2Simple NatureAn Introduction to Physics for Engineeringand Physical Science StudentsBenjamin Crowellwww.lightandmatter.comFullerton,Californiawww.lightandmatter.comCopyright c _2001-2008BenjaminCrowellrev. October9,2010Permissionisgrantedtocopy, distributeand/ormodifythisdocu-mentunderthetermsoftheCreativeCommonsAttributionShare-Alike License, whichcanbe foundat creativecommons.org. Thelicenseappliestotheentiretextof thisbook, plusall theillustra-tionsthatarebyBenjaminCrowell. (Atyouroption,youmayalsocopythis bookunder theGNUFreeDocumentationLicensever-sion1.2, withnoinvariant sections, nofront-cover texts, andnoback-covertexts.) AlltheillustrationsarebyBenjaminCrowellex-ceptasnotedinthephotocreditsorinparenthesesinthecaptionof the gure. This bookcanbe downloadedfree of charge fromwww.lightandmatter.cominavarietyofformats,includingeditableformats.Brief Contents0 Introduction 131 Conservation of Mass 532 Conservation of Energy 733 Conservation of Momentum 1274 Conservation of Angular Momentum 2375 Thermodynamics 2896 Waves 3297 Relativity 3718 Atoms and Electromagnetism 4039 DC Circuits 45910 Fields 50311 Electromagnetism 58912 Optics 67513 Quantum Physics 75956Contents0Introduction and Review0.1Introduction and Review. . . . . . . . . . . . . . 13Thescienticmethod, 13.Whatisphysics?, 16.Howtolearnphysics, 19.Velocity and acceleration, 21.Self-evaluation, 23.Basics of themetricsystem, 24.Less commonmetricprexes,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-solvingtechniques,56.Deltanotation,57.1.2Equivalence of Gravitational and Inertial Mass . . . . . 581.3Galilean Relativity . . . . . . . . . . . . . . . . . 60Applicationsofcalculus,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.Gravitationalenergy,81.Equilibriumandstability,86.Predictingthedirectionofmotion,88.2.2Numerical Techniques. . . . . . . . . . . . . . . 902.3Gravitational Phenomena. . . . . . . . . . . . . . 95Keplers laws, 95.Circular orbits, 97.Thesuns gravitationaleld, 98.Gravitational energy in general, 98.The shell theorem,101.Evidenceforrepulsivegravity,107.2.4Atomic Phenomena. . . . . . . . . . . . . . . . 108Heat is kinetic energy., 109.All energy comes from particles mov-ingorinteracting.,110.2.5Oscillations . . . . . . . . . . . . . . . . . . . 112Problems . . . . . . . . . . . . . . . . . . . . . . 117Exercises . . . . . . . . . . . . . . . . . . . . . . 1253Conservation of Momentum3.1Momentum In One Dimension. . . . . . . . . . . . 128Mechanicalmomentum,128.Nonmechanicalmomentum,131.Momentumcomparedtokinetic energy, 132.Collisions inonedimension, 133.Thecenter of mass, 138.Thecenter of massframeofreference,142.3.2Force In One Dimension. . . . . . . . . . . . . . 143Momentumtransfer, 143.Newtons laws, 145.What force isnot, 148.Forces between solids, 150.Fluid friction, 153.Analysisof forces, 154.Transmission of forces by low-mass objects, 156.Work, 158.SimpleMachines, 165.Forcerelatedtointeractionenergy,166.3.3Resonance. . . . . . . . . . . . . . . . . . . . 168Damped, free motion, 169.The quality factor, 172.Driven motion,173.3.4Motion In Three Dimensions . . . . . . . . . . . . 183The Cartesian perspective, 183.Rotational invariance, 186.Vectors,189.Calculus with vectors, 203.The dot product, 207.Gradientsandlineintegrals(optional),210.Problems . . . . . . . . . . . . . . . . . . . . . . 213Exercises . . . . . . . . . . . . . . . . . . . . . . 2304Conservation of Angular Momentum4.1Angular Momentum In Two Dimensions . . . . . . . . 237Angular momentum, 237.Application to planetary motion, 242.Twotheorems about angular momentum, 243.Torque, 246.Applications to statics, 250.Proof of Keplers elliptical orbit law,254.4.2Rigid-Body Rotation . . . . . . . . . . . . . . . . 257Kinematics, 257.Relationsbetweenangularquantitiesandmo-tionof apoint, 258.Dynamics, 260.Iteratedintegrals, 262.Findingmomentsofinertiabyintegration,265.4.3Angular Momentum In Three Dimensions. . . . . . . 270Rigid-body kinematics inthree dimensions, 270.Angular mo-mentuminthreedimensions,272.Rigid-bodydynamicsinthreedimensions,277.Problems . . . . . . . . . . . . . . . . . . . . . . 280Exercises . . . . . . . . . . . . . . . . . . . . . . 2885Thermodynamics5.1Pressure and Temperature. . . . . . . . . . . . . 290Pressure,290.Temperature,294.5.2Microscopic Description of An Ideal Gas . . . . . . . 297Evidence for the kinetic theory, 297.Pressure, volume, and temperature,298.5.3Entropy As a Macroscopic Quantity . . . . . . . . . 300Eciency and grades of energy, 300.Heat engines, 301.Entropy,303.5.4Entropy As a Microscopic Quantity . . . . . . . . . . 307A microscopic view of entropy, 307.Phase space, 308.Microscopicdenitionsofentropyandtemperature, 309.Thearrowoftime,or this way to the Big Bang, 317.Quantum mechanics and zeroentropy,318.Summaryofthelawsofthermodynamics,319.8 Contents5.5More About Heat Engines . . . . . . . . . . . . . 319Problems . . . . . . . . . . . . . . . . . . . . . . 3266Waves6.1Free Waves . . . . . . . . . . . . . . . . . . . 330Wave motion, 330.Waves ona string, 336.Soundandlightwaves,339.Periodicwaves,341.TheDopplereect,344.6.2Bounded Waves . . . . . . . . . . . . . . . . . 350Reection, transmission, and absorption, 350.Quantitative treat-ment of reection, 355.Interference eects, 358.Waves boundedonbothsides,360.Problems . . . . . . . . . . . . . . . . . . . . . . 3677Relativity7.1Time Is Not Absolute . . . . . . . . . . . . . . . 371The correspondence principle, 371.Causality, 371.Time distor-tionarisingfrommotionandgravity,372.7.2Distortion of Space and Time. . . . . . . . . . . . 374TheLorentztransformation, 374.Thefactor, 378.Theuni-versalspeedc,382.7.3Dynamics . . . . . . . . . . . . . . . . . . . . 385Momentum,385.Equivalenceofmassandenergy,388.Problems . . . . . . . . . . . . . . . . . . . . . . 394Exercises . . . . . . . . . . . . . . . . . . . . . . 3998Atoms and Electromagnetism8.1The Electric Glue. . . . . . . . . . . . . . . . . 403The quest for the atomic force, 404.Charge, electricity and magnetism,405.Atoms, 410.Quantizationof charge, 415.Theelectron,418.Theraisincookiemodeloftheatom,422.8.2The Nucleus. . . . . . . . . . . . . . . . . . . 425Radioactivity, 425.The planetary model of the atom, 428.Atomicnumber, 432.Thestructureof nuclei, 437.Thestrongnuclear force, alpha decayandssion, 440.The weaknuclearforce; beta decay, 443.Fusion, 446.Nuclear energy and bindingenergies, 448.Biological eectsof ionizingradiation, 449.Thecreationoftheelements,453.Problems . . . . . . . . . . . . . . . . . . . . . . 4559Circuits9.1Current and Voltage. . . . . . . . . . . . . . . . 460Current, 460.Circuits, 463.Voltage, 464.Resistance, 469.Current-conductingpropertiesofmaterials,476.9.2Parallel and Series Circuits. . . . . . . . . . . . . 480Schematics, 480.Parallel resistances and the junction rule, 481.Seriesresistances,485.Problems . . . . . . . . . . . . . . . . . . . . . . 492Exercises . . . . . . . . . . . . . . . . . . . . . . 499Contents 910Fields10.1Fields of Force. . . . . . . . . . . . . . . . . . 503Why elds?, 504.The gravitational eld, 505.The electric eld,509.10.2Voltage Related To Field . . . . . . . . . . . . . 513Onedimension,513.Twoorthreedimensions,516.10.3Fields by Superposition . . . . . . . . . . . . . . 518Electriceldof acontinuouschargedistribution, 518.Theeldnearachargedsurface,524.10.4Energy In Fields. . . . . . . . . . . . . . . . . 526Electric eld energy, 526.Gravitational eld energy, 531.Magneticeldenergy,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.Impedancesinseriesandparallel,558.10.6Fields by Gauss Law . . . . . . . . . . . . . . . 560Gausslaw,560.Additivityofux,564.Zerouxfromoutsidecharges, 564.Proof of Gauss theorem, 567.Gauss lawas afundamentallawofphysics,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.Nomagneticmonopoles, 599.Symmetryandhandedness,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.Aquick anddirty proof, 617.Maxwellsequationsforstaticelds,618.11.4Amp` eres Law In Differential Form (Optional) . . . . . 620Thecurloperator,620.Propertiesofthecurloperator,621.11.5Induced Electric Fields . . . . . . . . . . . . . . 626Faradays experiment, 626.Why induction?, 630.Faradays law,632.11.6Maxwells Equations . . . . . . . . . . . . . . . 637Inducedmagneticelds,637.Lightwaves,639.11.7Electromagnetic Properties of Materials . . . . . . . 649Conductors,649.Dielectrics,650.Magneticmaterials,652.Problems . . . . . . . . . . . . . . . . . . . . . . 659Exercises . . . . . . . . . . . . . . . . . . . . . . 67310 Contents12Optics12.1The Ray Model of Light . . . . . . . . . . . . . . 675Thenatureoflight,676.Interactionoflightwithmatter,679.The ray model of light, 681.Geometry of specular reection,684.Theprincipleofleasttimeforreection,688.12.2Images by Reection. . . . . . . . . . . . . . . 690A virtual image, 690.Curved mirrors, 693.A real image, 694.Imagesofimages,695.12.3Images, Quantitatively . . . . . . . . . . . . . . 699Areal imageformedbyaconvergingmirror, 699.Other caseswithcurvedmirrors,702.Aberrations,707.12.4Refraction. . . . . . . . . . . . . . . . . . . . 710Refraction,711.Lenses,717.Thelensmakersequation,719.Theprincipleofleasttimeforrefraction,720.12.5Wave Optics. . . . . . . . . . . . . . . . . . . 721Diraction, 721.Scaling of diraction, 723.The correspondenceprinciple, 723.Huygens principle, 724.Double-slitdiraction,725.Repetition, 729.Single-slit diraction, 730.The principleofleasttime,732.Problems . . . . . . . . . . . . . . . . . . . . . . 734Exercises . . . . . . . . . . . . . . . . . . . . . . 74913Quantum Physics13.1Rules of Randomness . . . . . . . . . . . . . . 759Randomnessisntrandom.,760.Calculatingrandomness,761.Probabilitydistributions, 765.Exponential decayandhalf-life,767.Applicationsofcalculus,772.13.2Light As a Particle . . . . . . . . . . . . . . . . 774Evidence for light as a particle, 775.How much light is one photon?,777.Wave-particleduality, 781.Photons inthreedimensions,786.13.3Matter As a Wave . . . . . . . . . . . . . . . . 787Electronsaswaves, 788.Dispersivewaves, 792.Boundstates,795.The uncertainty principle and measurement, 797.Electronsinelectricelds,803.TheSchrodingerequation,804.13.4The Atom. . . . . . . . . . . . . . . . . . . . 810Classifyingstates,810.Angularmomentuminthreedimensions,812.Thehydrogenatom, 813.Energiesof statesinhydrogen,816.Electronspin, 822.Atoms withmore thanone electron,824.Problems . . . . . . . . . . . . . . . . . . . . . . 827Exercises . . . . . . . . . . . . . . . . . . . . . . 836Appendix 1: Programming with Python 837Appendix 2: Miscellany 840Appendix 3: Photo Credits 848Appendix 4: Hints and Solutions 851Contents 11Appendix 5: Useful Data 872Notation and terminology, compared with other books, 872.Notationandunits, 873.Fundamental constants, 873.Metricprexes, 874.Nonmetric units, 874.The Greek alphabet, 874.Subatomicparticles, 874.Earth, moon, andsun, 875.Thepe-riodictable,875.Atomicmasses,875.Appendix 6: Summary 87612 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 ispullingharderonit?Howdoesthelensofyoureyework,andwhydo your eyes muscles need to squash its lens into dierent shapes inordertofocusonobjectsnearbyorfaraway? Thesearethekindsof questions that physics tries to answer about the behavior of lightandmatter,thetwothingsthattheuniverseismadeof.0.1.1 The scientic methodUntil very recently in history, no progress was made in answeringquestionslikethese. Worsethanthat, thewrong answerswrittenbythinkersliketheancientGreekphysicistAristotlewereacceptedwithoutquestionforthousandsof years. WhyisitthatscienticknowledgehasprogressedmoresincetheRenaissancethanithadinall theprecedingmillenniasincethebeginningof recordedhis-tory? Undoubtedlytheindustrial revolutionispartoftheanswer.13a / Science is a cycle of the-ory and experiment.b / A satirical drawing of analchemistslaboratory. H. Cock,after a drawing by Peter Brueghelthe Elder (16th century).Buildingitscenterpiece, thesteamengine, requiredimprovedtech-niquesforpreciseconstructionandmeasurement. (Earlyon,itwasconsidered a major advance when English machine shops learned tobuildpistons andcylinders that t together withagapnarrowerthanthethicknessofapenny.) Butevenbeforetheindustrialrev-olution,thepaceofdiscoveryhadpickedup,mainlybecauseoftheintroductionofthemodernscienticmethod. Althoughitevolvedovertime,mostscientiststodaywouldagreeonsomethinglikethefollowinglistofthebasicprinciplesofthescienticmethod:(1)Scienceisacycleof theoryandexperiment. Scienticthe-ories are createdtoexplainthe results of experiments that werecreated under certain conditions. A successful theory will also makenew predictions about new experiments under new conditions. Even-tually, though, italwaysseemstohappenthatanewexperimentcomes along, showingthat under certainconditions thetheoryisnotagoodapproximationorisnotvalidatall. Theball isthenbackinthetheorists court. If anexperiment disagrees withthecurrenttheory,thetheoryhastobechanged,nottheexperiment.(2) Theories should both predict and explain. The requirement ofpredictive power means that a theory is only meaningful if it predictssomethingthatcanbecheckedagainstexperimentalmeasurementsthatthetheoristdidnotalreadyhaveathand. Thatis, atheoryshould be testable. Explanatory value means that many phenomenashouldbeaccountedfor withfewbasicprinciples. If youanswerevery why question with because thats the way it is, then yourtheoryhas noexplanatoryvalue. Collectinglots of datawithoutbeingabletondanybasicunderlyingprinciplesisnotscience.(3)Experimentsshouldbereproducible. Anexperimentshouldbetreatedwithsuspicionif itonlyworksforoneperson, oronlyinone part of the world. Anyone withthe necessaryskills andequipment shouldbeabletoget thesameresults fromthesameexperiment. Thisimpliesthatsciencetranscendsnationalandeth-nic boundaries;you can be sure that nobody is doing actual sciencewhoclaimsthattheirworkisAryan, notJewish,Marxist, notbourgeois,orChristian,notatheistic.Anexperimentcannotbereproduced 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 elements couldnot betransformedintoeachother, e.g.,leadcouldnot be turnedintogold. This ledtothe theorythatchemical reactions consistedof rearrangements of theelements indierentcombinations, withoutanychangeintheidentitiesoftheelements themselves. The theoryworked for hundreds of years,andwas conrmedexperimentallyover awiderangeof pressures andtemperatures andwithmanycombinations of elements. Onlyin14 Chapter 0 Introduction and Reviewthe twentieth century did we learn that one element could be trans-formedinto one another under the conditions of extremelyhighpressure and temperature existing in a nuclear bomb or inside a star.That observation didnt completely invalidate the original theory oftheimmutabilityoftheelements,butitshowedthatitwasonlyanapproximation,validatordinarytemperaturesandpressures.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. 854Thescienticmethodasdescribedhereisanidealization, andshouldnotbeunderstoodasasetprocedurefordoingscience. Sci-entistshaveasmanyweaknessesandcharacterawsasanyothergroup,anditisverycommonforscientiststotrytodiscreditotherpeoplesexperimentswhentheresultsruncontrarytotheirownfa-voredpoint of view. Successful sciencealsohas moretodowithluck, intuition, andcreativitythanmost people realize, andtherestrictionsof thescienticmethoddonotstieindividualityandself-expressionanymore thanthe fugueandsonataforms stiedBachandHaydn. Thereisarecenttendencyamongsocial scien-tiststogoevenfurtherandtodenythatthescienticmethodevenexists, 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,itwouldseemdiculttoexplainitseectivenessinbuildingsuchuseful itemsasairplanes, CDplayers, andsewers. Ifalchemyandastrologywerenoless scienticintheir methods thanchem-istryandastronomy, whatwasitthatkeptthemfromproducinganythinguseful?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?Section 0.1 Introduction and Review 15B 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 meaningof 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?Givenfor oneinstant anintelligencewhichcouldcomprehendalltheforcesbywhichnatureisanimatedandtherespectivepositionsof thethingswhichcomposeit...nothingwouldbeuncertain, andthefutureasthepastwouldbelaidoutbeforeitseyes.PierreSimondeLaplacePhysicsistheuseofthescienticmethodtondoutthebasicprinciplesgoverninglightandmatter, andtodiscovertheimplica-tionsofthoselaws. Partofwhatdistinguishesthemodernoutlookfromtheancientmind-setistheassumptionthattherearerulesbywhich the universe functions, and that those laws can be at least par-tiallyunderstoodbyhumans. FromtheAgeofReasonthroughthenineteenthcentury,manyscientistsbegantobeconvincedthatthelawsofnaturenotonlycouldbeknownbut,asclaimedbyLaplace,thoselawscouldinprinciplebeusedtopredicteverythingabouttheuniverses futureif completeinformationwas availableaboutthepresent stateof all light andmatter. Insubsequent sections,Ill describe two general types of limitations on prediction using thelaws of physics, which were only recognized in the twentieth century.Matter canbedenedas anythingthat is aectedbygravity,i.e., thathasweightorwouldhaveweightifitwasneartheEarthor another star or planet massiveenoughtoproducemeasurablegravity. Lightcanbedenedasanythingthatcantravelfromoneplace to another through empty space and can inuence matter, buthasnoweight. Forexample, sunlightcaninuenceyourbodybyheatingitorbydamagingyourDNAandgivingyouskincancer.Thephysicistsdenitionof lightincludesavarietyof phenomenathatarenotvisibletotheeye, includingradiowaves, microwaves,x-rays, and gamma rays. These are the colors of light that do not16 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.happentofallwithinthenarrowviolet-to-redrangeoftherainbowthatwecansee.self-check BAt 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. 854Manyphysical phenomenaarenot themselveslight ormatter,butarepropertiesof lightormatterorinteractionsbetweenlightand matter. For instance, motion is a property of all light and somematter,butitisnotitselflightormatter. Thepressurethatkeepsabicycletireblownupis aninteractionbetweentheair andthetire. Pressure is not aformof matter inandof itself. It is asmuchapropertyof thetireasof theair. Analogously, sisterhoodandemploymentarerelationships amongpeoplebutarenotpeoplethemselves.Some things that appear weightless actually do have weight, andsoqualifyasmatter. Airhasweight, andisthusaformofmattereventhoughacubicinchofairweighslessthanagrainofsand. Ahelium balloon has weight, but is kept from falling by the force of thesurroundingmoredenseair, whichpushesuponit. AstronautsinorbitaroundtheEarthhaveweight,andarefallingalongacurvedarc, but theyaremovingsofast that thecurvedarcof their fallisbroadenoughtocarrythemall thewayaroundtheEarthinacircle. Theyperceivethemselvesasbeingweightlessbecausetheirspace capsule is falling along with them, and the oor therefore doesnotpushupontheirfeet.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 speedSection 0.1 Introduction and Review 17d / Reductionism.of 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 fundamentaldistinctionbetween 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.The boundarybetweenphysics andthe other sciences is notalways clear. For instance, chemists studyatoms andmolecules,whicharewhatmatterisbuiltfrom, andtherearesomescientistswhowouldbeequallywillingtocall themselvesphysical chemistsorchemical physicists. Itmightseemthatthedistinctionbetweenphysicsandbiologywouldbeclearer, sincephysicsseemstodealwithinanimateobjects. Infact, almostall physicistswouldagreethat the basic laws of physics that apply to molecules in a test tubeworkequallywellforthecombinationofmoleculesthatconstitutesabacterium. (Somemightbelievethatsomethingmorehappensinthemindsofhumans,oreventhoseofcatsanddogs.) Whatdier-entiatesphysicsfrombiologyisthatmanyofthescientictheoriesthatdescribelivingthings,whileultimatelyresultingfromthefun-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 wantstostudythemotionofarotatinggyroscopewouldprobablypreferthat it be isolated from vibrations and air currents. Even in biology,where eld work is indispensable for understanding how living thingsrelatetotheirentireenvironment,itisinterestingtonotethevitalhistorical roleplayedbyDarwinsstudyof theGalapagosIslands,whichwereconvenientlyisolatedfromtherestof theworld. Anypartof theuniversethatisconsideredapartfromtherestcanbecalledasystem.Physicshashadsomeof itsgreatestsuccessesbycarryingthisprocess of isolation to extremes, subdividing the universe into smallerandsmallerparts. Mattercanbedividedintoatoms, andthebe-havior of individual atoms can be studied. Atoms can be split apartintotheirconstituentneutrons,protonsandelectrons. Protonsandneutrons appear tobe made out of evensmaller particles calledquarks, andtherehaveevenbeensomeclaimsof experimental ev-idencethat quarks havesmaller parts insidethem. This method18 Chapter 0 Introduction and Reviewofsplittingthingsintosmallerandsmallerpartsandstudyinghowthosepartsinuenceeachotheriscalledreductionism. Thehopeisthattheseeminglycomplexrulesgoverningthelargerunitscanbebetterunderstoodintermsof simplerrulesgoverningthesmallerunits. Toappreciatewhatreductionismhasdoneforscience, itisonlynecessarytoexaminea19th-centurychemistrytextbook. Atthattime, theexistenceof atomswasstill doubtedbysome, elec-trons were not evensuspectedtoexist, andalmost nothingwasunderstoodofwhatbasicrulesgovernedthewayatomsinteractedwitheachotherinchemical reactions. Studentshadtomemorizelonglistsofchemicalsandtheirreactions,andtherewasnowaytounderstandanyofitsystematically. Today,thestudentonlyneedstorememberasmall setofrulesabouthowatomsinteract, forin-stancethatatomsofoneelementcannotbeconvertedintoanotherviachemicalreactions,orthatatomsfromtherightsideofthepe-riodic table tendtoformstrongbonds withatoms fromthe leftside.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 physicsForasknowledgesarenowdelivered, thereisakindofcontractoferrorbetweenthedelivererandthereceiver; forhethatdeliverethknowledgedesirethtodeliveritinsuchaformasmaybebestbe-lieved, andnot as maybe best examined; andhe that receivethknowledge desirethrather present satisfactionthanexpectant in-quiry.FrancisBaconMany students approach a science course with the idea that theycansucceedbymemorizingtheformulas, sothatwhenaproblemisassignedonthehomeworkoranexam,theywillbeabletoplugnumbersintotheformulaandgetanumerical resultontheircal-culator. Wrong! Thatsnotwhatlearningscienceisabout! Thereisabigdierencebetweenmemorizingformulasandunderstandingconcepts. Tostartwith, dierentformulasmayapplyindierentsituations. Oneequationmightrepresentadenition, whichisal-waystrue. Anothermightbeaveryspecicequationforthespeedof an object sliding down an inclined plane, which would not be trueSection 0.1 Introduction and Review 19iftheobjectwasarockdriftingdowntothebottomoftheocean.Ifyoudontworktounderstandphysicsonaconceptuallevel,youwontknowwhichformulascanbeusedwhen.Most students takingcollegesciencecourses for therst timealsohaveverylittleexperiencewithinterpretingthemeaningofanequation. Considertheequationn=,/relatingthewidthof arectangletoitsheightandarea. Astudentwhohasnotdevelopedskill at interpretationmight viewthis as yet another equationtomemorizeandplugintowhenneeded. Aslightlymoresavvystu-dent might realizethat it issimplythefamiliarformula=n/inadierent form. Whenaskedwhether arectanglewouldhaveagreater or smaller widththananother withthe same areabutasmaller height, the unsophisticatedstudent might be at aloss,nothavinganynumberstopluginonacalculator. Themoreex-periencedstudent wouldknowhowtoreasonabout anequationinvolvingdivisionif/issmaller, andstaysthesame, thennmust be bigger. Often, students fail to recognize a sequence of equa-tionsasaderivationleadingtoanal result, sotheythinkall theintermediatestepsareequallyimportantformulasthattheyshouldmemorize.Whenlearninganysubjectatall, itisimportanttobecomeasactivelyinvolvedas possible, rather thantryingtoreadthroughall theinformationquicklywithoutthinkingaboutit. Itisagoodidea to read and think about the questions posed at the end of eachsection of these notes as you encounter them,so that you know youhaveunderstoodwhatyouwerereading.Manystudents dicultiesinphysicsboil downmainlytodi-culties with math. Suppose you feel condent that you have enoughmathematical preparationtosucceedinthis course, but youarehavingtroublewithafewspecicthings. Insomeareas, thebriefreviewgiveninthischaptermaybesucient, butinotherareasitprobablywillnot. Onceyouidentifytheareasofmathinwhichyouarehavingproblems,gethelpinthoseareas. Dontlimpalongthrough the whole course with a vague feeling of dread about some-thinglikescienticnotation. Theproblemwill notgoawayifyouignore it. The same applies to essential mathematical skills that youare learning in this course for the rst time, such as vector addition.Sometimes students tell metheykeeptryingtounderstandacertain topic in the book, and it just doesnt make sense. The worstthingyoucanpossiblydointhat situationis tokeeponstaringatthesamepage. Everytextbookexplainscertainthingsbadlyevenmine! sothebestthingtodointhissituationistolookatadierentbook. Insteadofcollegetextbooksaimedatthesamemathematical level as thecourseyouretaking, youmayinsomecases ndthat highschool books or books at alower mathlevelgiveclearerexplanations.20 Chapter 0 Introduction and ReviewFinally, whenreviewingfor anexam, dont simplyreadbackoverthetextandyourlecturenotes. Instead, trytouseanactivemethodofreviewing,forinstancebydiscussingsomeofthediscus-sionquestionswithanotherstudent, ordoinghomeworkproblemsyouhadntdonethersttime.0.1.4 Velocity and accelerationCalculus was inventedbyaphysicist, Isaac Newton, becauseheneededitasatool forcalculatingvelocityandacceleration; inyour introductorycalculus course, velocityandaccelerationwereprobablypresentedassomeoftherstapplications.If an objects position as a function of time is given by the func-tionr(t), thenitsvelocityandaccelerationaregivenbytherstandsecondderivativeswithrespecttotime,=drdtando =d2rdt2.Thenotationrelatesinalogicalwaytotheunitsofthequantities.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-inglyweirdandinconsistentplacementofthesuperscriptedtwosinthenotationfor theaccelerationis likewisemeant tosuggest theunits: something on top with units of meters, and something on thebottomwithunitsofsecondssquared.Velocityandaccelerationhavecompletelydierentphysical in-terpretations. Velocityisamatterofopinion. Rightnowasyousitinachairandreadthisbook,youcouldsaythatyourvelocitywaszero, but an observer watching the Earth rotate would say that youhad a velocity of hundreds of miles an hour. Acceleration representsa changein velocity, and its not a matter of opinion. Accelerationsproducephysical eects, anddontoccurunlesstheresaforcetocause them. For example, gravitational forces on Earth cause fallingobjectstohaveanaccelerationof9.8m,s2.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 m,s2.We need to nd a connection between the distance she travelsandtimeittakes. Inotherwords, werelookingforinformationSection 0.1 Introduction and Review 21about the function x(t ), given information about the acceleration.To 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 m,s2_(1.0 s)2= 4.9 mNote that when we put in the numbers,we check that the unitswork out correctly, _m,s2_(s)2=m. We should also check thatthe result makes sense:4.9 meters is pretty high, but not unrea-sonable.Thenotationdincalculusrepresentsaninnitesimallysmallchangeinthevariable. Thecorrespondingnotationforanitechangeinavariableis. Forexample, if representsthevalueof acertainstockonthe stockmarket, andthe value falls fromo=5dollars initiallytof=3dollars nally, then = 2dollars. When we study linear functions,whose slopes are constant,thederivativeissynonymouswiththeslopeoftheline,anddj,dristhesamethingasj,r,theriseovertherun.Under conditions of constant acceleration, we can relate velocityandtime,o =t,or,asintheexample1,positionandtime,r =12ot2+ ot + ro.22 Chapter 0 Introduction and ReviewItcanalsobehandytohavearelationinvolvingvelocityandposi-tion,eliminatingtime. Straightforwardalgebragives2f= 2o + 2or ,wherefis thenal velocity, otheinitial velocity, andrthedistancetraveled.> Solved problem: Dropping a rock on Mars page 48, problem 17> Solved problem: The Dodge Viper page 49, 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. OnestudentmayhavegrownupoutsidetheU.S.andsomaybecompletelycomfortablewiththemetricsystem, butmayhavehadanalgebracourseinwhichtheinstructor passedtooquicklyoverscienticnotation. Anotherstudentmayhavealreadytakenvector calculus, but may have never learned the metric system. Thefollowingself-evaluationisachecklisttohelpyougureoutwhatyouneedtostudytobepreparedfortherestofthecourse.If youdisagreewiththisstate-ment. . .youshouldstudythissection:Iamfamiliarwiththebasicmetricunitsofmeters, kilograms, andsec-onds,andthemostcommonmetricprexes: milli- (m), kilo- (k), andcenti-(c).subsection0.1.6BasicoftheMetricSystemI amfamiliar withthese less com-mon metric prexes: mega- (M),micro-(j),andnano-(n).subsection0.1.7LessCommonMet-ricPrexesIamcomfortablewithscienticno-tation.subsection0.1.8ScienticNotationI cancondentlydometricconver-sions.subsection0.1.9ConversionsI understand the purpose and use ofsignicantgures.subsection 0.1.10 Signicant FiguresIt wouldnt hurt youtoskimthe sections youthinkyoualreadyknowabout,andtodotheself-checksinthosesections.Section 0.1 Introduction and Review 230.1.6 Basics of the metric systemThemetricsystemUnitswerenotstandardizeduntil fairlyrecentlyinhistory, sowhenthephysicistIsaacNewtongavetheresultof anexperimentwithapendulum,hehadtospecifynotjustthatthestringwas377,8incheslongbutthatitwas377,8Londonincheslong. TheinchasdenedinYorkshirewouldhavebeendierent. EvenaftertheBritishEmpirestandardizeditsunits,itwasstillveryinconve-nient to do calculations involving money,volume,distance,time,orweight, becauseofall theoddconversionfactors, like16ouncesinapound,and5280feetinamile. Throughthenineteenthcentury,schoolchildrensquanderedmostoftheirmathematicaleducationinpreparing to do calculations such as making change when a customerinashopoeredaone-crownnoteforabookcostingtwopounds,thirteen shillings and tuppence. The dollar has always been decimal,and British money went decimal decades ago, but the United Statesisstill saddledwiththeantiquatedsystemof feet, inches, pounds,ouncesandsoon.EverycountryintheworldbesidestheU.S.hasadoptedasys-tem ofunits knownin English as the metric system.This systemis entirely decimal, thanks to the same eminently logical people whobroughtabouttheFrenchRevolution. IndeferencetoFrance, thesystemsocialnameistheSyst`emeInternational,orSI,meaningInternational System. (ThephraseSIsystemisthereforeredun-dant.)The wonderful thingabout the SI is that people wholive incountries more modernthanours donot needtomemorize howmanyouncesthereareinapound, howmanycupsinapint, howmanyfeet inamile, etc. Thewholesystemworks withasingle,consistent set of prexes (derived from Greek) that modify the basicunits. Each prex stands for a power of ten, and has an abbreviationthatcanbecombinedwiththesymbol fortheunit. Forinstance,themeterisaunitofdistance. Theprexkilo-standsfor103,soakilometer,1km,isathousandmeters.Thebasicunitsofthemetricsystemarethemeterfordistance,thesecondfortime,andthegramformass.The following are the most common metric prexes. You shouldmemorizethem.prex meaning examplekilo- k 10360kg =apersonsmasscenti- c 10228cm =heightofapieceofpapermilli- m 1031ms =timeforonevibrationofaguitarstringplayingthenoteDTheprexcenti-,meaning102,isonlyusedinthecentimeter;ahundredthofagramwouldnotbewrittenas1cgbutas10mg.24 Chapter 0 Introduction and ReviewThecenti-prexcanbeeasilyrememberedbecauseacentis102dollars. Theocial SIabbreviationforsecondsiss(notsec)andgramsareg(notgm).ThesecondThesunstoodstillandthemoonhalteduntilthenationhadtakenvengeanceonitsenemies. . .Joshua10:12-14Absolute, true, andmathematical time, ofitself, andfromitsownnature,owsequablywithoutrelationtoanythingexternal. . .IsaacNewtonWhenIstatedbrieyabovethatthesecondwasaunitoftime,itmaynothaveoccurredtoyouthatthiswasnotreallymuchofadenition. Thetwoquotesabovearemeanttodemonstratehowmuch room for confusion exists among people who seem to mean thesame thing by a word such as time.The rst quote has been inter-pretedbysomebiblicalscholarsasindicatinganancientbeliefthatthemotionof thesunacrosstheskywasnotjustsomethingthatoccurredwiththepassageoftimebutthatthesunactuallycausedtime to pass by its motion, so that freezing it in the sky would havesomekindofasupernatural deceleratingeectoneveryoneexcepttheHebrewsoldiers. Manyancientculturesalsoconceivedoftimeascyclical, ratherthanproceedingalongastraightlineasin1998,1999, 2000, 2001,... Thesecondquote, fromarelativelymodernphysicist, maysoundalotmorescientic, butmostphysiciststo-daywouldconsider it useless as adenitionof time. Today, thephysicalsciencesarebasedonoperationaldenitions,whichmeansdenitionsthatspell outtheactual steps(operations)requiredtomeasuresomethingnumerically.Now in an era when our toasters, pens, and coee pots tell us thetime,itisfarfromobvioustomostpeoplewhatisthefundamentaloperational denition of time. Until recently, the hour, minute, andsecondweredenedoperationallyintermsofthetimerequiredfortheearthtorotateabout itsaxis. Unfortunately, theEarthsro-tationisslowingdownslightly, andby1967thiswasbecominganissueinscienticexperimentsrequiringprecisetimemeasurements.Thesecondwasthereforeredenedasthetimerequiredforacer-tainnumberof vibrationsof thelightwavesemittedbyacesiumatomsinalampconstructedlikeafamiliarneonsignbutwiththeneonreplacedbycesium. Thenewdenitionnotonlypromisestostayconstant indenitely, but for scientists is amore convenientwayof calibratingaclockthanhavingtocarryout astronomicalmeasurements.self-check CWhat is a possible operational denition of how strong a person is? >Section 0.1 Introduction and Review 25e / The original denition ofthe meter.Answer, p. 854ThemeterTheFrenchoriginallydenedthemeteras107timesthedis-tance from the equator to the north pole, as measured through Paris(ofcourse). Evenifthedenitionwasoperational,theoperationoftraveling to the north pole and laying a surveying chain behind youwas not one that most working scientists wanted to carry out. Fairlysoon, astandardwascreatedintheformof ametal barwithtwoscratchesonit. Thiswasreplacedbyanatomicstandardin1960,andnallyin1983bythecurrentdenition, whichisthattheme-teristhedistancetraveledbylightinavacuumoveraperiodof(1/299792458)seconds.ThekilogramThethirdbaseunit of theSI isthekilogram, aunit of mass.Mass is intendedtobeameasureof the amount of asubstance,butthatisnotanoperationaldenition. Bathroomscalesworkbymeasuringourplanetsgravitationalattractionfortheobjectbeingweighed, butusingthattypeof scaletodenemassoperationallywouldbeundesirablebecausegravityvariesinstrengthfromplacetoplaceontheearth.Theres a surprising amount of disagreement among physics text-books about how mass should be dened, but heres how its actuallyhandledbythefewworkingphysicistswhospecializeinultra-high-precisionmeasurements. TheymaintainaphysicalobjectinParis,which is the standard kilogram, a cylinder made of platinum-iridiumalloy. Duplicatesarecheckedagainstthismotherof all kilogramsbyputtingtheoriginalandthecopyonthetwooppositepansofabalance. Althoughthismethodof comparisondependsongravity,theproblemsassociatedwithdierencesingravityindierentgeo-graphicallocationsarebypassed,becausethetwoobjectsarebeingcomparedinthesameplace. TheduplicatescanthenberemovedfromtheParisiankilogramshrineandtransportedelsewhereintheworld. Itwouldbedesirabletoreplacethisatsomepointwithauniversallyaccessibleatomicstandardratherthanonebasedonaspecic artifact, but as of 2010 the technology for automated count-ingoflargenumbersofatomshasnotgottengoodenoughtomakethatworkwiththedesiredprecision.CombinationsofmetricunitsJust about anything you want to measure can be measured withsomecombinationofmeters,kilograms,andseconds. Speedcanbemeasured in m/s, volume in m3, and density in kg,m3. Part of whatmakes the SI great is this basic simplicity. No more funny units likeacordof wood, abolt of cloth, or ajigger of whiskey. Nomoreliquidanddrymeasure. Justasimple,consistentsetofunits. TheSI measures put together from meters, kilograms, and seconds make26 Chapter 0 Introduction and Reviewupthemkssystem. Forexample,themksunitofspeedism/s,notkm/hr.CheckingunitsAuseful technique for ndingmistakes inones algebrais toanalyzetheunitsassociatedwiththevariables.Checking units example 2> Jae starts from the formula V=13Ah for the volume of a cone,where A is the area of its base, and h is its height. He wants tond an equation that will tell him how tall a conical tent has to bein order to have a certain volume,given its radius. His algebragoes like this:V=13Ah [1]A = r2[2]V=13r2h [3]h =r23V[4]Is his algebra correct? If not, nd the mistake.> Line 4 is supposed to be an equation for the height, so the unitsof the expression on the right-hand side had better equal meters.The pi and the 3 are unitless, so we can ignore them. In terms ofunits, line 4 becomesm =m2m3=1m.This is false, so there must be a mistake in the algebra. The unitsof lines 1, 2, and 3 check out, so the mistake must be in the stepfrom line 3 to line 4. In fact the result should have beenh =3Vr2.Now the units check: m = m3,m2.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, wherebytime 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?Section 0.1 Introduction and Review 27f / This is a mnemonic to help youremember the most importantmetricprexes. Thewordlittleis to remind you that the list startswiththeprexesusedfor smallquantities and builds upward.The exponent changes by 3,exceptthatofcoursethatwedonot need a special prex for 100,which equals one.0.1.7 Less common metric prexesThe following are three metric prexes which, while less commonthantheonesdiscussedpreviously,arewellworthmemorizing.prex meaning examplemega- M 1066.4Mm =radiusoftheearthmicro- j 10610jm =sizeofawhitebloodcellnano- n 1090.154nm =distance between carbonnucleiinanethanemoleculeNotethattheabbreviationformicroistheGreeklettermu, jacommonmistakeistoconfuseitwithm(milli)orM(mega).Thereareotherprexesevenlesscommon, usedforextremelylarge and small quantities. For instance, 1 femtometer = 1015m isaconvenientunitof distanceinnuclearphysics, and1gigabyte=109bytesisusedforcomputersharddisks. Theinternationalcom-mitteethatmakesdecisionsabouttheSIhasrecentlyevenaddedsome new prexes that sound like jokes, e.g., 1 yoctogram = 1024gisabouthalf themassof aproton. Intheimmediatefuture, how-ever,youreunlikelytoseeprexeslikeyocto-andzepto-usedexcept perhaps intriviacontests at science-ctionconventions orothergeekfests.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. 8540.1.8 Scientic notationMostof theinterestingphenomenainouruniversearenotonthe human scale. It would take about 1,000,000,000,000,000,000,000bacteriatoequal themassof ahumanbody. WhenthephysicistThomasYoungdiscoveredthatlightwasawave,itwasbackinthebadolddaysbeforescienticnotation,andhewasobligedtowritethatthetimerequiredforonevibrationof thewavewas1/500ofamillionthof amillionthof asecond. Scienticnotationisalessawkwardwaytowriteverylargeandverysmall numberssuchasthese. Heresaquickreview.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.Forinstance,32 = 3.2 101320 = 3.2 1023200 = 3.2 103. . .Eachnumberistentimesbiggerthanthepreviousone.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 exponentsareusedforsmallnumbers:3.2 = 3.2 1000.32 = 3.2 1010.032 = 3.2 102. . .Acommonsourceof confusionisthenotationusedonthedis-playsofmanycalculators. Examples:3.2 106(writtennotation)3.2E+6 (notationonsomecalculators)3.26(notationonsomeothercalculators)The last example is particularlyunfortunate, because 3.26reallystandsforthenumber3.23.23.23.23.23.2=1074, atotallydierentnumberfrom3.2106=3200000. Thecalculatornotationshouldnever beusedinwriting. Itsjust awayfor themanufacturertosavemoneybymakingasimplerdisplay.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. 8540.1.9 ConversionsI suggest youavoidmemorizinglots of conversionfactors be-tweenSIunitsandU.S.units,buttwothatdocomeinhandyare:1inch=2.54cmAnobjectwithaweightonEarthof 2.2pounds-forcehasamassof1kg.Therstoneisthepresentdenitionoftheinch,soitsexact. Thesecond one is not exact, but is good enough for most purposes. (U.S.unitsof forceandmassareconfusing, soitsagoodthingtheyrenotusedinscience. InU.S. units, theunitof forceisthepound-force, andthebestunittouseformassistheslug, whichisabout14.6kg.)Section 0.1 Introduction and Review 29More important thanmemorizingconversionfactors is under-standingtherightmethodfordoingconversions. EvenwithintheSI,youmayneedtoconvert, say, fromgramstokilograms. Dier-entpeoplehavedierentwaysof thinkingaboutconversions, butthemethodIlldescribehereissystematicandeasytounderstand.Theideaisthatif1kgand1000grepresentthesamemass, thenwecanconsiderafractionlike103g1kgto be a way of expressing the number one. This may bother you. Forinstance,ifyoutype1000/1intoyourcalculator,youwillget1000,not one. Again, dierent people have dierent ways of thinkingaboutit, butthejusticationisthatithelpsustodoconversions,anditworks! Nowifwewanttoconvert0.7kgtounitsofgrams,wecanmultiplykgbythenumberone:0.7kg 103g1kgIfyourewillingtotreatsymbolssuchaskgasiftheywerevari-ables as usedinalgebra(whichtheyrereallynot), youcanthencancelthekgontopwiththekgonthebottom,resultingin0.7

kg 103g1

kg= 700g .Toconvertgramstokilograms, youwouldsimplyipthefractionupsidedown.One advantage of this method is that it can easily be applied toaseriesofconversions. Forinstance,toconvertoneyeartounitsofseconds,1$$$year 365days1$$$year

24 $$$hours1day

60min1 $$$hour 60s1min== 3.15 107s .Shouldthatexponentbepositive,ornegative?A common mistake is to write the conversion fraction incorrectly.Forinstancethefraction103kg1g(incorrect)doesnotequalone, because103kgisthemassofacar, and1gisthemassofaraisin. Onecorrectwayofsettinguptheconversionfactorwouldbe103kg1g(correct) .30 Chapter 0 Introduction and ReviewYoucanusuallydetectsuchamistakeifyoutakethetimetocheckyouranswerandseeifitisreasonable.Ifcommonsensedoesntruleouteitherapositiveoranegativeexponent, heresanotherwaytomakesureyougetitright. Therearebigprexesandsmallprexes:bigprexes: k Msmallprexes: m j n(Itsnothardtokeepstraightwhicharewhich, sincemegaandmicroareevocative, anditseasytorememberthatakilometerisbiggerthanameterandamillimeterissmaller.) Intheexampleabove, we want the top of the fraction to be the same as the bottom.Since/isabigprex, weneedtocompensate byputtingasmallnumberlike103infrontofit,notabignumberlike103.> 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.0.1.10 Signicant guresAn engineer is designing a car engine, and has been told that thediameterofthepistons(whicharebeingdesignedbysomeoneelse)is 5 cm. He knows that 0.02 cm of clearance is required for a pistonofthissize,sohedesignsthecylindertohaveaninsidediameterof5.04cm. Luckily, hissupervisorcatcheshismistakebeforethecargoesintoproduction. Sheexplainshiserrortohim, andmentallyputshiminthedonotpromotecategory.Whatwashismistake? Thepersonwhotoldhimthepistonswere5cmindiameterwaswisetothewaysof signicantgures,aswashisboss, whoexplainedtohimthatheneededtogobackandget amoreaccuratenumber for thediameter of thepistons.That person said 5 cm rather than 5.00 cm specically to avoidcreating the impression that the number was extremely accurate. Inreality, thepistonsdiameterwas5.13cm. Theywouldneverhavetinthe5.04-cmcylinders.Thenumberofdigitsofaccuracyinanumberisreferredtoasthenumberofsignicantgures, orsiggsforshort. Asintheexampleabove,siggsprovideawayofshowingtheaccuracyofanumber. Inmostcases,theresultofacalculationinvolvingseveralSection 0.1 Introduction and Review 31pieces of data can be no more accurate than the least accurate pieceof data. Inother words, garbagein, garbageout. Sincethe5cmdiameterofthepistonswasnotveryaccurate,theresultoftheengineers calculation, 5.04cm, was reallynot as accurate as hethought. Ingeneral, your result shouldnot have more thanthenumber of siggs intheleast accuratepieceof datayoustartedwith. Thecalculationaboveshouldhavebeendoneasfollows:5cm (1sigg)+0.04cm (1sigg)=5cm (roundedoto1sigg)Thefact that thenal result onlyhas onesignicant gurethenalerts you to the fact that the result is not very accurate, and wouldnotbeappropriateforuseindesigningtheengine.Notethattheleadingzeroesinthenumber0.04donotcountassignicantgures, becausetheyareonlyplaceholders. Ontheother hand, a number such as 50 cm is ambiguous the zero couldbeintendedas asignicant gure, or it might just bethereas aplaceholder. The ambiguity involving trailing zeroes can be avoidedbyusingscienticnotation, inwhich5101cmwouldimplyonesiggofaccuracy,while5.0 101cmwouldimplytwosiggs.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. 854Dealingcorrectlywithsignicantgurescansaveyoutime! Of-ten, studentscopydownnumbersfromtheircalculatorswitheightsignicantguresof precision, thentypethembackinforalatercalculation. Thatsawasteof time, unlessyouroriginal datahadthatkindofincredibleprecision.Therulesaboutsignicantguresareonlyrulesofthumb,andarenot asubstitutefor careful thinking. For instance, $20.00+$0.05is$20.05. Itneednotandshouldnotberoundedoto$20.In general, the sig g rules work best for multiplication and division,andwealsoapplythemwhendoingacomplicatedcalculationthatinvolves many types of operations. For simple addition and subtrac-tion,itmakesmoresensetomaintainaxednumberofdigitsafterthedecimalpoint.Whenindoubt, dontusethesiggrulesatall. Instead, in-32 Chapter 0 Introduction and Reviewtentionallychangeonepieceof yourinitial databythemaximumamountbywhichyouthinkitcouldhavebeeno, andrecalculatethe nal result. The digits on the end that are completely reshuedaretheonesthataremeaningless,andshouldbeomitted.self-check GHowmanysignicantguresarethereineachofthefollowingmea-surements?(1) 9.937 m(2) 4.0 s(3) 0.0000000000000037 kg > Answer, p. 854Section 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 youreallyneedisthehumbleconversion,appliedtoareaandvolume.AreaandvolumeAreacanbedenedbysayingthat wecancopytheshapeofinterestontographpaperwith1cm 1cmsquaresandcountthenumberofsquaresinside. Fractionsofsquarescanbeestimatedbyeye. Wethensaytheareaequalsthenumberofsquares,inunitsofsquarecm. Althoughthismightseemlesspurethancomputingareasusingformulaelike=:2foracircleor=n/,2foratriangle, those formulae are not useful as denitions of area becausetheycannotbeappliedtoirregularlyshapedareas.Units of square cm are more commonly written as cm2in science.Of course, theunitof measurementsymbolizedbycmisnotanalgebra symbol standing for a number that can be literally multipliedbyitself. Butitisadvantageoustowritetheunitsofareathatwayandtreattheunitsasif theywerealgebrasymbols. Forinstance,if youhavearectanglewithanareaof 6m2andawidthof 2m,thencalculatingits lengthas (6m2),(2m) =3mgives aresultthat makes sense bothnumericallyandinterms of units. Thisalgebra-styletreatmentoftheunitsalsoensuresthatourmethodsof convertingunitsworkoutcorrectly. Forinstance, if weacceptthefraction100cm1mas a valid way of writing the number one, then one times one equalsone,soweshouldalsosaythatonecanberepresentedby100cm1m

100cm1m,whichisthesameas10000cm21m2.That means the conversion factor from square meters to square cen-timetersisafactorof104, i.e., asquaremeterhas104squarecen-timetersinit.All oftheabovecanbeeasilyappliedtovolumeaswell, usingone-cubic-centimeterblocksinsteadofsquaresongraphpaper.Tomanypeople, itseemshardtobelievethatasquaremeterequals 10000square centimeters, or that acubic meter 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. Theexamplesshowningurebaimtomakethecorrectanswermorebelievable,usingthetraditionalU.S.unitsoffeetandyards. (Onefootis12inches,andoneyardisthreefeet.)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.854> 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 volumeGreateashavelessereasUpontheirbackstobiteem.Andlessereashavelesserstill,Andsoadinnitum.JonathanSwiftNowhowdotheseconversionsofareaandvolumerelatetothequestionsIposedaboutsizesof livingthings? Well, imaginethatyouareshrunklikeAliceinWonderlandtothesizeof aninsect.Onewayof thinkingaboutthechangeof scaleisthatwhatusedtolooklikeacentimeter nowlooks likeperhaps ameter toyou,because youre so much smaller. If area and volume scaled accordingtomostpeoplesintuitive, incorrectexpectations, with1m2beingthe same as 100 cm2, thenthere wouldbe no particular reasonwhynature shouldbehave anydierentlyonyour new, reducedscale. But naturedoes behavedierentlynowthat youresmall.Forinstance, youwill ndthatyoucanwalkonwater, andjumpSection 0.2 Scaling and Order-of-Magnitude Estimates 35c / Galileo Galilei (1564-1642).d / 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.tomanytimesyourownheight. ThephysicistGalileoGalilei hadthebasicinsight that thescalingof areaandvolumedetermineshownatural phenomenabehavedierentlyondierentscales. Herstreasonedaboutmechanical structures, butlaterextendedhisinsights to living things, taking the then-radical point of view that atthe fundamental level, a living organism should follow the same lawsof natureasamachine. Wewill followhisleadbyrstdiscussingmachinesandthenlivingthings.GalileoonthebehaviorofnatureonlargeandsmallscalesOne of the worlds most famous pieces of scientic writingisGalileosDialoguesConcerningtheTwoNewSciences. Galileowasan 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 dialogueamongthreepeople. SalviatiisreallyGalileosalterego. Simpliciois the stupid character, and one of the reasons Galileo got in troublewiththeChurchwasthattherewererumorsthatSimpliciorepre-sented the Pope. Sagredo is the earnest and intelligent student, withwhomthereaderissupposedtoidentify. (Thefollowingexcerptsarefromthe1914translationbyCrewanddeSalvio.)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.SalviaticontradictsSagredo: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 givenoperationaldenitionsof thingslikestrength,i.e., denitionsthatspell outhowtomeasurethemnumerically.Also, acat is shapeddierentlyfromahorseanenlargedphotographofacatwouldnotbemistakenforahorse,evenifthephoto-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 wholeargumentwouldbealotmoreconvincingifwecoulddosomeiso-lationofvariables,ascientictermthatmeanstochangeonlyonething at a time, isolating it from the other variables that might haveaneect. Ifsizeisthevariablewhoseeectwereinterestedinsee-ing,thenwedontreallywanttocomparethingsthataredierentinsizebutalsodierentinotherways.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?Afterthisentertainingbutnotscienticallyrigorousbeginning,Galileostarts todosomethingworthwhile bymodernstandards.Hesimplies everythingbyconsideringthestrengthof awoodenplank. Thevariablesinvolvedcanthenbenarroweddowntothetypeof wood, thewidth, thethickness, andthelength. Healsogives anoperational denitionof what it means for theplanktohaveacertainstrengthinproportiontoitssize,byintroducingtheconceptofaplankthatisthelongestonethatwouldnotsnapunder its ownweight if supportedat one end. If youincreaseditslengthbytheslightestamount, withoutincreasingitswidthorthickness, it wouldbreak. Hesays that if oneplankis thesameshapeasanotherbutadierentsize, appearinglikeareducedorenlargedphotographoftheother,thentheplankswouldbestrongin proportion to their sizes if both were just barely able to supporttheirownweight.Also, Galileoisdoingsomethingthatwouldbefrownedoninmodernscience: heismixingexperimentswhoseresultshehasac-tually observed (building boats of dierent sizes),with experimentsthat hecouldnot possiblyhavedone(droppinganant fromtheheight of themoon). Henowrelates howhehas doneactual ex-perimentswithsuchplanks, andfoundthat, accordingtothisop-erationaldenition,theyarenotstronginproportiontotheirsizes.Thelargeronebreaks. Hemakessuretotellthereaderhowimpor-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.tanttheresultis,viaSagredosastonishedresponse: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, thatnatureactsdierentlyondierentscales.Tonishthediscussion, Galileogivesanexplanation. Hesaysthat the strength of a plank (dened as, say, the weight of the heav-iestboulderyoucouldputontheendwithoutbreakingit)ispro-portionaltoitscross-sectionalarea,thatis,thesurfaceareaofthefreshwoodthatwouldbeexposedif yousawedthroughitinthemiddle. Itsweight,however,isproportionaltoitsvolume.1Howdothevolumeandcross-sectionalareaofthelongerplankcomparewiththoseof theshorter plank? Wehavealreadyseen,whilediscussingconversionsof theunitsof areaandvolume, thatthesequantitiesdontactthewaymostpeoplenaivelyexpect. Youmight think that the volume and area of the longer plank would bothbedoubledcomparedtotheshorterplank, sotheywouldincreaseinproportiontoeachother,andthelongerplankwouldbeequallyabletosupportitsweight. Youwouldbewrong,butGalileoknowsthat this is a common misconception, so he has Salviati address thepointspecically: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.Thesamereasoningapplies totheplanks. Eventhoughtheyarenotcubes, thelargeonecouldbesawedintoeightsmall ones,eachwithhalf thelength, half thethickness, andhalf thewidth.Thesmall plank, therefore, hasmoresurfaceareainproportiontoits weight,and is therefore able to support its own weight while thelargeonebreaks.ScalingofareaandvolumeforirregularlyshapedobjectsYouprobablyarenotgoingtobelieveGalileosclaimthatthishasdeepimplicationsforallofnatureunlessyoucanbeconvincedthatthesameistrueforanyshape. Everydrawingyouveseensofarhasbeenofsquares, rectangles, andrectangularsolids. Clearlythereasoningaboutsawingthingsupintosmallerpieceswouldnotprove anything about, say, an egg, which cannot be cut up into eightsmalleregg-shapedobjectswithhalfthelength.Isit alwaystruethat somethinghalf thesizehasonequarterthesurfaceareaandoneeighththevolume,evenifithasanirreg-ularshape? Taketheexampleofachildsviolin. Violinsaremadeforsmallchildreninsmallersizetoaccomodatetheirsmallbodies.Figurei showsafull-sizeviolin, alongwithtwoviolinsmadewithhalfand3/4ofthenormallength.2Letsstudythesurfaceareaofthefrontpanelsofthethreeviolins.Considerthesquareintheinteriorof thepanel of thefull-sizeviolin. Inthe3/4-sizeviolin,itsheightandwidtharebothsmallerbyafactorof3/4,sotheareaofthecorresponding,smallersquarebecomes 3,43,4 = 9,16 of the original area, not 3/4 of the originalarea. Similarly,thecorrespondingsquareonthesmallestviolinhashalftheheightandhalfthewidthoftheoriginalone,soitsareais2Thecustomarytermshalf-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.1/4theoriginalarea,nothalf.Thesamereasoningworksforpartsofthepanelneartheedge,suchasthepartthatonlypartiallyllsintheothersquare. Theentiresquarescalesdownthesameasasquareintheinterior,andin each violin the same fraction (about 70%) of the square is full, sothecontributionofthisparttothetotal areascalesdownjustthesame.Sinceanysmallsquareregionoranysmallregioncoveringpartofasquarescalesdownlikeasquareobject,theentiresurfaceareaofanirregularlyshapedobjectchangesinthesamemannerasthesurface area of a square: scaling it down by 3/4 reduces the area byafactorof9/16,andsoon.Ingeneral,wecanseethatanytimetherearetwoobjectswiththe same shape, but dierent linear dimensions (i.e., one looks like areduced photo of the other), the ratio of their areas equals the ratioofthesquaresoftheirlineardimensions:12=_1112_2.Notethatitdoesntmatterwherewechoosetomeasurethelinearsize, 1, of an object. In the case of the violins, for instance, it couldhave been measured vertically, horizontally, diagonally, or even fromthebottomof theleftf-holetothemiddleof therightf-hole. Wejusthavetomeasureitinaconsistentwayoneachviolin. Sinceallthepartsareassumedtoshrinkorexpandinthesamemanner,theratio11,12isindependentofthechoiceofmeasurement.Itisalsoimportanttorealizethatitiscompletelyunnecessarytohaveaformulafor theareaof aviolin. It is onlypossibletoderivesimpleformulas for theareas of certainshapes likecircles,rectangles, trianglesandsoon, butthatisnoimpedimenttothetypeofreasoningweareusing.Sometimesitisinconvenienttowritealltheequationsintermsof ratios, especially when more than two objects are being compared.Amorecompactwayofrewritingthepreviousequationis 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.Alloftheabovereasoningworksjustaswellinthecaseofvol-ume. Volumegoeslikelengthcubed:\ 13.If dierent objects aremadeof thesamematerial withthesamedensity, =:,\ , thentheir masses, :=\ , areproportional40 Chapter 0 Introduction and Reviewk / Example 3. The big trian-gle has four times more area thanthe little one.l / A tricky way of solving ex-ample 3, explained in solution #2.to13, andsoaretheirweights. (Thesymbol fordensityis, thelower-caseGreekletterrho.)Animportant point is that all of the above reasoningaboutscaling only applies to objects that are the same shape. For instance,apiece of paper is larger thanapencil, but has amuchgreatersurface-to-volumeratio.OneoftherstthingsIlearnedasateacherwasthatstudentswere 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 3>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 = bh,2, where b is the base and h is the height. The areas ofthe triangles areA1 = b1h1,2A2 = b2h2,2= (2b1)(2h1),2= 2b1h1A2,A1 = (2b1h1),(b1h1,2)= 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=bh,2. 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 A2,A1 = 4.00.(Thesolutioniscorrect, but it wouldnt workwithashapeforwhoseareawedont haveaformula. Also, thenumerical cal-Section 0.2 Scaling and Order-of-Magnitude Estimates 41m / Example4. Thebigspherehas 125 times more volume thanthe little one.n / Example 5. The 48-pointS has 1.78 times more areathan the 36-point S.culation might make the answer of 4.00 appear inexact, whereassolution #1 makes it clear that it is exactly 4.)Incorrect solution: The area of a triangle is A = bh,2, and if youpluginb=2.00mandh=2.00m, yougetA=2.00m2, sothebiggertrianglehas2.00timesmorearea. (Thissolutionisincorrect because no comparison has been made with the smallertriangle.)Scaling of the volume of a sphere example 4>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= (4,3)r3, soV1 =43r31V2 =43r32=43(5r1)3=5003r31V2,V1 =_5003r31_,_43r31_= 125Incorrect solution: The volume of a sphere is V= (4,3)r3, soV1 =43r31V2 =43r32=435r31=203r31V2,V1 =_203r31_,_43r31_= 5(The solution is incorrect because (5r1)3is not the same as 5r31.)Scaling of a more complex shape example 5> 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 (48,36)2= 1.78.Incorrect solution: The length of the curve of the second S islongerbyafactorof 48,36=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 (48,36)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 degreeofprecisionthatthenatureofthesubjectpermitsandnottoseekanexactnesswhereonlyanapproximationofthetruthispossible.AristotleItisacommonmisconceptionthatsciencemustbeexact. Forinstance, intheStar TrekTVseries, it wouldoftenhappenthatCaptainKirkwouldaskMr. Spock, Spock, wereinaprettybadsituation. What doyouthinkare our chances of gettingout ofhere?ThescienticMr. Spockwouldanswerwithsomethinglike,Captain, I estimate the odds as 237.345toone. Inreality, hecouldnot have estimatedthe odds withsixsignicant gures ofaccuracy, butneverthelessoneofthehallmarksofapersonwithagoodeducationinscienceistheabilitytomakeestimatesthatarelikelytobeatleastsomewhereintherightballpark. Inmanysuchsituations, 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 dierbyafactoroftenaresaidtodierbyoneorderofmagnitude,suchSection 0.2 Scaling and Order-of-Magnitude Estimates 43anestimate is calledanorder-of-magnitude estimate. The tilde,, is usedtoindicate that things are onlyof the same order ofmagnitude,butnotexactlyequal,asinoddsofsurvival 100toone .Thetildecanalsobeusedinfrontofanindividual numbertoem-phasizethatthenumberisonlyoftherightorderofmagnitude.Although making order-of-magnitude estimates seems simple andnatural toexperiencedscientists, its amodeof reasoningthat iscompletely unfamiliar to most college students. Some of the typicalmentalstepscanbeillustratedinthefollowingexample.Cost of transporting tomatoes example 6> 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 tomatoesttinginthetruckiswayo. Thatswhypeoplehaveahardtimeat those contests where you are supposed to estimate the number ofjellybeansinabigjar. Anotherexampleisthatmostpeoplethinktheirfamiliesuseabout10gallonsof waterperday, butinrealitytheaverageis about 300gallons per day. Whenestimatingareaor volume, youaremuchbetter oestimatinglinear dimensions,andcomputingvolumefromthelineardimensions. Heresabettersolution:Bettersolution: Asintheprevioussolution,saythecostofthetrip is $2000. The dimensions of the bin are probably 4 m2 m 1m, foravolumeof8m3. Sincethewholethingisjustanorder-of-magnitudeestimate, letsroundthatotothenearestpoweroften, 10 m3. The shape of a tomato is complicated, and I dont knowanyformulaforthevolumeofatomatoshape,butsincethis isjustan 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 numberoftomatoes bydividing the volume of the bin by the volume of onetomato: 10m3,104m3=105tomatoes. Thetransportationcost44 Chapter 0 Introduction and Reviewo / Consider a spherical cow.per tomato is $2000,105tomatoes=$0.02/tomato. That means thattransportationreallydoesntcontributeverymuchtothecostofatomato.Approximating the shape of a tomato as a cube is an example ofanothergeneral strategyformakingorder-of-magnitudeestimates.Asimilarsituationwouldoccurifyouweretryingtoestimatehowmanym2ofleathercouldbeproducedfromaherdoftenthousandcattle. There is no point in trying to take into account the shape ofthecowsbodies. Areasonableplanofattackmightbetoconsiderasphericalcow. Probablyacowhasroughlythesamesurfaceareaasaspherewitharadiusofabout1m, whichwouldbe4(1m)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 about10m2,sotheherdasawholemightyield105m2ofleather.Thefollowinglistsummarizesthestrategiesforgettingagoodorder-of-magnitudeestimate.1. Dontevenattemptmorethanonesignicantgureofpreci-sion.2. Dontguessarea, volume, ormassdirectly. Guesslineardi-mensionsandgetarea,volume,ormassfromthem.3. Whendealingwithareasorvolumesofobjectswithcomplexshapes, idealizethemasif theyweresomesimplershape, acubeorasphere,forexample.4. Checkyournal answertoseeifitisreasonable. Ifyouesti-matethataherdoftenthousandcattlewouldyield0.01m2ofleather, thenyouhaveprobablymadeamistakewithcon-versionfactorssomewhere.Section 0.2 Scaling and Order-of-Magnitude Estimates 45ProblemsThesymbols, ,etc. areexplainedonpage52.1 Correct use of a calculator: (a) Calculate7465853222+97554on a cal-culator. [Self-check: The most common mistake results in 97555.40.](b)WhichwouldbemorelikethepriceofaTV,andwhichwouldbemorelikethepriceofahouse,$3.5 105or$3.55?2 Computethefollowingthings. If theydontmakesensebe-causeofunits,sayso.(a)3cm+5cm(b)1.11m+22cm(c)120miles+2.0hours(d)120miles/2.0hours3 Yourbackyardhasbrickwallsonbothends. Youmeasureadistanceof 23.4mfromtheinsideof onewall totheinsideof theother. Eachwall is29.4cmthick. Howfarisitfromtheoutsideofonewalltotheoutsideoftheother?Payattentiontosignicantgures.4 Thespeedoflightis3.0108m/s. Convertthistofurlongsper fortnight. A furlong is 220 yards, and a fortnight is 14 days. Aninchis2.54cm.5 Expresseachofthefollowingquantitiesinmicrograms:(a)10mg,(b)104g,(c)10kg,(d)100 103g,(e)1000ng.6 Convert 134 mg to units of kg, writing your answer in scienticnotation. >Solution,p. 8637 In the last century, the average age of the onset of puberty forgirlshasdecreasedbyseveralyears. Urbanfolklorehasitthatthisisbecauseofhormonesfedtobeefcattle,butitismorelikelytobebecausemoderngirlshavemorebodyfatontheaverageandpos-siblybecauseof estrogen-mimickingchemicalsintheenvironmentfromthebreakdownof pesticides. Ahamburgerfromahormone-implantedsteer has about 0.2 ng of estrogen(about double theamount of natural beef). Aservingof peas contains about 300ng of estrogen. An adult woman produces about 0.5 mg of estrogenper day (note the dierent unit!). (a) How many hamburgers wouldagirl havetoeatinonedaytoconsumeasmuchestrogenasanadultwomansdailyproduction? (b)Howmanyservingsof peas?8 Theusualdenitionofthemean(average)oftwonumbersoand / is (o+/),2. This is called the arithmetic mean. The geometric46 Chapter 0 Introduction and ReviewProblem 10.Problem 12.mean, however, is dened as (o/)1/2(i.e., the square root of o/). Forthesakeofdeniteness,letssaybothnumbershaveunitsofmass.(a)Computethearithmeticmeanof twonumbersthathaveunitsof grams. Thenconvert the numbers to units of kilograms andrecomputetheirmean. Istheanswerconsistent? (b)Dothesameforthegeometricmean. (c)If oand/bothhaveunitsof grams,whatshouldwecalltheunitsofab? Doesyouranswermakesensewhenyoutakethesquareroot? (d)Supposesomeoneproposestoyouathirdkindof mean, calledthesuperdupermean, denedas(o/)1/3. Isthisreasonable? >Solution,p. 8639 InanarticleontheSARSepidemic, theMay7, 2003NewYorkTimesdiscussesconictingestimatesof thediseasesincuba-tion period (the average time that elapses from infection to the rstsymptoms). Thestudyestimatedittobe6.4days. Butothersta-tisticalcalculations... showedthattheincubationperiodcouldbeaslongas14.22days.Whatswronghere?10 Thephotoshowsthecorner of abagof pretzels. Whatswronghere?11 The distance to the horizon is given by the expression2:/,where:istheradiusof theEarth, and/istheobserversheightabove the Earths surface. (This can be proved using the Pythagoreantheorem.) Show that the units of this expression make sense. Donttrytoprovetheresult,justcheckitsunits.12 (a)Basedonthedenitionsofthesine,cosine,andtangent,whatunitsmusttheyhave? (b)Acuteformulafromtrigonometrylets youndanyangle of atriangle if youknowthe lengths ofitssides. Usingthenotationshowninthegure, andletting:=(o + / + c),2behalftheperimeter,wehavetan ,2 =(: /)(: c):(: o).Showthattheunitsof thisequationmakesense. Inotherwords,checkthat theunits of theright-handsidearethesameas youranswertopartaofthequestion. >Solution,p. 86313 Aphysicshomeworkquestionasks, If youstartfromrestand accelerate at 1.54 m,s2for 3.29 s, how far do you travel by theendofthattime?Astudentanswersasfollows:1.54 3.29 = 5.07mHis Aunt Wanda is good with numbers, but has never taken physics.Shedoesntknowtheformulaforthedistancetraveledundercon-stant accelerationover agivenamount of time, but shetells hernephewhisanswercannotberight. Howdoessheknow?14 Youarelookingintoadeepwell. Itisdark,andyoucannotseethebottom. Youwanttondouthowdeepitis, soyoudropProblems 47arockin,andyouhearasplash3.0secondslater. Howdeepisthewell?15 You take a trip in your spaceship to another star. Setting o,youincreaseyourspeedataconstantacceleration. Onceyougethalf-waythere,youstartdecelerating,atthesamerate,sothatbythe time you get there, you have slowed down to zero speed. You seethetouristattractions,andthenheadhomebythesamemethod.(a)Findaformulaforthetime, T, requiredfortheroundtrip, intermsofd,thedistancefromoursuntothestar,ando,themagni-tudeoftheacceleration. Notethattheaccelerationisnotconstantoverthewholetrip, butthetripcanbebrokenupintoconstant-accelerationparts.(b) The nearest star to the Earth (other than our own sun) is Prox-imaCentauri,atadistanceofd = 4 1016m. Supposeyouuseanacceleration of o = 10 m,s2,just enough to compensate for the lackoftruegravityandmakeyoufeel comfortable. Howlongdoestheroundtriptake,inyears?(c) Using the same numbers for d and o,nd your maximum speed.Comparethistothespeedoflight,whichis3.0 108m/s. (Laterinthiscourse, youwilllearnthattherearesomenewthingsgoingoninphysicswhenonegetsclosetothespeedoflight,andthatitis impossible to exceed the speed of light. For now, though, just usethesimplerideasyouvelearnedsofar.)16 Youclimbhalf-wayupatree, anddroparock. Thenyouclimbtothetop, anddropanotherrock. Howmanytimesgreateristhevelocityofthesecondrockonimpact?Explain. (Theanswerisnottwotimesgreater.)17 IftheaccelerationofgravityonMarsis1/3thatonEarth,howmanytimes longer does it takefor arocktodropthesamedistanceonMars?Ignoreairresistance. >Solution,p. 86318 Apersonisparachutejumping. Duringthetimebetweenwhensheleapsoutoftheplaneandwhensheopensherchute,heraltitudeisgivenbyanequationoftheformj= / c_t + /ct/k_,wherecisthebaseofnatural logarithms, and/, c, and/arecon-stants. Because of air resistance,her velocity does not increase at asteadyrateasitwouldforanobjectfallinginvacuum.(a)Whatunitswould/, c, and/havetohavefortheequationtomakesense?(b)Findthepersonsvelocity, , asafunctionof time. [Youwillneedtousethechainrule,andthefactthatd(cx),dr = cx.](c)Useyouranswerfrompart(b)togetaninterpretationof theconstantc. [Hint: cxapproacheszeroforlargevaluesofr.](d)Findthepersonsacceleration,o,asafunctionoftime.(e)Useyouranswerfrompart(b)toshowthatif shewaitslong48 Chapter 0 Introduction and Reviewenoughtoopenherchute, heraccelerationwillbecomeverysmall.19 InJuly1999, Popular Mechanicscarriedout