2Simple NatureAn Introduction to Physics for Engineeringand Physical Science StudentsBenjamin Crowellwww.lightandmatter.comFullerton, Californiawww.lightandmatter.comCopyright c (2001-2008 Benjamin Crowellrev. May 14, 2008Permission is granted to copy,distribute and/or modify this docu-ment under the terms of the Creative Commons Attribution Share-AlikeLicense, whichcanbefoundatcreativecommons.org. Thelicenseappliestotheentiretextofthisbook,plusalltheillustra-tions that are by Benjamin Crowell. (At your option, you may alsocopythisbookundertheGNUFreeDocumentationLicensever-sion1.2, withnoinvariantsections, nofront-covertexts, andnoback-cover texts.)All the illustrations are by Benjamin Crowell ex-cept as noted in the photo credits or in parentheses in the captionof thegure. Thisbookcanbedownloadedfreeof chargefromwww.lightandmatter.com in a variety of formats, including editableformats.Brief Contents0 Introduction 131 Conservation of Mass 532 Conservation of Energy 733 Conservation of Momentum 1254 Conservation of Angular Momentum 2255 Thermodynamics 2776 Waves 3177 Relativity 3598 Atoms and Electromagnetism 4039 DC Circuits 45910 Fields 50311 Electromagnetism 58912 Optics 67513 Quantum Physics 75756Contents0Introduction and Review0.1Introduction and Review. . . . . . . . . . . . . . 13Thescienticmethod, 13.Whatisphysics?, 16.Howtolearnphysics, 19.Velocity and acceleration, 21.Self-evaluation, 23.Basics of the metric system, 24.The newton, the metric unit offorce, 27.Less common metric prexes, 28.Scientic notation,28.Conversions, 29.Signicant gures, 31.0.2Scaling and Order-of-Magnitude Estimates . . . . . . 34Introduction, 34.Scaling of area and volume, 35.Order-of-magnitudeestimates, 43.Problems . . . . . . . . . . . . . . . . . . . . . . 461Conservation of Mass1.1Mass . . . . . . . . . . . . . . . . . . . . . . 54Problem-solving techniques, 56.Delta notation, 57.1.2Equivalence of Gravitational and Inertial Mass . . . . . 581.3Galilean Relativity . . . . . . . . . . . . . . . . . 60Applications of calculus, 65.1.4A Preview of Some Modern Physics . . . . . . . . . 66Problems . . . . . . . . . . . . . . . . . . . . . . 692Conservation of Energy2.1Energy . . . . . . . . . . . . . . . . . . . . . 73The energy concept, 73.Logical issues, 75.Kinetic energy, 76.Power, 80.Gravitational energy, 81.Equilibrium and stability,86.Predicting the direction of motion, 88.2.2Numerical Techniques. . . . . . . . . . . . . . . 892.3Gravitational Phenomena. . . . . . . . . . . . . . 94Keplerslaws, 94.Circularorbits, 96.Thesunsgravitationaleld, 97.Gravitational energy in general, 98.The shell theorem,101.Evidence for repulsive gravity, 106.2.4Atomic Phenomena. . . . . . . . . . . . . . . . 107Heat is kinetic energy., 108.All energy comes from particles mov-ing or interacting., 110.2.5Oscillations . . . . . . . . . . . . . . . . . . . 111Problems . . . . . . . . . . . . . . . . . . . . . . 117Exercises . . . . . . . . . . . . . . . . . . . . . . 1243Conservation of Momentum3.1Momentum in One Dimension . . . . . . . . . . . . 126Mechanical momentum, 126.Nonmechanical momentum, 129.Momentumcomparedtokineticenergy, 130.Collisionsinonedimension, 131.Thecenterof mass, 136.Thecenterof massframe of reference, 140.3.2Force in One Dimension . . . . . . . . . . . . . . 141Momentumtransfer, 141.Newtons laws, 143.What force isnot, 145.Forces between solids, 147.Fluid friction, 150.Analysisof forces, 151.Transmission of forces by low-mass objects, 153.Work, 155.Simplemachines, 161.Forcerelatedtointeractionenergy, 162.3.3Resonance. . . . . . . . . . . . . . . . . . . . 164Damped, free motion, 166.The quality factor, 168.Driven motion,169.3.4Motion in Three Dimensions . . . . . . . . . . . . 176The Cartesian perspective, 176.Rotational invariance, 180.Vectors,181.Calculus with vectors, 195.The dot product, 199.Gradientsand line integrals (optional), 202.Problems . . . . . . . . . . . . . . . . . . . . . . 205Exercises . . . . . . . . . . . . . . . . . . . . . . 2184Conservation of Angular Momentum4.1Angular Momentum in Two Dimensions . . . . . . . . 225Angular momentum, 225.Application to planetary motion, 230.Twotheorems about angular momentum, 231.Torque, 234.Applications to statics, 238.Proof of Keplers elliptical orbit law,242.4.2Rigid-Body Rotation . . . . . . . . . . . . . . . . 245Kinematics, 245.Relationsbetweenangularquantitiesandmo-tionofapoint, 246.Dynamics, 248.Iteratedintegrals, 250.Finding moments of inertia by integration, 253.4.3Angular Momentum in Three Dimensions. . . . . . . 258Rigid-bodykinematics inthree dimensions, 258.Angular mo-mentum in three dimensions, 260.Rigid-body dynamics in threedimensions, 265.Problems . . . . . . . . . . . . . . . . . . . . . . 269Exercises . . . . . . . . . . . . . . . . . . . . . . 2755Thermodynamics5.1Pressure and Temperature. . . . . . . . . . . . . 278Pressure, 278.Temperature, 282.5.2Microscopic Description of an Ideal Gas . . . . . . . 285Evidence for the kinetic theory, 285.Pressure, volume, and temperature,286.5.3Entropy as a Macroscopic Quantity. . . . . . . . . . 288Eciency and grades of energy, 288.Heat engines, 289.Entropy,291.5.4Entropy as a Microscopic Quantity . . . . . . . . . . 295A microscopic view of entropy, 295.Phase space, 296.Microscopicdenitions of entropy and temperature, 297.The arrow of time,or this way to the Big bang, 305.Quantum mechanics and zeroentropy, 306.Summary of the laws of thermodynamics, 306.8 Contents5.5More about Heat Engines. . . . . . . . . . . . . . 307Problems . . . . . . . . . . . . . . . . . . . . . . 3146Waves6.1Free Waves . . . . . . . . . . . . . . . . . . . 318Wave motion, 318.Waves onastring, 324.Soundandlightwaves, 327.Periodic waves, 329.The doppler eect, 332.6.2Bounded Waves . . . . . . . . . . . . . . . . . 338Reection, transmission, and absorption, 338.Quantitative treat-ment of reection, 343.Interference eects, 346.Waves boundedon both sides, 348.Problems . . . . . . . . . . . . . . . . . . . . . . 3557Relativity7.1Basic Relativity. . . . . . . . . . . . . . . . . . 360Theprincipleof relativity, 360.Distortionof timeandspace,361.Applications, 366.7.2The Lorentz transformation. . . . . . . . . . . . . 371Coordinate transformations ingeneral, 371.Derivationof thelorentz transformation, 372.Spacetime, 377.7.3Dynamics . . . . . . . . . . . . . . . . . . . . 385Invariants, 385.Combination of velocities, 385.Momentum andforce, 386.Kinetic energy, 388.Equivalence of mass and energy,390.Problems . . . . . . . . . . . . . . . . . . . . . . 395Exercises . . . . . . . . . . . . . . . . . . . . . . 4008Atoms and Electromagnetism8.1The Electric Glue. . . . . . . . . . . . . . . . . 403The quest for the atomic force, 404.Charge, electricity and magnetism,405.Atoms, 410.Quantizationof charge, 415.Theelectron,418.The raisin cookie model of the atom, 422.8.2The Nucleus. . . . . . . . . . . . . . . . . . . 425Radioactivity, 425.The planetarymodel of the atom, 428.Atomicnumber,432.Thestructureofnuclei,437.Thestrongnuclear force, alphadecayandssion, 440.Theweaknuclearforce; beta decay, 443.Fusion, 446.Nuclear energy and bindingenergies, 448.Biologicaleectsofionizingradiation, 449.Thecreation of the elements, 452.Problems . . . . . . . . . . . . . . . . . . . . . . 4559Circuits9.1Current and Voltage. . . . . . . . . . . . . . . . 460Current, 460.Circuits, 463.Voltage, 464.Resistance, 469.Current-conducting properties of materials, 476.9.2Parallel and Series Circuits. . . . . . . . . . . . . 480Schematics, 480.Parallel resistances and the junction rule, 481.Series resistances, 485.Problems . . . . . . . . . . . . . . . . . . . . . . 492Contents 9Exercises . . . . . . . . . . . . . . . . . . . . . . 49910Fields10.1Fields of Force. . . . . . . . . . . . . . . . . . 503Why elds?, 504.The gravitational eld, 505.The electric eld,509.10.2Voltage Related to Field . . . . . . . . . . . . . . 513One dimension, 513.Two or three dimensions, 516.10.3Fields by Superposition . . . . . . . . . . . . . . 518Electriceldofacontinuouschargedistribution,518.Theeldnear a charged surface, 524.10.4Energy in Fields. . . . . . . . . . . . . . . . . 526Electric eld energy, 526.Gravitational eld energy, 531.Magneticeld energy, 531.10.5LRC Circuits . . . . . . . . . . . . . . . . . . 533Capacitance and inductance, 533.Oscillations, 537.Voltage andcurrent, 539.Decay, 544.Reviewof complexnumbers, 547.Impedance, 550.Power, 553.Impedance matching, 556.Impedancesin series and parallel, 558.10.6Fields by Gauss Law . . . . . . . . . . . . . . . 560Gauss law, 560.Additivity of ux, 564.Zero ux from outsidecharges, 564.Proof of Gauss theorem, 567.Gauss lawas afundamental law of physics, 567.Applications, 568.10.7Gauss Law in Differential Form. . . . . . . . . . . 571Problems . . . . . . . . . . . . . . . . . . . . . . 576Exercises . . . . . . . . . . . . . . . . . . . . . . 58611Electromagnetism11.1More About the Magnetic Field . . . . . . . . . . . 589Magnetic forces, 589.The magnetic eld, 593.Some applications,597.No magnetic monopoles, 599.Symmetry and handedness,601.11.2Magnetic Fields by Superposition . . . . . . . . . . 603Superposition of straight wires, 603.Energy in the magnetic eld,607.Superposition of dipoles, 607.The biot-savart law (optional),611.11.3Magnetic Fields by Amp` eres Law. . . . . . . . . . 615Amp`eres law, 615.Aquickanddirtyproof, 617.Maxwellsequations for static elds, 618.11.4Amp` eres Law in Differential Form (optional) . . . . . 620The curl operator, 620.Properties of the curl operator, 621.11.5Induced Electric Fields . . . . . . . . . . . . . . 626Faradays experiment, 626.Why induction?, 630.Faradays law,632.11.6Maxwells Equations . . . . . . . . . . . . . . . 637Induced magnetic elds, 637.Light waves, 639.11.7Electromagnetic Properties of Materials . . . . . . . 649Conductors, 649.Dielectrics, 650.Magnetic materials, 652.Problems . . . . . . . . . . . . . . . . . . . . . . 65910 ContentsExercises . . . . . . . . . . . . . . . . . . . . . . 67312Optics12.1The Ray Model of Light . . . . . . . . . . . . . . 675The nature of light, 676.Interaction of light with matter, 679.The raymodel of light, 681.Geometryof specular reection,684.The principle of least time for reection, 688.12.2Images by Reection. . . . . . . . . . . . . . . 690A virtual image, 690.Curved mirrors, 693.A real image, 694.Images of images, 695.12.3Images, Quantitatively . . . . . . . . . . . . . . 699Areal imageformedbyaconvergingmirror, 699.Othercaseswith curved mirrors, 702.Aberrations, 707.12.4Refraction. . . . . . . . . . . . . . . . . . . . 710Refraction, 711.Lenses, 717.The lensmakers equation, 719.The principle of least time for refraction, 720.12.5Wave Optics. . . . . . . . . . . . . . . . . . . 721Diraction, 721.Scaling of diraction, 723.The correspondenceprinciple, 723.Huygensprinciple, 724.Double-slitdiraction,725.Repetition, 729.Single-slit diraction, 730.The principleof least time, 732.Problems . . . . . . . . . . . . . . . . . . . . . . 734Exercises . . . . . . . . . . . . . . . . . . . . . . 74713Quantum Physics13.1Rules of Randomness . . . . . . . . . . . . . . 757Randomness isnt random., 758.Calculating randomness, 759.Probabilitydistributions, 763.Exponential decayandhalf-life,765.Applications of calculus, 770.13.2Light as a Particle . . . . . . . . . . . . . . . . 772Evidence for light as a particle, 773.How much light is one photon?,775.Wave-particleduality, 779.Photonsinthreedimensions,784.13.3Matter as a Wave . . . . . . . . . . . . . . . . 785Electronsaswaves, 786.Dispersivewaves, 790.Boundstates,793.The uncertainty principle and measurement, 795.Electronsin electric elds, 801.The Schrodinger equation, 802.13.4The Atom. . . . . . . . . . . . . . . . . . . . 808Classifying states, 808.Angular momentum in three dimensions,810.Thehydrogenatom, 811.Energiesofstatesinhydrogen,814.Electronspin, 820.Atomswithmorethanoneelectron,822.Problems . . . . . . . . . . . . . . . . . . . . . . 825Exercises . . . . . . . . . . . . . . . . . . . . . . 834Appendix 1: Programming with Python 835Appendix 2: Miscellany 837Appendix 3: Photo Credits 844Appendix 4: Hints and Solutions 846Contents 11Appendix 5: Useful Data 867Notationandterminology, comparedwithother books, 867.Notationandunits, 868.Fundamental constants, 868.Metricprexes, 869.Nonmetric units, 869.The Greek alphabet, 869.Subatomicparticles,869.Earth,moon,andsun,870.Thepe-riodic table, 870.Atomic masses, 870.Appendix 6: Summary 87112 ContentsThe Mars Climate Orbiter is pre-paredfor itsmission. Thelawsof physicsarethesameevery-where, even on Mars, so theprobecouldbedesignedbasedon the laws of physics as discov-eredonearth. Thereisunfor-tunately another reason why thisspacecraftisrelevanttothetop-ics of this chapter: it was de-stroyed attempting to enter Marsatmosphere because engineersat Lockheed Martin forgot to con-vertdataonenginethrustsfrompounds into the metric unit offorce (newtons) before giving theinformation to NASA. Conver-sions are important!Chapter 0Introduction and Review0.1 Introduction and ReviewIf you drop your shoe and a coin side by side, they hit the ground atthe same time. Why doesnt the shoe get there rst, since gravity ispulling harder on it?How does the lens of your eye work, and whydo your eyes muscles need to squash its lens into dierent shapes inorder to focus on objects nearby or far away? These are the kindsof questions that physics tries to answer about the behavior of lightand matter, the two things that the universe is made of.0.1.1 The scientic methodUntil very recently in history, no progress was made in answeringquestionslikethese. Worsethanthat, thewronganswerswrittenby thinkers like the ancient Greek physicist Aristotle were acceptedwithoutquestionforthousandsofyears. WhyisitthatscienticknowledgehasprogressedmoresincetheRenaissancethanithadinall theprecedingmillenniasincethebeginningofrecordedhis-tory? Undoubtedly the industrial revolution is part of the answer.13b / A satirical drawing of analchemistslaboratory. H. Cock,after a drawing by Peter Brueghelthe Elder (16th century).a / Science is a cycle of the-ory and experiment.Building its centerpiece, the steam engine, required improved tech-niques for precise construction and measurement. (Early on, it wasconsidered a major advance when English machine shops learned tobuildpistonsandcylindersthatttogetherwithagapnarrowerthan the thickness of a penny.) But even before the industrial rev-olution, the pace of discovery had picked up, mainly because of theintroduction of the modern scientic method. Although it evolvedover time, most scientists today would agree on something like thefollowing list of the basic principles of the scientic method:(1)Scienceisacycleoftheoryandexperiment. Scienticthe-oriesarecreatedtoexplaintheresultsof experimentsthat werecreated under certain conditions. A successful theory will also makenew predictions about new experiments under new conditions. Even-tually, though, italwaysseemstohappenthatanewexperimentcomesalong, showingthatundercertainconditionsthetheoryisnotagoodapproximationorisnotvalidatall. Theball isthenbackinthetheorists court. If anexperimentdisagreeswiththecurrent theory, the theory has to be changed, not the experiment.(2) Theories should both predict and explain.The requirement ofpredictive power means that a theory is only meaningful if it predictssomething that can be checked against experimental measurementsthatthetheoristdidnotalreadyhaveathand. Thatis, atheoryshould be testable. Explanatory value means that many phenomenashouldbeaccountedforwithfewbasicprinciples. If youanswerevery why question with because thats the way it is, then yourtheoryhasnoexplanatoryvalue. Collectinglotsof datawithoutbeing able to nd any basic underlying principles is not science.(3)Experimentsshouldbereproducible. Anexperimentshouldbetreatedwithsuspicionif itonlyworksforoneperson, oronlyinonepart of theworld. Anyonewiththenecessaryskills andequipmentshouldbeabletogetthesameresultsfromthesameexperiment. This implies that science transcends national and eth-nic boundaries; you can be sure that nobody is doing actual sciencewhoclaimsthattheirworkisAryan, notJewish,Marxist, notbourgeois, or Christian, not atheistic.An experiment cannot bereproduced if it is secret, so science is necessarily a public enterprise.As an example of the cycle of theory and experiment, a vital steptoward modern chemistry was the experimental observation that thechemical elementscouldnotbetransformedintoeachother, e.g.,leadcouldnotbeturnedintogold. Thisledtothetheorythatchemical reactionsconsistedof rearrangementsof theelementsindierent combinations,without any change in the identities of theelements themselves. The theory worked for hundreds of years, andwasconrmedexperimentallyoverawiderangeof pressuresandtemperaturesandwithmanycombinationsof elements. Onlyin14 Chapter 0 Introduction and Reviewthe twentieth century did we learn that one element could be trans-formedintooneanother under theconditions of extremelyhighpressure and temperature existing in a nuclear bomb or inside a star.That observation didnt completely invalidate the original theory ofthe immutability of the elements, but it showed that it was only anapproximation, valid at ordinary temperatures and pressures.self-check AA psychic conducts seances in which the spirits of the dead speak tothe participants. He says he has special psychic powers not possessedby other people, which allow him to channel the communications withthe spirits. What part of the scientic method is being violated here?> Answer, p. 849Thescienticmethodasdescribedhereisanidealization, andshould not be understood as a set procedure for doing science. Sci-entistshaveasmanyweaknessesandcharacterawsasanyothergroup, and it is very common for scientists to try to discredit otherpeoples experiments when the results run contrary to their own fa-voredpointof view. Successful sciencealsohasmoretodowithluck, intuition, andcreativitythanmost peoplerealize, andtherestrictionsofthescienticmethoddonotstieindividualityandself-expressionanymorethanthefugueandsonataformsstiedBachandHaydn. Thereisarecenttendencyamongsocial scien-tists to go even further and to deny that the scientic method evenexists, claiming that science is no more than an arbitrary social sys-temthatdetermineswhatideastoacceptbasedonanin-groupscriteria. I think thats going too far. If science is an arbitrary socialritual, it would seem dicult to explain its eectiveness in buildingsuchusefulitemsasairplanes, CDplayersandsewers. Ifalchemyandastrologywerenolessscienticintheirmethodsthanchem-istryandastronomy, whatwasitthatkeptthemfromproducinganything useful?Discussion QuestionsConsider whether or not the scientic method is being applied in the fol-lowingexamples. Ifthescienticmethodisnotbeingapplied, arethepeoplewhoseactionsarebeingdescribedperformingauseful humanactivity, albeit an unscientic one?A Acupunctureisatraditional medical techniqueofAsianorigininwhichsmall needlesareinsertedinthepatientsbodytorelievepain.Many doctors trained in the west consider acupuncture unworthy of ex-perimental study because if it had therapeutic effects, such effects couldnot be explained by their theories of the nervous system. Who is beingmore scientic, the western or eastern practitioners?B Goethe, a German poet, is less well known for his theory of color.Hepublishedabookonthesubject, inwhichhearguedthatscienticapparatusformeasuringandquantifyingcolor, suchasprisms, lensesand colored lters, could not give us full insight into the ultimate meaningSection 0.1 Introduction and Review 15of color, forinstancethecoldfeelingevokedbyblueandgreenortheheroic sentiments inspired by red. Was his work scientic?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 withthe earth. Are these explanations scientic?D Buddhism is partly a psychological explanation of human suffering,andpsychologyisof courseascience. TheBuddhacouldbesaidtohave engaged in a cycle of theory and experiment, since he worked bytrial and error, and even late in his life he asked his followers to challengehisideas. Buddhismcouldalsobeconsideredreproducible, sincetheBuddhatoldhisfollowerstheycouldndenlightenmentforthemselvesif they followed a certain course of study and discipline. Is Buddhism ascientic pursuit?0.1.2 What is physics?Givenforoneinstantanintelligencewhichcouldcomprehendallthe forces by which nature is animated and the respective positionsof thethingswhichcomposeit...nothingwouldbeuncertain, andthe future as the past would be laid out before its eyes.Pierre Simon de LaplacePhysics is the use of the scientic method to nd out the basicprinciples governing light and matter,and to discover the implica-tions of those laws. Part of what distinguishes the modern outlookfrom the ancient mind-set is the assumption that there are rules bywhich the universe functions, and that those laws can be at least par-tially understood by humans. From the Age of Reason through thenineteenth century, many scientists began to be convinced that thelaws of nature not only could be known but, as claimed by Laplace,thoselawscouldinprinciplebeusedtopredicteverythingabouttheuniversesfutureif completeinformationwasavailableaboutthepresentstateof all lightandmatter. Insubsequentsections,Ill describe two general types of limitations on prediction using thelaws of physics, which were only recognized in the twentieth century.Mattercanbedenedasanythingthatisaectedbygravity,i.e., that has weight or would have weight if it was near the Earthoranotherstarorplanetmassiveenoughtoproducemeasurablegravity. Light can be dened as anything that can travel from oneplace to another through empty space and can inuence matter, buthasnoweight. Forexample, sunlightcaninuenceyourbodybyheatingitorbydamagingyourDNAandgivingyouskincancer.Thephysicistsdenitionoflightincludesavarietyofphenomenathatarenotvisibletotheeye, includingradiowaves, microwaves,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 rainbowthat we can see.self-check B16 Chapter 0 Introduction and Reviewc / Thistelescopepictureshowstwoimagesof thesamedistantobject, anexotic, veryluminousobject calledaquasar. Thisisinterpreted as evidence that amassive, dark object, possiblya black hole, happens to bebetween us and it. Light rays thatwould otherwise have missed theearthoneither sidehavebeenbent bythedarkobjectsgravityso that they reach us. The actualdirection to the quasar is presum-ablyinthecenter of theimage,but the light along that central linedoesnt get tous becauseit isabsorbed by the dark object.Thequasar is known by its catalognumber, MG1131+0456, or moreinformally as Einsteins Ring.At the turn of the 20th century, a strange new phenomenon was discov-ered in vacuum tubes: mysterious rays of unknown origin and nature.These rays are the same as the ones that shoot from the back of yourTVspicturetubeandhitthefronttomakethepicture. Physicistsin1895 didnt have the faintest idea what the rays were,so they simplynamedthemcathoderays, afterthenamefortheelectrical contactfrom which they sprang. A erce debate raged, complete with national-istic overtones, over whether the rays were a form of light or of matter.What would they have had to do in order to settle the issue? >Answer, p. 849Manyphysical phenomenaarenotthemselveslightormatter,butarepropertiesoflightormatterorinteractionsbetweenlightand matter. For instance, motion is a property of all light and somematter, but it is not itself light or matter. The pressure that keepsabicycletireblownupisaninteractionbetweentheairandthetire. Pressureis not aformof matter inandof itself. It is asmuchapropertyofthetireasoftheair. Analogously, sisterhoodand employment are relationships among people but are not peoplethemselves.Some things that appear weightless actually do have weight, andso qualify as matter. Air has weight, and is thus a form of mattereven though a cubic inch of air weighs less than a grain of sand. Ahelium balloon has weight, but is kept from falling by the force of thesurrounding more dense air,which pushes up on it. Astronauts inorbit around the Earth have weight, and are falling along a curvedarc, buttheyaremovingsofastthatthecurvedarcof theirfallisbroadenoughtocarrythemall thewayaroundtheEarthinacircle. Theyperceivethemselvesasbeingweightlessbecausetheirspace capsule is falling along with them, and the oor therefore doesnot push up on their feet.Optional Topic: ModernChangesintheDenitionofLightandMatterEinstein predicted as a consequence of his theory of relativity that lightwould after all be affected by gravity, although the effect would be ex-tremelyweakundernormal conditions. Hispredictionwasborneoutby observations of the bending of light rays from stars as they passedclose to the sun on their way to the Earth. Einsteins theory also impliedthe existence of black holes, stars so massive and compact that theirintense gravity would not even allow light to escape. (These days thereis strong evidence that black holes exist.)Einsteins interpretation was that light doesnt really have mass, butthat energy is affected by gravity just like mass is. The energy in a lightbeam is equivalent to a certain amount of mass, given by the famousequation E=mc2, where cis the speed of light. Because the speedof light is such a big number, a large amount of energy is equivalent toonly a very small amount of mass, so the gravitational force on a lightray can be ignored for most practical purposes.There is however a more satisfactory and fundamentaldistinctionSection 0.1 Introduction and Review 17d / Reductionism.between light and matter, which should be understandable to you if youhave had a chemistry course. In chemistry,one learns that electronsobey the Pauli exclusion principle, which forbids more than one electronfrom occupying the same orbital if they have the same spin. The Pauliexclusion principle is obeyed by the subatomic particles of which matteris composed, but disobeyed by the particles, called photons, of which abeam of light is made.Einsteins theory of relativity is discussed more fully in book 6 of thisseries.Theboundarybetweenphysics andtheother sciences is notalwaysclear. Forinstance, chemistsstudyatomsandmolecules,which are what matter is built from,and there are some scientistswhowouldbeequallywillingtocall themselvesphysical chemistsor chemical physicists. It might seem that the distinction betweenphysicsandbiologywouldbeclearer, sincephysicsseemstodealwithinanimateobjects. Infact, almostall physicistswouldagreethat the basic laws of physics that apply to molecules in a test tubework equally well for the combination of molecules that constitutesa bacterium. (Some might believe that something more happens inthe minds of humans, or even those of cats and dogs.)What dier-entiates physics from biology is that many of the scientic theoriesthat describe living things, while ultimately resulting from the fun-damental laws of physics, cannot be rigorously derived from physicalprinciples.IsolatedsystemsandreductionismTo avoid having to study everything at once, scientists isolate thethings they are trying to study. For instance, a physicist who wantsto study the motion of a rotating gyroscope would probably preferthat it be isolated from vibrations and air currents. Even in biology,where eld work is indispensable for understanding how living thingsrelate to their entire environment, it is interesting to note the vitalhistorical roleplayedbyDarwinsstudyoftheGalapagosIslands,whichwereconvenientlyisolatedfromtherestoftheworld. Anypartoftheuniversethatisconsideredapartfromtherestcanbecalled a system.Physicshashadsomeofitsgreatestsuccessesbycarryingthisprocess of isolation to extremes, subdividing the universe into smallerandsmallerparts. Mattercanbedividedintoatoms, andthebe-havior of individual atoms can be studied. Atoms can be split apartinto their constituent neutrons, protons and electrons. Protons andneutrons appear tobemadeout of evensmaller particles calledquarks,andtherehaveevenbeensomeclaimsofexperimentalev-idencethatquarkshavesmallerpartsinsidethem. Thismethodof splitting things into smaller and smaller parts and studying howthose parts inuence each other is called reductionism. The hope isthat the seemingly complex rules governing the larger units can bebetterunderstoodintermsof simplerrulesgoverningthesmaller18 Chapter 0 Introduction and Reviewunits. Toappreciatewhatreductionismhasdoneforscience, itisonlynecessarytoexaminea19th-centurychemistrytextbook. Atthattime, theexistenceofatomswasstill doubtedbysome, elec-trons werenot evensuspectedtoexist, andalmost nothingwasunderstood of what basic rules governed the way atoms interactedwitheachotherinchemical reactions. Studentshadtomemorizelong lists of chemicals and their reactions, and there was no way tounderstand any of it systematically. Today, the student only needsto remember a small set of rules about how atoms interact,for in-stance that atoms of one element cannot be converted into anothervia chemical reactions, or that atoms from the right side of the pe-riodictabletendtoformstrongbondswithatomsfromtheleftside.Discussion QuestionsA Ive suggested replacing the ordinary dictionary denition of lightwith a more technical, more precise one that involves weightlessness. Itsstill possible, though, that thestuff alightbulbmakes, ordinarilycalledlight, does have some small amount of weight. Suggest an experimentto attempt to measure whether it does.B Heat is weightless (i.e., an object becomes no heavier when heated),andcantravel acrossanemptyroomfromthereplacetoyour skin,whereitinuencesyoubyheatingyou. Shouldheatthereforebecon-sidered a form of light by our denition? Why or why not?C Similarly, should sound be considered a form of light?0.1.3 How to learn physicsFor as knowledges are now delivered, there is a kind of contract oferror between the deliverer and the receiver;for he that deliverethknowledge desireth to deliver it in such a form as may be best be-lieved, andnotasmaybebestexamined; andhethatreceivethknowledgedesirethrather present satisfactionthanexpectant in-quiry.Francis BaconMany students approach a science course with the idea that theycansucceedbymemorizingtheformulas, sothatwhenaproblemis assigned on the homework or an exam, they will be able to plugnumbers in to the formula and get a numerical result on their cal-culator. Wrong! Thats not what learning science is about! Thereis a big dierence between memorizing formulas and understandingconcepts. Tostartwith, dierentformulasmayapplyindierentsituations. Oneequationmightrepresentadenition,whichisal-ways true. Another might be a very specic equation for the speedof 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 you dont work to understand physics on a conceptual level, youwont know which formulas can be used when.Section 0.1 Introduction and Review 19Moststudentstakingcollegesciencecoursesforthersttimealso have very little experience with interpreting the meaning of anequation. Considertheequationn=/relatingthewidthofarectangle to its height and area. A student who has not developedskill atinterpretationmightviewthisasyetanotherequationtomemorize and plug in to when needed. A slightly more savvy stu-dentmightrealizethatitissimplythefamiliarformula=n/inadierentform. Whenaskedwhetherarectanglewouldhaveagreaterorsmallerwidththananotherwiththesameareabutasmallerheight, theunsophisticatedstudentmightbeataloss,nothavinganynumberstopluginonacalculator. Themoreex-periencedstudent wouldknowhowtoreasonabout anequationinvolving division if/ is smaller, and stays the same, thennmust be bigger. Often, students fail to recognize a sequence of equa-tions as a derivation leading to a nal result, so they think all theintermediate steps are equally important formulas that they shouldmemorize.When learning any subject at all, it is important to become asactivelyinvolvedaspossible, ratherthantryingtoreadthroughall the information quickly without thinking about it. It is a goodidea to read and think about the questions posed at the end of eachsection of these notes as you encounter them, so that you know youhave understood what you were reading.Many students diculties in physics boil down mainly to di-culties with math. Suppose you feel condent that you have enoughmathematical preparationtosucceedinthis course, but youarehaving trouble with a few specic things. In some areas,the briefreviewgiveninthischaptermaybesucient, butinotherareasit probably will not. Once you identify the areas of math in whichyou are having problems, get help in those areas. Dont limp alongthrough the whole course with a vague feeling of dread about some-thing like scientic notation. The problem will not go away if youignore it. The same applies to essential mathematical skills that youare learning in this course for the rst time, such as vector addition.Sometimesstudentstell metheykeeptryingtounderstandacertain topic in the book, and it just doesnt make sense. The worstthingyoucanpossiblydointhatsituationistokeeponstaringat the same page. Every textbook explains certain things badly evenmine! sothebestthingtodointhissituationistolookat a dierent book. Instead of college textbooks aimed at the samemathematical level asthecourseyouretaking, youmayinsomecasesndthathighschool booksorbooksatalowermathlevelgive clearer explanations.Finally, whenreviewingforanexam, dontsimplyreadbackover the text and your lecture notes. Instead,try to use an activemethod of reviewing, for instance by discussing some of the discus-20 Chapter 0 Introduction and Reviewsionquestionswithanotherstudent,ordoinghomeworkproblemsyou hadnt done the rst time.0.1.4 Velocity and accelerationCalculus was inventedbyaphysicist, IsaacNewton, becauseheneededitasatoolforcalculatingvelocityandacceleration; inyour introductorycalculus course, velocityandaccelerationwereprobably presented as some of the rst applications.If an objects position as a function of time is given by the func-tionr(t), thenitsvelocityandaccelerationaregivenbytherstand second derivatives with respect to time, =drdtando =d2rdt2.The notation relates in a logical way to the units of the quantities.Velocity has units of m/s, and that makes sense because dr is inter-preted as an innitesimally small distance, with units of meters, anddt as an innitesimally small time, with units of seconds. The seem-ingly weird and inconsistent placement of the superscripted twos inthenotationfortheaccelerationislikewisemeanttosuggesttheunits: something on top with units of meters, and something on thebottom with units of seconds squared.Velocity and acceleration have completely dierent physical in-terpretations. Velocity is a matter of opinion. Right now as you sitin a chair and read this book, you could say that your velocity waszero, but an observer watching the Earth rotate would say that youhad a velocity of hundreds of miles an hour. Acceleration representsa change in velocity, and its not a matter of opinion. Accelerationsproducephysical eects, anddontoccurunlesstheresaforcetocause them. For example, gravitational forces on Earth cause fallingobjects to have an acceleration of 9.8 ms2.Constant acceleration example 1> How high does a diving board have to be above the water if thediver is to have as much as 1.0 s in the air?> The diver starts at rest, and has an acceleration of 9.8 ms2.We need to nd a connection between the distance she travelsandtimeittakes. Inotherwords, werelookingforinformationabout the function x(t ), given information about the acceleration.Section 0.1 Introduction and Review 21To go from acceleration to position, we need to integrate twice:x = a dt dt=

(at + vo) dt [vo is a constant of integration.]=

at dt [vo is zero because shes dropping from rest.]=12at2+ xo[xo is a constant of integration.]=12at2[xo can be zero if we dene it that way.]Note some of the good problem-solving habits demonstrated here.We solve the problem symbolically, and only plug in numbers atthe very end,once all the algebra and calculus are done. Oneshould also make a habit, after nding a symbolic result, of check-ingwhetherthedependenceonthevariablesmakesense. Agreater value of t in this expression would lead to a greater valueforx; thatmakessense, becauseifyouwantmoretimeintheair,youre going to have to jump from higher up. A greater ac-celeration also leads to a greater height; this also makes sense,because the stronger gravity is, the more height youll need in or-der to stay in the air for a given amount of time. Now we plug innumbers.x =12

9.8 ms2

(1.0 s)2= 4.9 mNote that when we put in the numbers,we check that the unitswork out correctly, ms2

(s)2=m. We should also check thatthe result makes sense:4.9 meters is pretty high, but not unrea-sonable.Thenotationdcincalculusrepresentsaninnitesimallysmallchangeinthevariablec. Thecorrespondingnotationforanitechangeinavariableisc. Forexample, if crepresentsthevalueof acertainstockonthestockmarket, andthevaluefalls fromco=5dollarsinitiallytocf=3dollarsnally, thenc = 2dollars. When we study linear functions, whose slopes are constant,the derivative is synonymous with the slope of the line, and dndris the same thing as nr, the rise over the run.Under conditions of constant acceleration, we can relate velocityand time,o =t,or, as in the example 1, position and time,r =12ot2+ ot + ro.22 Chapter 0 Introduction and ReviewIt can also be handy to have a relation involving velocity and posi-tion, eliminating time. Straightforward algebra gives2f= 2o + 2or ,wherefisthenal velocity, otheinitial velocity, andrthedistance traveled.> Solved problem: Dropping a rock on Mars page 48, problem 17> Solved problem: The Dodge Viper page 48, problem 190.1.5 Self-evaluationThe introductory part of a book like this is hard to write, becauseevery student arrives at this starting point with a dierent prepara-tion. One student may have grown up outside the U.S. and so maybecompletelycomfortablewiththemetricsystem, butmayhavehadanalgebracourseinwhichtheinstructorpassedtooquicklyoverscienticnotation. Anotherstudentmayhavealreadytakenvector calculus, but may have never learned the metric system. Thefollowingself-evaluationisachecklisttohelpyougureoutwhatyou need to study to be prepared for the rest of the course.Ifyoudisagreewiththisstate-ment. . .youshouldstudythissection:I am familiar with the basic metricunits of meters, kilograms, and sec-onds, and the most common metricprexes: milli- (m), kilo- (k), andcenti- (c).subsection 0.1.6 Basic of the MetricSystemI know about the newton, a unit offorcesubsection 0.1.7 The newton, theMetric Unit of ForceI amfamiliar withtheseless com-mon metric prexes: mega- (M),micro- (j), and nano- (n).subsection 0.1.8 Less Common Met-ric PrexesI am comfortable with scientic no-tation.subsection 0.1.9 Scientic NotationIcancondentlydometricconver-sions.subsection 0.1.10 ConversionsI understand the purpose and use ofsignicant gures.subsection 0.1.11 Signicant FiguresItwouldnthurtyoutoskimthesectionsyouthinkyoualreadyknow about, and to do the self-checks in those sections.Section 0.1 Introduction and Review 230.1.6 Basics of the metric systemThemetricsystemUnitswerenotstandardizeduntil fairlyrecentlyinhistory, sowhenthephysicistIsaacNewtongavetheresultofanexperimentwith a pendulum, he had to specify not just that the string was 3778incheslongbutthatitwas3778Londonincheslong. Theinch as dened in Yorkshire would have been dierent. Even afterthe British Empire standardized its units, it was still very inconve-nient to do calculations involving money, volume, distance, time, orweight, because of all the odd conversion factors, like 16 ounces ina pound, and 5280 feet in a mile. Through the nineteenth century,schoolchildren squandered most of their mathematical education inpreparing to do calculations such as making change when a customerin a shop oered a one-crown note for a book costing two pounds,thirteen shillings and tuppence. The dollar has always been decimal,and British money went decimal decades ago, but the United Statesisstillsaddledwiththeantiquatedsystemoffeet,inches,pounds,ounces and so on.Every country in the world besides the U.S. has adopted a sys-tem of units known in English as the metric system.This systemis entirely decimal, thanks to the same eminently logical people whobroughtabouttheFrenchRevolution. IndeferencetoFrance,thesystems ocial name is the Syst`eme International, or SI, meaningInternational System. (The phrase SI system is therefore redun-dant.)Thewonderful thingabout theSI is that peoplewholiveincountries moremodernthanours donot needtomemorizehowmanyouncesthereareinapound,howmanycupsinapint,howmanyfeetinamile, etc. Thewholesystemworkswithasingle,consistent set of prexes (derived from Greek) that modify the basicunits. Each prex stands for a power of ten, and has an abbreviationthatcanbecombinedwiththesymbolfortheunit. Forinstance,the meter is a unit of distance. The prex kilo- stands for 103, so akilometer, 1 km, is a thousand meters.The basic units of the metric system are the meter for distance,the second for time, and the gram for mass.The following are the most common metric prexes. You shouldmemorize them.prex meaning examplekilo- k 10360 kg = a persons masscenti- c 10228 cm = height of a piece of papermilli- m 1031 ms = time for one vibration of a guitarstring playing the note DThe prex centi-, meaning 102, is only used in the centimeter;a hundredth of a gram would not be written as 1 cg but as 10 mg.24 Chapter 0 Introduction and Reviewe / Pope Gregory created ourmodern Gregorian calendar, withits system of leap years, to makethelengthof thecalendar yearmatch the length of the cycleof seasons. Not until 1752didProtestant England switched tothe newcalendar. Some lesseducated citizens believed thattheshorteningof themonthbyelevendayswouldshortentheirlives by the same interval. In thisillustration by WilliamHogarth,the leaet lying on the groundreads, Give us our eleven days.The centi- prex can be easily remembered because a cent is 102dollars. TheocialSIabbreviationforsecondsiss(notsec)and grams are g (not gm).ThesecondThe sun stood still and the moon halted until the nation had takenvengeance on its enemies. . .Joshua 10:12-14Absolute,true,and mathematical time,of itself,and from its ownnature, ows equably without relation to anything external. . .Isaac NewtonWhen I stated briey above that the second was a unit of time,itmaynothaveoccurredtoyouthatthiswasnotreallymuchofa denition. The two quotes above are meant to demonstrate howmuch room for confusion exists among people who seem to mean thesame thing by a word such as time.The rst quote has been inter-preted by some biblical scholars as indicating an ancient belief thatthemotionofthesunacrosstheskywasnotjustsomethingthatoccurred with the passage of time but that the sun actually causedtime to pass by its motion, so that freezing it in the sky would havesome kind of a supernatural decelerating eect on everyone exceptthe Hebrew soldiers. Many ancient cultures also conceived of timeas cyclical, rather than proceeding along a straight line as in 1998,1999, 2000, 2001,... Thesecondquote, fromarelativelymodernphysicist, maysoundalotmorescientic, butmostphysiciststo-daywouldconsiderituselessasadenitionof time. Today, thephysical sciences are based on operational denitions, which meansdenitionsthatspellouttheactualsteps(operations)requiredtomeasure something numerically.Now in an era when our toasters, pens, and coee pots tell us thetime, it is far from obvious to most people what is the fundamentaloperational denition of time. Until recently, the hour, minute, andsecond were dened operationally in terms of the time required fortheearthtorotateaboutitsaxis. Unfortunately, theEarthsro-tation is slowing down slightly,and by 1967 this was becoming anissue in scientic experiments requiring precise time measurements.Thesecondwasthereforeredenedasthetimerequiredforacer-tainnumberof vibrationsof thelightwavesemittedbyacesiumatoms in a lamp constructed like a familiar neon sign but with theneon replaced by cesium. The new denition not only promises tostayconstantindenitely, but forscientistsisamoreconvenientwayof calibratingaclockthanhavingtocarryoutastronomicalmeasurements.self-check CWhat is a possible operational denition of how strong a person is? >Answer, p. 849Section 0.1 Introduction and Review 25f / The original denition ofthe meter.ThemeterTheFrenchoriginallydenedthemeteras107timesthedis-tance from the equator to the north pole, as measured through Paris(of course). Even if the denition was operational, the operation oftraveling to the north pole and laying a surveying chain behind youwas not one that most working scientists wanted to carry out. Fairlysoon,astandardwascreatedintheformofametalbarwithtwoscratches on it. This denition persisted until 1960, when the meterwas redened as the distance traveled by light in a vacuum over aperiod of (1/299792458) seconds.ThekilogramThethirdbaseunitof theSIisthekilogram, aunitof mass.Massisintendedtobeameasureof theamountof asubstance,but that is not an operational denition. Bathroom scales work bymeasuring our planets gravitational attraction for the object beingweighed, butusingthattypeofscaletodenemassoperationallywould be undesirable because gravity varies in strength from placeto place on the earth.Theres a surprising amount of disagreement among physics text-books about how mass should be dened, but heres how its actuallyhandled by the few working physicists who specialize in ultra-high-precision measurements. They maintain a physical object in Paris,which is the standard kilogram, a cylinder made of platinum-iridiumalloy. Duplicatesarecheckedagainstthismotherofall kilogramsby putting the original and the copy on the two opposite pans of abalance. Althoughthismethodofcomparisondependsongravity,the problems associated with dierences in gravity in dierent geo-graphical locations are bypassed, because the two objects are beingcomparedin thesameplace. Theduplicates canthenberemovedfrom the Parisian kilogram shrine and transported elsewhere in theworld.CombinationsofmetricunitsJust about anything you want to measure can be measured withsome combination of meters, kilograms, and seconds. Speed can bemeasured in m/s, volume in m3, and density in kgm3. Part of whatmakes the SI great is this basic simplicity. No more funny units likeacordof wood, aboltof cloth, orajiggerof whiskey. Nomoreliquid and dry measure. Just a simple, consistent set of units. TheSI measures put together from meters, kilograms, and seconds makeup the mks system. For example, the mks unit of speed is m/s, notkm/hr.Discussion QuestionA Isaac Newton wrote, . . . the natural days are truly unequal, thoughtheyarecommonlyconsideredasequal, andusedfor ameasureoftime. . . It may be that there is no such thing as an equable motion, whereby26 Chapter 0 Introduction and Reviewg / This is a mnemonic tohelp you remember the most im-portant metric prexes.The wordlittle istoremindyouthat thelist startswiththeprexesusedfor small quantities and buildsupward. Theexponent changesby3, except that of coursethatwedonot needaspecial prexfor 100, which equals one.time may be accurately measured. All motions may be accelerated or re-tarded. . . Newton was right. Even the modern denition of the secondin terms of light emitted by cesium atoms is subject to variation. For in-stance, magnetic elds could cause the cesium atoms to emit light witha slightly different rate of vibration. What makes us think, though, that apendulum clock is more accurate than a sundial, or that a cesium atomis a more accurate timekeeper than a pendulum clock?That is, how canonetest experimentallyhowtheaccuraciesof different timestandardscompare?0.1.7 The newton, the metric unit of forceA force is a push or a pull, or more generally anything that canchange an objects speed or direction of motion. A force is requiredto start a car moving,to slow down a baseball player sliding in tohome base, or to make an airplane turn. (Forces may fail to changeanobjectsmotionif theyarecanceledbyotherforces, e.g., theforce of gravity pulling you down right now is being canceled by theforceofthechairpushinguponyou.) ThemetricunitofforceistheNewton, denedastheforcewhich, ifappliedforonesecond,will cause a 1-kilogram object starting from rest to reach a speed of1 m/s. Later chapters will discuss the force concept in more detail.In fact, this entire book is about the relationship between force andmotion.In subsection 0.1.6, I gave a gravitational denition of mass, butby dening a numerical scale of force, we can also turn around anddene a scale of mass without reference to gravity. For instance, if aforce of two Newtons is required to accelerate a certain object fromrestto1m/sin1s, thenthatobjectmusthaveamassof 2kg.Fromthispointofview, masscharacterizesanobjectsresistancetoachangeinitsmotion, whichwecall inertiaorinertial mass.Although there is no fundamental reason why an objects resistancetoachangeinitsmotionmustberelatedtohowstronglygravityaectsit, carefulandpreciseexperimentshaveshownthatthein-ertial denition and the gravitational denition of mass are highlyconsistent for a variety of objects. It therefore doesnt really matterfor any practical purpose which denition one adopts.Discussion QuestionA Spending a long time in weightlessness is unhealthy. One of themostimportantnegativeeffectsexperiencedbyastronautsisalossofmuscle and bone mass.Since an ordinary scale wont work for an astro-naut in orbit, what is a possible way of monitoring this change in mass?(Measuring the astronauts waist or biceps with a measuring tape is notgood enough, because it doesnt tell anything about bone mass, or aboutthe replacement of muscle with fat.)Section 0.1 Introduction and Review 270.1.8 Less common metric prexesThe following are three metric prexes which, while less commonthan the ones discussed previously, are well worth memorizing.prex meaning examplemega- M 1066.4 Mm = radius of the earthmicro- j 10610jm = size of a white blood cellnano- n 1090.154 nm =distance betweencarbonnuclei in an ethane moleculeNote that the abbreviation for micro is the Greek letter mu, j a common mistake is to confuse it with m (milli) or M (mega).Thereareotherprexesevenlesscommon,usedforextremelylarge and small quantities. For instance, 1 femtometer = 1015m isaconvenientunitof distanceinnuclearphysics,and 1gigabyte =109bytes is used for computers hard disks. The international com-mitteethatmakesdecisionsabouttheSIhasrecentlyevenaddedsome new prexes that sound like jokes, e.g., 1 yoctogram = 1024gisabouthalf themassofaproton. Intheimmediatefuture,how-ever, youre unlikely to see prexes like yocto- and zepto- usedexceptperhapsintriviacontestsatscience-ctionconventionsorother geekfests.self-check DSuppose you could slowdown time so that according to your perception,a beam of light would move across a room at the speed of a slow walk.If you perceived a nanosecond as if it was a second,how would youperceive a microsecond? > Answer, p. 8490.1.9 Scientic notationMostoftheinterestingphenomenainouruniversearenotonthe human scale. It would take about 1,000,000,000,000,000,000,000bacteriatoequal themassofahumanbody. WhenthephysicistThomas Young discovered that light was a wave, it was back in thebad old days before scientic notation, and he was obliged to writethatthetimerequiredforonevibrationofthewavewas1/500ofamillionthofamillionthofasecond. Scienticnotationisalessawkwardwaytowriteverylargeandverysmallnumberssuchasthese. Heres a quick review.Scientic notation means writing a number in terms of a productof something from 1 to 10 and something else that is a power of ten.For instance,32 = 3.2 101320 = 3.2 1023200 = 3.2 103. . .Each number is ten times bigger than the previous one.Since101istentimessmallerthan102,itmakessensetouse28 Chapter 0 Introduction and Reviewthenotation100tostandforone, thenumberthatisinturntentimes smaller than 101. Continuing on, we can write 101to standfor 0.1, the number ten times smaller than 100. Negative exponentsare used for small numbers:3.2 = 3.2 1000.32 = 3.2 1010.032 = 3.2 102. . .Acommonsourceofconfusionisthenotationusedonthedis-plays of many calculators. Examples:3.2 106(written notation)3.2E+6 (notation on some calculators)3.26(notation on some other calculators)Thelast exampleis particularlyunfortunate, because3.26reallystandsforthenumber3.23.23.23.23.23.2=1074, atotally dierent number from 3.2106= 3200000. The calculatornotationshouldneverbeusedinwriting. Itsjustawayforthemanufacturer to save money by making a simpler display.self-check EA student learns that 104bacteria, standing in line to register for classesat Paramecium Community College, would form a queue of this size:The student concludes that 102bacteria would form a line of this length:Why is the student incorrect? > Answer, p. 8490.1.10 ConversionsI suggest youavoidmemorizinglots of conversionfactors be-tweenSIunitsandU.S.units. SupposetheUnitedNationssendsits blackhelicopters toinvadeCalifornia(after all whowouldntrather live here than in New York City?), and institutes water u-oridationandtheSI, makingtheuseofinchesandpoundsintoacrime punishable by death. I think you could get by with only twomental conversion factors:1 inch = 2.54 cmAnobjectwithaweightonEarthof2.2pounds-forcehasamass of 1 kg.The rst one is the present denition of the inch, so its exact. Thesecond one is not exact, but is good enough for most purposes. (U.S.Section 0.1 Introduction and Review 29unitsofforceandmassareconfusing, soitsagoodthingtheyrenotusedinscience. InU.S. units, theunitofforceisthepound-force, and the best unit to use for mass is the slug, which is about14.6 kg.)Moreimportantthanmemorizingconversionfactorsisunder-standingtherightmethodfordoingconversions. EvenwithintheSI, you may need to convert, say, from grams to kilograms. Dier-entpeoplehavedierentwaysof thinkingaboutconversions, butthe method Ill describe here is systematic and easy to understand.The idea is that if 1 kg and 1000 g represent the same mass,thenwe can consider a fraction like103g1 kgto 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,not one. Again, dierent peoplehavedierent ways of thinkingabout it, but the justication is that it helps us to do conversions,and it works! Now if we want to convert 0.7 kg to units of grams,we can multiply kg by the number one:0.7 kg 103g1 kgIf youre willing to treat symbols such as kg as if they were vari-ablesasusedinalgebra(whichtheyrereallynot), youcanthencancel the kg on top with the kg on the bottom, resulting in0.7

kg 103g1

kg= 700 g .Toconvertgramstokilograms, youwouldsimplyipthefractionupside down.One advantage of this method is that it can easily be applied toa series of conversions. For instance, to convert one year to units ofseconds,1$$$year 365days1$$$year

24 $$$hours1day

60min1 $$$hour 60 s1min== 3.15 107s .Shouldthatexponentbepositiveornegative?A common mistake is to write the conversion fraction incorrectly.For instance the fraction103kg1 g(incorrect)30 Chapter 0 Introduction and Reviewdoes not equal one, because 103kg is the mass of a car, and 1 g isthe mass of a raisin. One correct way of setting up the conversionfactor would be103kg1 g(correct) .You can usually detect such a mistake if you take the time to checkyour answer and see if it is reasonable.If common sense doesnt rule out either a positive or a negativeexponent, heres another way to make sure you get it right. Thereare big prexes and small prexes:big prexes: k Msmall prexes: m j n(Itsnothardtokeepstraightwhicharewhich,sincemegaandmicroareevocative, anditseasytorememberthatakilometeris bigger than a meter and a millimeter is smaller.)In the exampleabove, we want the top of the fraction to be the same as the bottom.Since/isabigprex, weneedtocompensatebyputtingasmallnumber like 103in front of it, not a big number like 103.> Solved problem: a simple conversion page 46, problem 6> Solved problem: the geometric mean page 46, problem 8Discussion QuestionA Each of the following conversions contains an error. In each case,explain what the error is.(a) 1000 kg 1kg1000g= 1 g(b) 50 m1cm100m= 0.5 cm(c) Nano is 109, so there are 109nm in a meter.(d) Micro is 106, so 1 kg is 106g.0.1.11 Signicant guresAn engineer is designing a car engine, and has been told that thediameter 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 pistonof this size, so he designs the cylinder to have an inside diameter of5.04 cm. Luckily, his supervisor catches his mistake before the cargoesintoproduction. Sheexplainshiserrortohim, andmentallyputs him in the do not promote category.Whatwashismistake? Thepersonwhotoldhimthepistonswere5cmindiameterwaswisetothewaysofsignicantgures,aswashisboss, whoexplainedtohimthatheneededtogobackandgetamoreaccuratenumberforthediameterof thepistons.That person said 5 cm rather than 5.00 cm specically to avoidcreating the impression that the number was extremely accurate. Inreality, the pistons diameter was 5.13 cm. They would never haveSection 0.1 Introduction and Review 31t in the 5.04-cm cylinders.The number of digits of accuracy in a number is referred to asthe number of signicant gures,or sig gs for short. As in theexample above, sig gs provide a way of showing the accuracy of anumber. In most cases, the result of a calculation involving severalpieces of data can be no more accurate than the least accurate pieceof data. Inotherwords, garbagein, garbageout. Sincethe5cm diameter of the pistons was not very accurate, the result of theengineers calculation, 5.04cm, was reallynot as accurateas hethought. Ingeneral, your result shouldnot havemorethanthenumberof siggsintheleastaccuratepieceof datayoustartedwith. The calculation above should have been done as follows:5 cm (1 sig g)+0.04 cm (1 sig g)=5 cm (rounded o to 1 sig g)Thefactthatthenal resultonlyhasonesignicantgurethenalerts you to the fact that the result is not very accurate, and wouldnot be appropriate for use in designing the engine.Notethattheleadingzeroesinthenumber0.04donotcountassignicantgures, becausetheyareonlyplaceholders. Ontheother hand, a number such as 50 cm is ambiguous the zero couldbeintendedasasignicantgure, oritmightjustbethereasaplaceholder. The ambiguity involving trailing zeroes can be avoidedbyusingscienticnotation,inwhich5101cmwouldimplyonesig g of accuracy, while 5.0 101cm would imply two sig gs.self-check FThe following quote is taken from an editorial by Norimitsu Onishi in theNew York Times, August 18, 2002.Consider Nigeria. Everyone agrees it is Africas most populousnation. But what isitspopulation? TheUnitedNationssays114 million; the State Department, 120 million. The World Banksays 126.9 million, while the Central Intelligence Agency puts itat 126,635,626.What should bother you about this? > Answer, p. 849Dealing correctly with signicant gures can save you time! Of-ten, students copy down numbers from their calculators with eightsignicantguresofprecision, thentypethembackinforalatercalculation. Thatsawasteoftime, unlessyouroriginaldatahadthat kind of incredible precision.The rules about signicant gures are only rules of thumb, andarenotasubstituteforcareful thinking. Forinstance, $20.00+$0.05 is $20.05. It need not and should not be rounded o to $20.In general, the sig g rules work best for multiplication and division,32 Chapter 0 Introduction and Reviewand we also apply them when doing a complicated calculation thatinvolves many types of operations. For simple addition and subtrac-tion, it makes more sense to maintain a xed number of digits afterthe decimal point.Whenindoubt, dontusethesiggrulesatall. Instead, in-tentionallychangeonepieceofyourinitialdatabythemaximumamount by which you think it could have been o, and recalculatethe nal result. The digits on the end that are completely reshuedare the ones that are meaningless, and should be omitted.self-check GHowmanysignicantguresarethereineachofthefollowingmea-surements?(1) 9.937 m(2) 4.0 s(3) 0.0000000000000037 kg > Answer, p. 849Section 0.1 Introduction and Review 33a / Amoebas this size areseldom encountered.0.2 Scaling and Order-of-Magnitude Estimates0.2.1 IntroductionWhy cant an insect be the size of a dog?Some skinny stretched-outcellsinyourspinal cordareametertall whydoesnaturedisplaynosinglecellsthatarenotjustametertall, butameterwide, and a meter thick as well?Believe it or not, these are questionsthat can be answered fairly easily without knowing much more aboutphysics than you already do. The only mathematical technique youreally need is the humble conversion, applied to area and volume.AreaandvolumeAreacanbedenedbysayingthatwecancopytheshapeofinterest onto graph paper with 1 cm1 cm squares and count thenumber of squares inside. Fractions of squares can be estimated byeye. We then say the area equals the number of squares, in units ofsquare cm. Although this might seem less pure than computingareasusingformulaelike=:2foracircleor=n/2foratriangle, those formulae are not useful as denitions of area becausethey cannot be applied to irregularly shaped areas.Units of square cm are more commonly written as cm2in science.Ofcourse,theunitofmeasurementsymbolizedbycmisnotanalgebra symbol standing for a number that can be literally multipliedby itself. But it is advantageous to write the units of area that wayandtreattheunitsasiftheywerealgebrasymbols. Forinstance,if youhavearectanglewithanareaof 6m2andawidthof 2m,thencalculatingitslengthas(6m2)(2m)=3mgivesaresultthat makes sensebothnumericallyandinterms of units. Thisalgebra-style treatment of the units also ensures that our methodsofconvertingunitsworkoutcorrectly. Forinstance, ifweacceptthe fraction100 cm1 mas a valid way of writing the number one, then one times one equalsone, so we should also say that one can be represented by100 cm1 m

100 cm1 m,which is the same as10000 cm21 m2.That means the conversion factor from square meters to square cen-timeters is a factor of 104,i.e.,a square meter has 104square cen-timeters in it.All of the above can be easily applied to volume as well,usingone-cubic-centimeter blocks instead of squares on graph paper.Tomanypeople, itseemshardtobelievethatasquaremeterequals 10000squarecentimeters, or that acubicmeter equals a34 Chapter 0 Introduction and Reviewmillion cubic centimeters they think it would make more sense ifthere were 100 cm2in 1 m2, and 100 cm3in 1 m3, but that would beincorrect. The examples shown in gure b aim to make the correctanswer more believable, using the traditional U.S. units of feet andyards. (One foot is 12 inches, and one yard is three feet.)b / Visualizing conversions ofarea and volume using traditionalU.S. units.self-check HBased on gure b, convince yourself that there are 9 ft2in a square yard,and 27 ft3in a cubic yard, then demonstrate the same thing symbolically(i.e., with the method using fractions that equal one). > Answer, p.849> Solved problem: converting mm2to cm2page 50, problem 31> Solved problem: scaling a liter page 51, problem 40Discussion QuestionA How many square centimeters are there in a square inch? (1 inch =2.54 cm) First nd an approximate answer by making a drawing, then de-rive the conversion factor more accurately using the symbolic method.0.2.2 Scaling of area and volumeGreat eas have lesser easUpon their backs to bite em.And lesser eas have lesser still,And so ad innitum.Jonathan SwiftNow how do these conversions of area and volume relate to thequestionsIposedaboutsizesoflivingthings? Well, imaginethatyouareshrunklikeAliceinWonderlandtothesizeof aninsect.Onewayof thinkingaboutthechangeof scaleisthatwhatusedtolooklikeacentimeternowlookslikeperhapsametertoyou,because youre so much smaller. If area and volume scaled accordingto most peoples intuitive, incorrect expectations, with 1 m2beingthesameas 100cm2, thentherewouldbenoparticular reasonwhynatureshouldbehaveanydierentlyonyour new, reducedscale. Butnaturedoesbehavedierentlynowthatyouresmall.Forinstance, youwillndthatyoucanwalkonwater, andjumpSection 0.2 Scaling and Order-of-Magnitude Estimates 35d / The small boat holds upjust ne.e / A larger boat built withthe same proportions as thesmall onewill collapseunderitsown weight.f / A boat this large needs tohave timbers that are thickercompared to its size.c / Galileo Galilei (1564-1642).tomanytimesyourownheight. ThephysicistGalileoGalileihadthebasicinsightthatthescalingof areaandvolumedetermineshownatural phenomenabehavedierentlyondierentscales. Herstreasonedaboutmechanicalstructures, butlaterextendedhisinsights to living things, taking the then-radical point of view that atthe fundamental level, a living organism should follow the same lawsofnatureasamachine. Wewillfollowhisleadbyrstdiscussingmachines and then living things.GalileoonthebehaviorofnatureonlargeandsmallscalesOneof theworldsmostfamouspiecesof scienticwritingisGalileos Dialogues Concerning the Two New Sciences. Galileo wasan entertaining writer who wanted to explain things clearly to laypeo-ple, and he livened up his work by casting it in the form of a dialogueamong three people. Salviati is really Galileos alter ego. Simpliciois the stupid character, and one of the reasons Galileo got in troublewiththeChurchwasthattherewererumorsthatSimpliciorepre-sented the Pope. Sagredo is the earnest and intelligent student, withwhomthereaderissupposedtoidentify. (Thefollowingexcerptsare from the 1914 translation by Crew and de Salvio.)SAGREDO: Yes, that is what I mean; and I refer especially tohis last assertion which I have always regarded as false. . . ;namely, that in speaking of these and other similar machinesone cannot argue from the small to the large, because manydeviceswhichsucceedonasmall scaledonot workonalarge scale. Now, since mechanics has its foundations in ge-ometry, where mere size [ is unimportant], I do not see thatthe properties of circles, triangles, cylinders, cones and othersolid gures will change with their size. If, therefore, a largemachine be constructed in such a way that its parts bear tooneanotherthesameratioasinasmallerone, andifthesmallerissufcientlystrongforthepurposeforwhichit isdesigned, I do not see why the larger should not be able towithstand any severe and destructive tests to which it may besubjected.Salviati contradicts Sagredo:SALVIATI: . . . Please observe, gentlemen, how facts whichat rst seem improbable will, even on scant explanation, dropthe cloak which has hidden themand stand forth in naked andsimple beauty. Who does not know that a horse falling from aheight of three or four cubits will break his bones, while a dogfalling from the same height or a cat from a height of eightor ten cubits will suffer no injury? Equally harmless would bethe fall of a grasshopper from a tower or the fall of an ant fromthe distance of the moon.The point Galileo is making here is that small things are sturdierin proportion to their size. There are a lot of objections that could be36 Chapter 0 Introduction and Reviewraised, however. After all, what does it really mean for something tobe strong, to be strong in proportion to its size, or to be strongout of proportiontoits size? Galileohasnt givenoperationaldenitionsofthingslikestrength,i.e., denitionsthatspell outhow to measure them numerically.Also, acatisshapeddierentlyfromahorseanenlargedphotograph of a cat would not be mistaken for a horse, even if thephoto-doctoring experts at the National Inquirer made it look like aperson was riding on its back. A grasshopper is not even a mammal,and it has an exoskeleton instead of an internal skeleton. The wholeargument would be a lot more convincing if we could do some iso-lation of variables, a scientic term that means to change only onething at a time, isolating it from the other variables that might havean eect. If size is the variable whose eect were interested in see-ing, then we dont really want to compare things that are dierentin size but also dierent in other ways.SALVIATI: . . . we asked the reason why [shipbuilders] em-ployed stocks, scaffolding, and bracing of larger dimensionsfor launching a big vessel than they do for a small one; and[an old man] answered that they did this in order to avoid thedanger of the ship parting under its own heavy weight, a dan-ger to which small boats are not subject?After this entertaining but not scientically rigorous beginning,Galileostarts todosomethingworthwhilebymodernstandards.Hesimplieseverythingbyconsideringthestrengthof awoodenplank. Thevariablesinvolvedcanthenbenarroweddowntothetypeof wood, thewidth, thethickness, andthelength. Healsogivesanoperational denitionof whatitmeansfortheplanktohaveacertainstrengthinproportiontoitssize,byintroducingthe concept of a plank that is the longest one that would not snapunder its ownweight if supportedat oneend. If youincreasedits length by the slightest amount,without increasing its width orthickness, itwouldbreak. Hesaysthatif oneplankisthesameshapeasanotherbutadierentsize, appearinglikeareducedorenlarged photograph of the other, then the planks would be strongin proportion to their sizes if both were just barely able to supporttheir own weight.Also, Galileoisdoingsomethingthatwouldbefrownedoninmodern science: he is mixing experiments whose results he has ac-tually observed (building boats of dierent sizes), with experimentsthathecouldnotpossiblyhavedone(droppinganantfromtheheightof themoon). Henowrelateshowhehasdoneactual ex-perimentswithsuchplanks, andfoundthat, accordingtothisop-erational denition, they are not strong in proportion to their sizes.The larger one breaks. He makes sure to tell the reader how impor-Section 0.2 Scaling and Order-of-Magnitude Estimates 37h / Galileo discusses planksmadeof wood, but theconceptmay be easier to imagine withclay. All threeclayrodsinthegure were originally the sameshape. Themedium-sizedonewastwicetheheight, twicethelength, and twice the width ofthesmall one, andsimilarlythelargeonewas twiceas bigasthemediumoneinall itslineardimensions. The big one hasfour timesthelinear dimensionsof thesmall one, 16times thecross-sectional area when cutperpendicular tothepage, and64 times the volume. That meansthat the big one has 64 times theweight to support, but only 16timesthestrengthcomparedtothe smallest one.g / 1. This plank is as long as it can be without collapsing underits own weight. If it was a hundredth of an inch longer, it would collapse.2. 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.tant the result is, via Sagredos astonished response:SAGREDO: My brain already reels. My mind, like a cloudmomentarily illuminated by a lightning ash, is for an instantlledwithanunusual light, whichnowbeckonstomeandwhichnowsuddenlyminglesandobscuresstrange, crudeideas. From what you have said it appears to me impossibletobuildtwosimilarstructuresof thesamematerial, but ofdifferent sizes and have them proportionately strong.In other words, this specic experiment, using things like woodenplanks that have no intrinsic scientic interest, has very wide impli-cationsbecauseitpointsoutageneral principle, thatnatureactsdierently on dierent scales.Tonishthediscussion,Galileogivesanexplanation. Hesaysthat the strength of a plank (dened as, say, the weight of the heav-iestboulderyoucouldputontheendwithoutbreakingit)ispro-portional to its cross-sectional area, that is, the surface area of thefreshwoodthatwouldbeexposedifyousawedthroughitinthemiddle. Its weight, however, is proportional to its volume.1How do the volume and cross-sectional area of the longer plankcomparewiththoseof theshorterplank? Wehavealreadyseen,whilediscussingconversionsoftheunitsofareaandvolume, thatthese quantities dont act the way most people naively expect. Youmight think that the volume and area of the longer plank would bothbe doubled compared to the shorter plank,so they would increasein proportion to each other, and the longer plank would be equallyable to support its weight. You would be wrong, but Galileo knowsthat this is a common misconception, so he has Salviati address thepoint specically:1Galileomakes aslightlymorecomplicatedargument, takingintoaccounttheeectofleverage(torque). TheresultImreferringtocomesoutthesameregardlessofthiseect.38 Chapter 0 Introduction and Reviewi / The area of a shape isproportional tothesquareof itslinear dimensions, even if theshape is irregular.SALVIATI: . . . Take, forexample, acubetwoinchesonasidesothat eachfacehasanareaof four squareinchesandthetotal area, i.e., thesumof thesixfaces, amountstotwenty-foursquareinches; nowimaginethiscubetobesawed through three times [with cuts in three perpendicularplanes] so as to divide it into eight smaller cubes, each oneinchontheside, eachfaceoneinchsquare, andthetotalsurfaceof eachcubesixsquareinchesinsteadof twenty-fourinthecaseof thelargercube. It isevident therefore,that thesurfaceof thelittlecubeisonlyone-fourththat ofthe larger, namely, the ratio of six to twenty-four; but the vol-ume 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, thatI was not mistaken when . . . I said that the surface of a smallsolid is comparatively greater than that of a large one.Thesamereasoningappliestotheplanks. Eventhoughtheyarenotcubes, thelargeonecouldbesawedintoeightsmallones,eachwithhalf thelength, half thethickness, andhalf thewidth.The small plank, therefore, has more surface area in proportion toits weight, and is therefore able to support its own weight while thelarge one breaks.ScalingofareaandvolumeforirregularlyshapedobjectsYou probably are not going to believe Galileos claim that thishas deep implications for all of nature unless you can be convincedthat the same is true for any shape. Every drawing youve seen sofar has been of squares, rectangles, and rectangular solids. Clearlythe reasoning about sawing things up into smaller pieces would notprove anything about, say, an egg, which cannot be cut up into eightsmaller egg-shaped objects with half the length.Isitalwaystruethatsomethinghalf thesizehasonequarterthe surface area and one eighth the volume, even if it has an irreg-ular shape?Take the example of a childs violin. Violins are madefor small children in smaller size to accomodate their small bodies.Figureishowsafull-sizeviolin,alongwithtwoviolinsmadewithhalf and 3/4 of the normal length.2Lets study the surface area ofthe front panels of the three violins.Considerthesquareintheinteriorofthepanelofthefull-sizeviolin. In the 3/4-size violin, its height and width are both smallerby a factor of 3/4, so the area of the corresponding, smaller squarebecomes 3434 = 916 of the original area, not 3/4 of the originalarea. Similarly, the corresponding square on the smallest violin hashalf the height and half the width of the original one, so its area is2Thecustomarytermshalf-sizeand3/4-sizeactuallydontdescribethesizes inanyaccurate way. Theyre reallyjust standard, arbitrarymarketinglabels.Section 0.2 Scaling and Order-of-Magnitude Estimates 39j / The mufn comes out ofthe oven too hot to eat. Breakingit upintofour piecesincreasesits surface area while keepingthe total volume the same. Itcools faster because of thegreater surface-to-volume ratio.Ingeneral, smaller things havegreater surface-to-volumeratios,but inthisexamplethereisnoeasywaytocomputetheeffectexactly, because the small piecesarent the same shape as theoriginal mufn.1/4 the original area, not half.The same reasoning works for parts of the panel near the edge,suchasthepartthatonlypartiallyllsintheothersquare. Theentire square scales down the same as a square in the interior, andin each violin the same fraction (about 70%) of the square is full, sothe contribution of this part to the total area scales down just thesame.Since any small square region or any small region covering partof a square scales down like a square object, the entire surface areaof an irregularly shaped object changes in the same manner as thesurface area of a square: scaling it down by 3/4 reduces the area bya factor of 9/16, and so on.In general, we can see that any time there are two objects withthe same shape, but dierent linear dimensions (i.e., one looks like areduced photo of the other), the ratio of their areas equals the ratioof the squares of their linear dimensions:12=

1112

2.Note that it doesnt matter where we choose to measure the linearsize, 1, of an object. In the case of the violins, for instance, it couldhave been measured vertically, horizontally, diagonally, or even fromthebottomoftheleftf-holetothemiddleoftherightf-hole. Wejust have to measure it in a consistent way on each violin. Since allthe parts are assumed to shrink or expand in the same manner, theratio1112 is independent of the choice of measurement.It is also important to realize that it is completely unnecessarytohaveaformulafortheareaof aviolin. Itisonlypossibletoderivesimpleformulasfortheareasof certainshapeslikecircles,rectangles, trianglesandsoon, butthatisnoimpedimenttothetype of reasoning we are using.Sometimes it is inconvenient to write all the equations in termsof ratios, especially when more than two objects are being compared.A more compact way of rewriting the previous equation is 12.Thesymbol meansisproportional to. Scientistsandengi-neers often speak about such relationships verbally using the phrasesscales like or goes like, for instance area goes like length squared.All of the above reasoning works just as well in the case of vol-ume. Volume goes like length cubed:\ 13.If dierentobjectsaremadeof thesamematerial withthesamedensity, =:\ , thentheirmasses, :=\ , areproportional40 Chapter 0 Introduction and Reviewk / Example 2. The big trian-gle has four times more area thanthe little one.l / A tricky way of solving ex-ample 2, explained in solution #2.to13,andsoaretheirweights. (Thesymbolfordensityis,thelower-case Greek letter rho.)Animportant point is that all of theabovereasoningaboutscaling only applies to objects that are the same shape. For instance,apieceof paperislargerthanapencil, buthasamuchgreatersurface-to-volume ratio.One of the rst things I learned as a teacher was that studentswere not very original about their mistakes. Every group of studentstendstocomeupwiththesamegoofsasthepreviousclass. Thefollowing are some examples of correct and incorrect reasoning aboutproportionality.Scaling of the area of a triangle example 2>Ingurek, thelargertrianglehassidestwiceaslong. Howmany 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 ofthesmallersize, asshowning. (b), soitsareaisfourtimesgreater. (This solution is correct, but it would not work for a shapelike a circle, which cant be cut up into smaller circles.)Correct solution #3: The area of a triangle is given byA = bh2, where b is the base and h is the height. The areas ofthe triangles areA1 = b1h12A2 = b2h22= (2b1)(2h1)2= 2b1h1A2A1 = (2b1h1)(b1h12)= 4(Although this solution is correct, it is a lot more work than solution#1, and it can only be used in this case because a triangle is asimple geometric shape, and we happen to know a formula for itsarea.)Correct solution#4: Theareaof atriangleisA=bh2. Thecomparison of the areas will come out the same as long as theratios of the linear sizes of the triangles is as specied,so letsjust say b1 = 1.00 m and b2 = 2.00 m.The heights are then alsoh1=1.00 m and h2=2.00 m,giving areas A1=0.50 m2andA2 = 2.00 m2, so A2A1 = 4.00.(Thesolutioniscorrect, but it wouldnt workwithashapeforwhoseareawedont haveaformula. Also, thenumerical cal-Section 0.2 Scaling and Order-of-Magnitude Estimates 41n / Example 4. The 48-pointS has 1.78 times more areathan the 36-point S.m / Example3. Thebigspherehas 125 times more volume thanthe little one.culation might make the answer of 4.00 appear inexact, whereassolution #1 makes it clear that it is exactly 4.)Incorrectsolution: TheareaofatriangleisA=bh, andifyoupluginb=2.00mandh=2.00m, youget A=2.00m2, sothebiggertrianglehas2.00timesmorearea. (Thissolutionisincorrect because no comparison has been made with the smallertriangle.)Scaling of the volume of a sphere example 3>Ingurem, thelargerspherehasaradiusthat isvetimesgreater. How many times greater is its volume?Correct solution#1: Volumescaleslikethethirdpowerof 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)r3, soV1 =43r31V2 =43r32=43(5r1)3=5003r31V2V1 =

5003r31

43r31

= 125Incorrect solution: The volume of a sphere is V= (43)r3, soV1 =43r31V1 =43r32=435r31=203r31V2V1 =

203r31

43r31

= 5(The solution is incorrect because (5r1)3is not the same as 5r31.)Scaling of a more complex shape example 4> The rst letter S in gure n is in a 36-point font, the second in48-point. How many times more ink is required to make the largerS? (Points are a unit of length used in typography.)Correct solution: The amount of ink depends on the area to becoveredwithink, andareaisproportional tothesquareofthe42 Chapter 0 Introduction and Reviewlinear dimensions, so the amount of ink required for the secondS is greater by a factor of (4836)2= 1.78.Incorrect solution: The length of the curve of the second S islongerbyafactorof 4836=1.33, so1.33timesmoreinkisrequired.(Thesolutioniswrongbecauseit assumesincorrectlythat thewidth of the curve is the same in both cases. Actually both thewidth and the length of the curve are greater by a factor of 48/36,so the area is greater by a factor of (4836)2= 1.78.)> Solved problem: a telescope gathers light page 50, problem 32> Solved problem: distance from an earthquake page 50, problem 33Discussion QuestionsA A toy re engine is 1/30 the size of the real one, but is constructedfrom the same metal with the same proportions. How many times smalleris its weight?How many times less red paint would be needed to paintit?B Galileospendsalotoftimeinhisdialogdiscussingwhatreallyhappens when things break. He discusses everything in terms of Aristo-tles now-discredited explanation that things are hard to break, becauseif something breaks, there has to be a gap between the two halves withnothing in between, at least initially. Nature, according to Aristotle, ab-hors a vacuum, i.e., nature doesnt like empty space to exist. Of course,air will rush into the gap immediately, but at the very moment of breaking,Aristotle imagined a vacuum in the gap. Is Aristotles explanation of whyit is hard to break things an experimentally testable statement? If so, howcould it be tested experimentally?0.2.3 Order-of-magnitude estimatesIt is the mark of an instructed mind to rest satised with the degreeof precision that the nature of the subject permits and not to seekan exactness where only an approximation of the truth is possible.AristotleIt is a common misconception that science must be exact. Forinstance, intheStarTrekTVseries, itwouldoftenhappenthatCaptain Kirk would ask Mr. Spock, Spock, were in a pretty badsituation. What doyouthinkareour chances of gettingout ofhere?The scientic Mr. Spock would answer with something like,Captain, I estimatetheoddsas237.345toone. Inreality, hecouldnot haveestimatedtheodds withsixsignicant guresofaccuracy, but nevertheless one of the hallmarks of a person with agood education in science is the ability to make estimates that arelikely to be at least somewhere in the right ballpark. In many suchsituations, it is often only necessary to get an answer that is o by nomore than a factor of ten in either direction. Since things that dierby a factor of ten are said to dier by one order of magnitude, suchSection 0.2 Scaling and Order-of-Magnitude Estimates 43anestimateis calledanorder-of-magnitudeestimate. Thetilde,, isusedtoindicatethat thingsareonlyof thesameorder ofmagnitude, but not exactly equal, as inodds of survival 100 to one .The tilde can also be used in front of an individual number to em-phasize that the number is only of the right order of magnitude.Although making order-of-magnitude estimates seems simple andnatural toexperiencedscientists, itsamodeof reasoningthatiscompletely unfamiliar to most college students. Some of the typicalmental steps can be illustrated in the following example.Cost of transporting tomatoes example 5> Roughly what percentage of the price of a tomato comes fromthe cost of transporting it in a truck?> The following incorrect solution illustrates one of the main waysyou can go wrong in order-of-magnitude estimates.Incorrectsolution: Letssaythetruckerneedstomakea$400prot on the trip. Taking into account her benets, the cost of gas,and maintenance and payments on the truck,lets say the totalcost is more like $2000.Id guess about 5000 tomatoes would tin the back of the truck, so the extra cost per tomato is 40 cents.That means the cost of transporting one tomato is comparable tothe cost of the tomato itself. Transportation really adds a lot to thecost of produce, I guess.Theproblemisthatthehumanbrainisnotverygoodatesti-mating area or volume, so it turns out the estimate of 5000 tomatoestting in the truck is way o. Thats why people have a hard timeat those contests where you are supposed to estimate the number ofjellybeans in a big jar. Another example is that most people thinktheirfamiliesuseabout10gallonsofwaterperday,butinrealitytheaverageisabout300gallonsperday. Whenestimatingareaorvolume, youaremuchbetteroestimatinglineardimensions,and computing volume from the linear dimensions. Heres a bettersolution:Better solution: As in the previous solution, say the cost of thetrip is $2000. The dimensions of the bin are probably 4 m2 m 1 m,for a volume of 8 m3. Since the whole thing is just an order-of-magnitude estimate, lets round that o to the nearest power often, 10 m3. The shape of a tomato is complicated, and I dont knowany formula for the volume of a tomato shape, but since this is justan estimate, lets pretend that a tomato is a cube, 0.05 m0.05 m 0.05 m, for a volume of 1.25 104m3. Since this is just a roughestimate, lets round that to 104m3. We can nd the total numberof tomatoes by dividing the volume of the bin by the volume of onetomato: 10 m3104m3= 105tomatoes. The transportation cost44 Chapter 0 Introduction and Reviewo / Consider a spherical cow.per tomato is $2000105tomatoes=$0.02/tomato. That means thattransportation really doesnt contribute very much to the cost of atomato.Approximating the shape of a tomato as a cube is an example ofanothergeneralstrategyformakingorder-of-magnitudeestimates.A similar situation would occur if you were trying to estimate howmany m2of leather could be produced from a herd of ten thousandcattle. There is no point in trying to take into account the shape ofthe cows bodies. A reasonable plan of attack might be to considera spherical cow. Probably a cow has roughly the same surface areaas a sphere with a radius of about 1 m, which would be 4(1 m)2.Using the well-known facts that pi equals three, and four times threeequals about ten, we can guess that a cow has a surface area of about10 m2, so the herd as a whole might yield 105m2of leather.The following list summarizes the strategies for getting a goodorder-of-magnitude estimate.1. Dont even attempt more than one signicant gure of preci-sion.2. Dont guess area or volume directly. Guess linear dimensionsand get area or volume from them.3. When dealing with areas or volumes of objects with complexshapes, idealizethemasif theyweresomesimplershape, acube or a sphere, for example.4. Check your nal answer to see if it is reasonable. If you esti-mate that a herd of ten thousand cattle would yield 0.01 m2of leather, then you have probably made a mistake with con-version factors somewhere.Section 0.2 Scaling and Order-of-Magnitude Estimates 45ProblemsThe symbols , , etc. are explained on page 52.1 Correct use of a calculator: (a) Calculate7465853222+97554on a cal-culator. [Self-check: The most common mistake results in 97555.40.](b) Which would be more like the price of a TV, and which wouldbe more like the price of a house, $3.5 105or $3.55?2 Computethefollowingthings. Iftheydontmakesensebe-cause 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 hours3 Your backyard has brick walls on both ends. You measure adistanceof23.4mfromtheinsideofonewalltotheinsideoftheother. Eachwallis29.4cmthick. Howfarisitfromtheoutsideof one wall to the outside of the other?Pay attention to signicantgures.4 The speed of light is 3.0108m/s. Convert this to furlongsper fortnight. A furlong is 220 yards, and a fortnight is 14 days. Aninch is 2.54 cm.5 Express each of the following quantities in micrograms:(a) 10 mg, (b) 104g, (c) 10 kg, (d) 100 103g, (e) 1000 ng.6 Convert 134 mg to units of kg, writing your answer in scienticnotation. > Solution, p. 8587 In the last century, the average age of the onset of puberty forgirls has decreased by several years. Urban folklore has it that thisis because of hormones fed to beef cattle, but it is more likely to bebecausemoderngirlshavemorebodyfatontheaverageandpos-siblybecauseofestrogen-mimickingchemicalsintheenvironmentfromthebreakdownofpesticides. Ahamburgerfromahormone-implantedsteer has about 0.2ngof estrogen(about doubletheamount of nat