x and y directions must be zero.There are three forces acting on the wagon: F

T, F

W,

and the normal force from the ground, FN. If we pick a

coordinate system with x being horizontal and yvertical, then the angles of these forces measuredcounterclockwise from the x axis are 90°-ϕ, 270°, and90°+θ, respectively. We have

Fx,total

= FTcos(90°-ϕ) + F

Wcos(270°) + F

Ncos(90°+θ)

Fy,total

= FTsin(90°-ϕ) + F

Wsin(270°) + F

Nsin(90°+θ) ,

which simplifies to

0 = FT sin ϕ – F

N sin θ

0 = FT cos ϕ – F

W + F

N cos θ .

The normal force is a quantity that we are not givenand do not with to find, so we should choose it toeliminate. Solving the first equation for F

N=(sin ϕ/sin

θ)FT, we eliminate F

N from the second equation,

0 = FT cos ϕ – F

W + F

T sin ϕ cos θ/sin θ

and solve for FT, finding

FT =FW

cos ϕ + sin ϕ cos θ / sin θ .

Multiplying both the top and the bottom of the fractionby sin θ, and using the trig identity for sin(θ+ϕ) givesthe desired result,

FT = sin θsin θ + ϕ

FW

(b) The case of ϕ=0, i.e. pulling straight up on thewagon, results in F

T=F

W: we simply support the

wagon and it glides up the slope like a chair-lift on aski slope. In the case of ϕ=180°-θ, F

T becomes

infinite. Physically this is because we are pullingdirectly into the ground, so no amount of force will

Solutions to Selected Problems

205

suffice.

11. (a) If there was no friction, the angle of reposewould be zero, so the coefficient of static friction, µs,will definitely matter. We also make up symbols θ, mand g for the angle of the slope, the mass of theobject, and the acceleration of gravity. The forcesform a triangle just like the one in section 8.3, butinstead of a force applied by an external object, wehave static friction, which is less than µsFN. As in thatexample, Fs=mg sin θ, and Fs<µsFN, so

mg sin θ<µsF

N

.

From the same triangle, we have FN=mg cos θ, so

mg sin θ < µsmg cos θ.

Rearranging,

θ < tan –1 µs .

(b) Both m and g canceled out, so the angle ofrepose would be the same on an asteroid.

Chapter 9

5. Each cyclist has a radial acceleration of v2/r=5 m/s2. The tangential accelerations of cyclists A and Bare 375 N/75 kg=5 m/s2.

A

B

C

scale:5 m/s2

6. (a) The inward normal force must be sufficient toproduce circular motion, so

FN

= mv 2 / r .

We are searching for the minimum speed, which isthe speed at which the static friction force is justbarely able to cancel out the downward gravitationalforce. The maximum force of static friction is

|Fs| = µsFN ,

and this cancels the gravitational force, so

|Fs| = mg .

Solving these three equations for v gives

v = grµs

.

Solutions to Selected Problems

(b) Greater by a factor of 3 .

7. The inward force must be supplied by the inwardcomponent of the normal force,

FN sin θ = mv 2 / r .

The upward component of the normal force mustcancel the downward force of gravity,

FN cos θ = mg .

Eliminating FN and solving for θ, we find

θ = tan– 1 v 2

gr .

Chapter 10

10. Newton’s law of gravity tells us that her weightwill be 6000 times smaller because of the asteroid’ssmaller mass, but 132=169 times greater because ofits smaller radius. Putting these two factors togethergives a reduction in weight by a factor of 6000/169,so her weight will be (400 N)(169)/(6000)=11 N.

11. Newton’s universal law of gravity says F=Gm1m

2/

r2, and Newton’s second law says F=m2a, so Gm

1m

2/

r2=m

2a. Since m

2 cancels, a is independent of m

2.

12. Newton’s second law gives

F = mD aD ,

where F is Ida’s force on Dactyl. Using Newton’suniversal law of gravity, F = Gm

Im

D/r 2,and the

equation a = v 2 / r for circular motion, we find

GmIm

D / r 2 = mDv 2 / r .

Dactyl’s mass cancels out, giving

GmI / r 2 = v 2 / r .

Dactyl’s velocity equals the circumference of its orbitdivided by the time for one orbit: v=2πr/T. Insertingthis in the above equation and solving for m

I, we find

mI

= 4π2r 3

GT 2 ,

so Ida’s density is

ρ = mI / V

= 4π2r 3

GVT 2 .

206

207

GlossaryAcceleration. The rate of change of velocity; the

slope of the tangent line on a v-t graph.

Attractive. Describes a force that tends to pull thetwo participating objects together. Cf. repulsive,oblique.

Center of mass. The balance point of an object.

Component. The part of a velocity, acceleration, orforce that is along one particular coordinate axis.

Displacement. (avoided in this book) A name forthe symbol ∆x .

Fluid. A gas or a liquid.

Fluid friction. A friction force in which at least oneof the object is is a fluid (i.e. either a gas or aliquid).

Gravity. A general term for the phenomenon ofattraction between things having mass. Theattraction between our planet and a human-sized object causes the object to fall.

Inertial frame. A frame of reference that is notaccelerating, one in which Newton’s first law istrue

Kinetic friction. A friction force between surfacesthat are slipping past each other.

Light. Anything that can travel from one place toanother through empty space and can influencematter, but is not affected by gravity.

Magnitude. The “amount” associated with a vector;the vector stripped of any information about itsdirection.

Mass. A numerical measure of how difficult it is tochange an object’s motion.

Matter. Anything that is affected by gravity.

Mks system. The use of metric units based on themeter, kilogram, and second. Example: metersper second is the mks unit of speed, not cm/s orkm/hr.

Noninertial frame. An accelerating frame ofreference, in which Newton’s first law is violated

Nonuniform circular motion. Circular motion inwhich the magnitude of the velocity vector

changes

Normal force. The force that keeps two objects fromoccupying the same space.

Oblique. Describes a force that acts at some other angle,one that is not a direct repulsion or attraction. Cf.attractive, repulsive.

Operational definition. A definition that states whatoperations should be carried out to measure the thingbeing defined.

Parabola. The mathematical curve whose graph has yproportional to x2.

Radial. Parallel to the radius of a circle; the in-outdirection. Cf. tangential.

Repulsive. Describes a force that tends to push the twoparticipating objects apart. Cf. attractive, oblique.

Scalar. A quantity that has no direction in space, only anamount. Cf. vector.

Significant figures. Digits that contribute to the accuracyof a measurement.

Speed. (avoided in this book) The absolute value of or, inmore then one dimension, the magnitude of thevelocity, i.e. the velocity stripped of any informationabout its direction

Spring constant. The constant of proportionality betweenforce and elongation of a spring or other object understrain.

Static friction. A friction force between surfaces that arenot slipping past each other.

Système International.. Fancy name for the metricsystem.

Tangential. Tangent to a curve. In circular motion, usedto mean tangent to the circle, perpendicular to theradial direction Cf. radial.

Uniform circular motion. Circular motion in which themagnitude of the velocity vector remains constant

Vector. A quantity that has both an amount (magnitude)and a direction in space. Cf. scalar.

Velocity. The rate of change of position; the slope of thetangent line on an x-t graph.

Weight. The force of gravity on an object, equal to mg.

208

209

Mathematical ReviewAlgebraQuadratic equation:

The solutions of ax 2 + bx + c = 0

are x = – b ± b2 – 4ac

2a .

Logarithms and exponentials:

ln (ab) = ln a + ln b

ea + b = eaeb

ln ex = eln x = x

ln ab = b ln a

Geometry, area, and volumearea of a triangle of base b and height h = 1

2bh

circumference of a circle of radius r = 2πrarea of a circle of radius r = πr 2

surface area of a sphere of radius r = 4πr 2

volume of a sphere of radius r = 43πr 3

Trigonometry with a right triangle

θ

h = hypotenuseo = opposite side

a = adjacent side

Definitions of the sine, cosine, and tangent:

sin θ = oh

cos θ = ah

tan θ = oa

Pythagorean theorem: h2 =a2 + o2

Trigonometry with any triangle

A

B

Cα

β

γ

Law of Sines:

sin αA

=sin β

B=

sin γC

Law of Cosines:

C2 = A2 + B 2 – 2AB cos γ

Properties of the derivative and integral(for students in calculus-based courses)Let f and g be functions of x, and let c be a constant.Linearity of the derivative:

ddx

c f = c dfdx

ddx

f + g = dfdx

+dgdx

The chain rule:

ddx

f(g(x)) =f ′(g(x))g ′(x)

Derivatives of products and quotients:

ddx

fg = dfdx

g +dgdx

f

d

dxfg =

f ′g –

fg ′g2

Some derivatives:

ddx

x m = mx m – 1 (except for m=0)

ddx

sin x = cos x

ddx

cos x = –sin x

ddx

ex = ex

ddx

ln x = 1x

The fundamental theorem of calculus:

dfdx

dx = f

Linearity of the integral:

cf(x)dx = c f(x)dx

f (x) + g(x) dx = f(x)dx + g(x)dx

Integration by parts:

f dg = fg – g df

210

Trig Tables θ sin θ cos θ tan θ θ sin θ cos θ tan θ θ sin θ cos θ tan θ

0° 0.000 1.000 0.000 30° 0.500 0.866 0.577 60° 0.866 0.500 1.732

1 0.017 1.000 0.017 31 0.515 0.857 0.601 61 0.875 0.485 1.804

2 0.035 0.999 0.035 32 0.530 0.848 0.625 62 0.883 0.469 1.881

3 0.052 0.999 0.052 33 0.545 0.839 0.649 63 0.891 0.454 1.963

4 0.070 0.998 0.070 34 0.559 0.829 0.675 64 0.899 0.438 2.050

5 0.087 0.996 0.087 35 0.574 0.819 0.700 65 0.906 0.423 2.145

6 0.105 0.995 0.105 36 0.588 0.809 0.727 66 0.914 0.407 2.246

7 0.122 0.993 0.123 37 0.602 0.799 0.754 67 0.921 0.391 2.356

8 0.139 0.990 0.141 38 0.616 0.788 0.781 68 0.927 0.375 2.475

9 0.156 0.988 0.158 39 0.629 0.777 0.810 69 0.934 0.358 2.605

10 0.174 0.985 0.176 40 0.643 0.766 0.839 70 0.940 0.342 2.747

11 0.191 0.982 0.194 41 0.656 0.755 0.869 71 0.946 0.326 2.904

12 0.208 0.978 0.213 42 0.669 0.743 0.900 72 0.951 0.309 3.078

13 0.225 0.974 0.231 43 0.682 0.731 0.933 73 0.956 0.292 3.271

14 0.242 0.970 0.249 44 0.695 0.719 0.966 74 0.961 0.276 3.487

15 0.259 0.966 0.268 45 0.707 0.707 1.000 75 0.966 0.259 3.732

16 0.276 0.961 0.287 46 0.719 0.695 1.036 76 0.970 0.242 4.011

17 0.292 0.956 0.306 47 0.731 0.682 1.072 77 0.974 0.225 4.331

18 0.309 0.951 0.325 48 0.743 0.669 1.111 78 0.978 0.208 4.705

19 0.326 0.946 0.344 49 0.755 0.656 1.150 79 0.982 0.191 5.145

20 0.342 0.940 0.364 50 0.766 0.643 1.192 80 0.985 0.174 5.671

21 0.358 0.934 0.384 51 0.777 0.629 1.235 81 0.988 0.156 6.314

22 0.375 0.927 0.404 52 0.788 0.616 1.280 82 0.990 0.139 7.115

23 0.391 0.921 0.424 53 0.799 0.602 1.327 83 0.993 0.122 8.144

24 0.407 0.914 0.445 54 0.809 0.588 1.376 84 0.995 0.105 9.514

25 0.423 0.906 0.466 55 0.819 0.574 1.428 85 0.996 0.087 11.430

26 0.438 0.899 0.488 56 0.829 0.559 1.483 86 0.998 0.070 14.301

27 0.454 0.891 0.510 57 0.839 0.545 1.540 87 0.999 0.052 19.081

28 0.469 0.883 0.532 58 0.848 0.530 1.600 88 0.999 0.035 28.636

29 0.485 0.875 0.554 59 0.857 0.515 1.664 89 1.000 0.017 57.290

90 1.000 0.000 ∞

211Index

IndexDialogues Concerning the Two New Sciences 37dynamics 51

E

elephant 46energy

distinguished from force 104

F

falling objects 71Feynman 73Feynman, Richard 73force

analysis of forces 122Aristotelian versus Newtonian 95as a vector 156attractive 117contact 97distinguished from energy 104frictional 119gravitational 118net 98noncontact 97normal 118oblique 117positive and negative signs of 97repulsive 117transmission of 124

forcesclassification of 116

frame of referencedefined 57inertial or noninertial 107

French Revolution 23friction

fluid 121kinetic 119, 120static 119, 120

G

Galilei, Galileo. See Galileo GalileiGalileo Galilei 37gamma rays 18grand jete 54graphing 59graphs

of position versus time 58velocity versus time 67

A

acceleration 74as a vector 153constant 85definition 80negative 77

alchemy 17area

operational definition 35scaling of 37

area under a curve 83area under a-t graph 84under v-t graph 83

astrology 17

B

Bacon, Sir Francis 20

C

calculusdifferential 68fundamental theorem of 89integral 89invention by Newton 67Leibnitz notation 68with vectors 157

cathode rays 18center of mass 53

motion of 53center-of-mass motion 53centi- (metric prefix) 23Challenger disaster 87circular motion 163

nonuniform 165uniform 165

co*ckroaches 44coefficient of kinetic friction 120coefficient of static friction 120component

defined 134conversions of units 28coordinate system

defined 57Copernicus 62

D

Darwin 19delta notation 55derivative 68

second 89

212 Index

H

high jump 54Hooke’s law 126

I

inertiaprinciple of 62

integral 89

K

kilo- (metric prefix) 23kilogram 25kinematics 51

L

Laplace 17Leibnitz 68light 18

M

magnitude of a vectordefined 142

matter 18mega- (metric prefix) 23meter (metric unit) 24metric prefixes. See metric system: prefixesmetric system 22

prefixes 23micro- (metric prefix) 23microwaves 18milli- (metric prefix) 23model

scientific 119models 54motion

rigid-body 52types of 52

Muybridge, Eadweard 151

N

nano- (metric prefix) 23Newton

first law of motion 98second law of motion 102

Newton, Isaac 22definition of time 25

Newton's laws of motionin three dimensions 136

Newton's third law 112normal force 118

O

operational definitions 24

order-of-magnitude estimates 47

P

parabolamotion of projectile on 135

Pauli exclusion principle 19period

of uniform circular motion 169photon 115physics 17POFOSTITO 114Pope 37prefixes, metric. See metric system: prefixesprojectiles 135pulley 127

R

radial componentdefined 171

radio waves 18reductionism 20Renaissance 15rotation 52

S

salamanders 44sans culottides 25scalar

defined 142scaling 37

applied to biology 44scientific method 15second (unit) 24significant figures 30simple machine

defined 127slam dunk 54Stanford, Leland 151strain 126Swift, Jonathan 37

T

timeduration 55point in 55

transmission of forces 124

U

unit vectors 148units, conversion of 28

V

vectordefined 142

213Index

vector additiondefined 142

vectors 51velocity

addition of velocities 65as a vector 152definition 59negative 66

vertebra 46volume

operational definition 35scaling of 37

W

weight forcedefined 97

weightlessnessbiological effects 87

X

x-rays 18

214 Index

215© 1998 Benjamin Crowell

Photo CreditsAll photographs are by Benjamin Crowell, except as noted below.

CoverMoon: Loewy and Puiseux, 1894.

Chapter 1Mars Climate Orbiter: NASA/JPL/Caltech. Red blood cell: C. Magowan et al.

Chapter 2High jumper: Dunia Young. Rocket sled: U.S. Air Force.

Chapter 3X-33 art: NASA. Astronauts aboard Mir: NASA. Construction of the International Space Station: NASA.Gravity map: Data from US Navy Geosat and European Space Agency ERS-1 satellites, analyzed by DavidSandwell and Walter Smith.

Chapter 4Isaac Newton: Painting by Sir Godfrey Kneller, National Portrait Gallery, London.

Chapter 5Space shuttle launch: NASA.

Chapter 6The Ring Toss: Clarence White, ca. 1903.

Chapter 7Aerial photo of Mondavi vineyards: NASA.

Chapter 8Galloping horse: Eadweard Muybridge, 1878.

216