Fullerton,Californiawww.lightandmatter.comcopyright1998-2010BenjaminCrowellrev. July29,2011This book is licensed under the Creative CommonsAttribution-ShareAlike license, version 3.0, http://cre-ativecommons.org/licenses/by-sa/3.0/, except for thosephotographsanddrawingsofwhichIamnottheauthor,aslistedinthephotocredits. Ifyouagreetothelicense,it grants you certain privileges that you would not other-wise have, such as the right to copy the book, or downloadthedigitalversionfreeofchargefromwww.lightandmat-ter.com.Brief Contents0 Introduction and review 151 Scaling and estimation 41Motion inone dimension2 Velocity and relative motion 673 Acceleration and free fall 914 Force and motion 1215 Analysis of forces 145Motion inthree dimensions6 Newtons laws inthree dimensions 1777 Vectors 1898 Vectors and motion 2019 Circular motion 22110 Gravity 237Conservation laws11 Conservation of energy 26512 Simplifying the energy zoo 28713 Work: the transfer ofmechanical energy 30114 Conservation of momentum 32915 Conservation ofangular momentum 35916 Thermodynamics 395Vibrations and waves17 Vibrations 41918 Resonance 43319 Free waves 45720 Bounded waves 485Relativity andelectromagnetism21 Electricity and circuits 53322 The nonmechanical universe 58923 Relativity and magnetism 62324 Electromagnetism 64525 Capacitance and inductance 67526 The atom and E=mc269327 General relativity 755Optics28 The ray model of light 77329 Images by reection 79130 Images, quantitatively 80931 Refraction 82732 Wave optics 849The modern revolutionin physics33 Rules of randomness 87734 Light as a particle 90135 Matter as a wave 91936 The atom 945Contents0Introduction and review0.1The scientic method . . . . . . 150.2What is physics? . . . . . . . . 18Isolatedsystemsandreductionism,20.0.3How to learn physics. . . . . . . 210.4Self-evaluation . . . . . . . . . 230.5Basics of the metric system. . . . 24The metric system, 24.The second,25.The meter, 26.The kilogram,26.Combinations of metric units,26.Checkingunits,27.0.6The newton, the metric unit of force 280.7Less common metric prexes. . . 290.8Scientic notation. . . . . . . . 290.9Conversions . . . . . . . . . . 30Should that exponent be positive, ornegative?,31.0.10Signicant gures . . . . . . . 32Summary . . . . . . . . . . . . . 35Problems . . . . . . . . . . . . . 37Exercise 0: Models and idealization . . 391Scaling and estimation1.1Introduction . . . . . . . . . . 41Areaandvolume,41.1.2Scaling of area and volume . . . . 43Galileoonthebehaviorofnatureonlargeandsmall scales, 44.Scalingofareaandvolumeforirregularlyshapedobjects,47.1.3 Scaling applied to biology . . . . 51Organismsofdierentsizeswiththesameshape, 51.Changes in shape to accommo-datechangesinsize,53.1.4Order-of-magnitude estimates. . . 55Summary . . . . . . . . . . . . . 58Problems . . . . . . . . . . . . . 59Exercise 1: Scaling applied to leaves. . 63Motion in one dimension2Velocity and relative motion2.1Types of motion . . . . . . . . . 67Rigid-body motion distinguished from mo-tion that changes an objects shape,67.Center-of-massmotionasopposedtorotation, 67.Center-of-mass motion inonedimension,71.2.2Describing distance and time. . . 71A point in time as opposed to duration, 72.2.3Graphs of motion; velocity . . . . 74Motion with constant velocity,74.Motion with changing velocity,75.Conventionsaboutgraphing,76.2.4The principle of inertia. . . . . . 78Physical eectsrelateonlytoachangeinvelocity,78.Motionisrelative,79.62.5Addition of velocities . . . . . . . 81Additionof velocities todescriberelativemotion, 81.Negative velocities in relativemotion,81.2.6Graphs of velocity versus time. . . 832.7 Applications of calculus. . . . . 84Summary . . . . . . . . . . . . . 85Problems . . . . . . . . . . . . . 873Acceleration and free fall3.1The motion of falling objects . . . 91Howthespeedofafallingobjectincreaseswithtime,93.AcontradictioninAristo-tlesreasoning,94.Whatisgravity?,94.3.2Acceleration . . . . . . . . . . 95Denitionof accelerationfor linear v tgraphs,95.Theaccelerationofgravityisdierentindierentlocations.,96.3.3Positive and negative acceleration. 983.4Varying acceleration . . . . . . . 1023.5The area under the velocity-timegraph . . . . . . . . . . . . . . . 1053.6Algebraic results for constantacceleration . . . . . . . . . . . . 1073.7 Applications of calculus. . . . . 110Summary . . . . . . . . . . . . . 112Problems . . . . . . . . . . . . . 1134Force and motion4.1Force. . . . . . . . . . . . . 122Weneedonlyexplainchanges inmotion,not motionitself., 122.Motionchangesdue to an interaction between two objects.,123.Forces canall be measuredonthesame numerical scale., 123.More thanoneforceonanobject, 124.Objectscanexert forces oneachother at adistance.,124.Weight,125.Positiveandnegativesignsofforce,125.4.2Newtons rst law. . . . . . . . 125Moregeneralcombinationsofforces,127.4.3Newtons second law . . . . . . 129Ageneralization, 130.The relationshipbetweenmassandweight,131.4.4What force is not . . . . . . . . 134Force is not a property of one object.,134.Force is not a measure of an ob-jectsmotion., 134.Forceisnotenergy.,134.Force is not stored or used up.,135.Forces need not be exerted by livingthings or machines., 135.Aforceis thedirectcauseofachangeinmotion.,135.4.5Inertial and noninertial frames ofreference . . . . . . . . . . . . . 136Summary . . . . . . . . . . . . . 139Problems . . . . . . . . . . . . . 140Exercise 4: Force and motion . . . . . 14375Analysis of forces5.1Newtons third law. . . . . . . . 145Amnemonicforusingnewtonsthirdlawcorrectly,148.5.2Classication and behavior of forces149Normal forces, 152.Gravitationalforces, 153.Static and kinetic friction,153.Fluidfriction,157.5.3Analysis of forces. . . . . . . . 1585.4Transmissionof forcesbylow-massobjects . . . . . . . . . . . . . . 1615.5Objects under strain . . . . . . . 1635.6Simple machines: the pulley . . . 164Summary . . . . . . . . . . . . . 166Problems . . . . . . . . . . . . . 168Motion in three dimensions6Newtons laws in threedimensions6.1Forces have no perpendicular effects 177Relationshiptorelativemotion,179.6.2Coordinates and components . . . 180Projectilesmovealongparabolas.,183.6.3Newtons laws in three dimensions . 183Summary . . . . . . . . . . . . . 185Problems . . . . . . . . . . . . . 1867Vectors7.1Vector notation. . . . . . . . . 189Drawingvectorsasarrows,191.7.2Calculations with magnitude anddirection. . . . . . . . . . . . . . 1927.3Techniques for adding vectors . . . 194Addition of vectors given theircomponents, 194.Addition of vectorsgiven their magnitudes and directions,194.Graphicaladditionofvectors,194.7.4 Unit vector notation . . . . . . 1967.5 Rotational invariance . . . . . . 196Summary . . . . . . . . . . . . . 198Problems . . . . . . . . . . . . . 1998Vectors and motion8.1The velocity vector . . . . . . . 2028.2The acceleration vector . . . . . 2048.3The force vector and simple machines2078.4 Calculus with vectors . . . . . 208Summary . . . . . . . . . . . . . 212Problems . . . . . . . . . . . . . 213Exercise 8: Vectors and motion. . . . 2189Circular motion9.1Conceptual framework . . . . . . 221Circularmotiondoesnotproduceanout-ward force, 221.Circular motion does notpersist without a force, 222.Uniform andnonuniform circular motion, 223.Only aninwardforceisrequiredforuniformcircu-larmotion.,224.Inuniformcircularmo-tion, the acceleration vector is inward, 225.9.2Uniform circular motion. . . . . . 2279.3Nonuniform circular motion. . . . 230Summary . . . . . . . . . . . . . 232Problems . . . . . . . . . . . . . 23310Gravity10.1Keplers laws . . . . . . . . . 23810.2Newtons law of gravity . . . . . 240The suns force onthe planets obeys aninverse square law., 240.The forces be-tween heavenly bodies are the same type offorce as terrestrial gravity., 241.Newtonslawofgravity,242.810.3Apparent weightlessness . . . . 24610.4Vector addition of gravitationalforces. . . . . . . . . . . . . . . 24610.5Weighing the earth . . . . . . . 24910.6 Dark energy. . . . . . . . . 251Summary . . . . . . . . . . . . . 253Problems . . . . . . . . . . . . . 255Exercise 10: The shell theorem. . . . 261Conservation laws11Conservation of energy11.1Thesearchfor aperpetual motionmachine. . . . . . . . . . . . . . 26511.2Energy . . . . . . . . . . . . 26611.3A numerical scale of energy. . . 270Hownewforms of energyarediscovered,273.11.4Kinetic energy . . . . . . . . . 275Energyandrelativemotion,276.11.5Power . . . . . . . . . . . . 277Summary . . . . . . . . . . . . . 280Problems . . . . . . . . . . . . . 28212Simplifying the energy zoo12.1Heat is kinetic energy . . . . . . 28812.2Potential energy: energy of distanceor closeness . . . . . . . . . . . . 290An equation for gravitational potentialenergy,291.12.3All energy is potential or kinetic. . 294Summary . . . . . . . . . . . . . 296Problems . . . . . . . . . . . . . 29713Work: thetransfer of me-chanical energy13.1Work: the transfer of mechanicalenergy . . . . . . . . . . . . . . 301The concept of work, 301.Calculatingwork as force multiplied by distance,302.machines can increase force, but notwork., 304.No work is done withoutmotion., 304.Positive and negative work,305.13.2Work in three dimensions . . . . 307Aforceperpendiculartothemotiondoesno work., 307.Forces at other angles, 308.13.3Varying force . . . . . . . . . 31113.4 Applications of calculus. . . . 31413.5Work and potential energy . . . . 31513.6 When does work equal force timesdistance? . . . . . . . . . . . . . 31713.7 The dot product . . . . . . . 319Summary . . . . . . . . . . . . . 321Problems . . . . . . . . . . . . . 32314Conservation of momentum14.1Momentum. . . . . . . . . . 330A conserved quantity of motion,330.Momentum, 331.Generalization ofthe momentum concept, 333.Momentumcomparedtokineticenergy,334.14.2Collisions in one dimension . . . 336Thediscoveryoftheneutron,339.14.3 Relationship of momentum to thecenter of mass. . . . . . . . . . . 341Momentum in dierent frames of reference,342.The center of mass frame ofreference,343.14.4Momentum transfer. . . . . . . 344The rate of change of momentum,344.The area under the force-time graph,346.14.5Momentum in three dimensions . 347Thecenterofmass, 348.Countingequa-tions and unknowns, 349.Calculationswiththemomentumvector,350.14.6 Applications of calculus. . . . 351Summary . . . . . . . . . . . . . 353Problems . . . . . . . . . . . . . 355915Conservation of angularmomentum15.1Conservation of angular momentum361Restrictiontorotationinaplane,365.15.2Angular momentum in planetarymotion . . . . . . . . . . . . . . 36515.3Two theorems about angularmomentum . . . . . . . . . . . . 36715.4Torque: the rate of transfer of angu-lar momentum. . . . . . . . . . . 372Torque distinguished from force,372.Relationship between force andtorque, 373.The torque due togravity,375.15.5Statics. . . . . . . . . . . . 379Equilibrium, 379.Stable and unstableequilibria,382.15.6Simple machines: the lever . . . 38315.7 Proof of Keplers elliptical orbit law385Summary . . . . . . . . . . . . . 387Problems . . . . . . . . . . . . . 389Exercise 15: Torque . . . . . . . . . 39416Thermodynamics16.1Pressure and temperature . . . . 396Pressure,396.Temperature,400.16.2Microscopicdescriptionof anidealgas . . . . . . . . . . . . . . . . 403Evidence for the kinetic theory,403.Pressure, volume, andtemperature,403.16.3Entropy. . . . . . . . . . . . 407Eciency and grades of energy,407.Heatengines,407.Entropy,409.Problems . . . . . . . . . . . . . 413Vibrations and waves17Vibrations17.1Period, frequency, and amplitude. 42017.2Simple harmonic motion. . . . . 423Why are sine-wave vibrations so common?,423.Period is approximately indepen-dent of amplitude, if the amplitude issmall.,424.17.3 Proofs . . . . . . . . . . . 425Summary . . . . . . . . . . . . . 428Problems . . . . . . . . . . . . . 429Exercise 17: Vibrations . . . . . . . 43118Resonance18.1Energy in vibrations . . . . . . 43418.2Energy lost from vibrations. . . . 43618.3Putting energy into vibrations . . 43818.4 Proofs . . . . . . . . . . . 446Statement 2: maximum amplitude atresonance, 447.Statement 3: ampli-tude at resonance proportional to Q,447.Statement4: FWHMrelatedtoQ,448.Summary . . . . . . . . . . . . . 449Problems . . . . . . . . . . . . . 451Exercise 18: Resonance . . . . . . . 45519Free waves19.1Wave motion . . . . . . . . . 4591. Superposition, 459.2. The mediumis not transported with the wave., 461.3.A waves velocity depends on the medium.,10462.Wavepatterns,463.19.2Waves on a string . . . . . . . 464Intuitive ideas, 464.Approximatetreatment, 465.Rigorous derivation us-ingcalculus(optional), 466.Signicanceoftheresult,468.19.3Sound and light waves . . . . . 468Soundwaves,468.Lightwaves,469.19.4Periodic waves. . . . . . . . . 471Period and frequency of a periodicwave, 471.Graphs of waves as afunction of position, 471.Wavelength,472.Wave velocityrelatedtofrequencyand wavelength, 472.Sinusoidal waves,474.19.5The Doppler effect . . . . . . . 475TheBigBang, 477.WhattheBigBangisnot,479.Summary . . . . . . . . . . . . . 481Problems . . . . . . . . . . . . . 48320Bounded waves20.1Reection, transmission, andabsorption . . . . . . . . . . . . . 486Reection and transmission, 486.Invertedand uninverted reections, 489.Absorption,489.20.2 Quantitative treatment of reection493Whyreectionoccurs, 493.Intensityofreection, 494.Invertedanduninvertedreectionsingeneral,495.20.3Interference effects . . . . . . . 49620.4Waves bounded on both sides. . 499Musical applications, 501.Standingwaves, 501.Standing-wave patterns ofaircolumns,503.Summary . . . . . . . . . . . . . 505Problems . . . . . . . . . . . . . 506Hints for volume 1. . . . . . . . . . 508Answers for volume 1 . . . . . . . . 508Photo credits for volume 1 . . . . . . 529Relativity and electromagnetism21Electricity and circuits21.1The quest for the atomic force . . 53421.2Electrical forces . . . . . . . . 535Charge, 535.Conservation of charge,537.Electrical forces involving neutralobjects,538.21.3Current . . . . . . . . . . . . 538Unity of all types of electricity,538.Electriccurrent,539.21.4Circuits . . . . . . . . . . . . 54121.5Voltage . . . . . . . . . . . . 543Thevoltunit, 543.Thevoltageconceptingeneral,543.21.6Resistance . . . . . . . . . . 548Resistance, 548.Superconductors,550.Constant voltage throughouta conductor, 551.Short circuits,552.Resistors,552.21.7 Applications of calculus. . . . 55521.8Series and parallel circuits . . . . 556Schematics, 556.Parallel resistancesandthejunctionrule,557.Seriesresistances,562.Summary . . . . . . . . . . . . . 569Problems . . . . . . . . . . . . . 573Exercise 21A: Electrical measurements. 581Exercise 21B: Voltage and current . . . 582Exercise 21C: Reasoning about circuits 58722The nonmechanical universe22.1The stage and the actors . . . . 590Newtons instantaneous action at adistance, 590.No absolute time,590.Causality, 591.Time delays inforces exertedat a distance, 592.Moreevidencethateldsofforcearereal: theycarryenergy.,592.22.2The gravitational eld . . . . . . 594Sourcesandsinks, 595.Superpositionofelds,595.Gravitationalwaves,596.22.3The electric eld . . . . . . . . 597Denition, 597.Dipoles, 600.Alternative11denition of the electric eld,601.Voltagerelatedtoelectriceld,601.22.4Calculating energy in elds . . . 60322.5 Voltage for nonuniform elds . . 60622.6Two or three dimensions . . . . 60722.7 Field lines and Gausss law. . . 61022.8 Electric eld of a continuouscharge distribution . . . . . . . . . 612Summary . . . . . . . . . . . . . 614Problems . . . . . . . . . . . . . 616Exercise 22: Field vectors . . . . . . 62023Relativity and magnetism23.1Relativisticdistortionof spaceandtime . . . . . . . . . . . . . . . 623Time distortion arising from mo-tion and gravity, 623.The Lorentztransformation,625.TheGfactor,631.23.2Magnetic interactions . . . . . . 636Relativityrequiresmagnetism,637.Summary . . . . . . . . . . . . . 641Problems . . . . . . . . . . . . . 642Exercise 23: Polarization. . . . . . . 64324Electromagnetism24.1The magnetic eld. . . . . . . 645Nomagneticmonopoles, 645.Denitionofthemagneticeld,646.24.2Calculating magnetic elds andforces. . . . . . . . . . . . . . . 648Magnetostatics, 648.Force on a chargemoving through a magnetic eld,650.Energyinthemagneticeld,651.24.3The universal speed c. . . . . . 652Velocitiesdontsimplyaddandsubtract.,652.A universal speed limit,653.Lighttravelsatc., 653.TheMichelson-Morleyexperiment,653.24.4Induction. . . . . . . . . . . 655Theprincipleofinduction,655.24.5Electromagnetic waves . . . . . 657Polarization, 659.Light is an electro-magnetic wave, 659.The electromagneticspectrum,659.Momentumoflight,660.24.6 Symmetry and handedness . . 66124.7 Doppler shifts and clock time . . 662Doppler shifts of light, 662.Clocktime,664.Summary . . . . . . . . . . . . . 667Problems . . . . . . . . . . . . . 66825Capacitance and inductance25.1Capacitance and inductance . . . 675Capacitors,676.Inductors,676.25.2Oscillations. . . . . . . . . . 67825.3Voltage and current. . . . . . . 68125.4Decay . . . . . . . . . . . . 685The RC circuit, 685.The RL circuit, 686.25.5Impedance . . . . . . . . . . 688Problems . . . . . . . . . . . . . 69126The atom and E=mc226.1Atoms . . . . . . . . . . . . 693The chemical elements, 693.Makingsenseof theelements, 694.Direct proofthatatomsexisted,695.26.2Quantization of charge . . . . . 69626.3The electron. . . . . . . . . . 699Cathode rays, 699.Were cathoderays a form of light, or of matter?,699.Thomsons experiments, 700.Thecathoderayas asubatomicparticle: theelectron, 702.The raisin cookie model,702.26.4The nucleus . . . . . . . . . . 705Radioactivity, 705.The planetary model,708.Atomic number, 711.The struc-ture of nuclei, 717.The strong nu-clear force, alpha decay and ssion,720.Theweaknuclearforce;betadecay,722.Fusion, 726.Nuclear energy andbinding energies, 728.Biological eects ofionizing radiation, 731.The creationoftheelements,734.26.5Relativistic mass and energy. . . 735Momentum, 736.Equivalence of massandenergy,739.Proofs,744.Summary . . . . . . . . . . . . . 747Problems . . . . . . . . . . . . . 749Exercise 26: Sports in slowlightland. . 75427General relativity27.1Our universe isnt Euclidean. . . 755Curvature, 756.Curvature doesnt re-quirehigherdimensions,757.27.2The equivalence principle. . . . 758Universality of free-fall, 758.GravitationalDoppler shifts and time dilation,12760.Local atness, 761.Inertial frames,761.27.3Black holes. . . . . . . . . . 762Information paradox, 764.Formation,764.27.4Cosmology . . . . . . . . . . 765TheBigBang, 765.Thecosmiccensor-ship hypothesis, 766.The advent of high-precision cosmology, 767.Dark energyanddarkmatter,768.Problems . . . . . . . . . . . . . 770Optics28The ray model of light28.1The nature of light . . . . . . . 774The cause and eect relationship in vision,774.Lightisathing, andittravelsfromone point to another., 775.Light cantravelthroughavacuum.,776.28.2Interaction of light with matter . . 777Absorption of light, 777.How we see non-luminous objects, 777.Numerical mea-surementofthebrightnessoflight,779.28.3The ray model of light . . . . . . 779Modelsoflight,779.Raydiagrams,781.28.4Geometry of specular reection . 782Reversibilityoflightrays,784.28.5 The principle of least time forreection . . . . . . . . . . . . . 786Summary . . . . . . . . . . . . . 788Problems . . . . . . . . . . . . . 78929Images by reection29.1A virtual image . . . . . . . . 79229.2Curved mirrors. . . . . . . . . 79529.3A real image . . . . . . . . . 79629.4Images of images . . . . . . . 797Summary . . . . . . . . . . . . . 801Problems . . . . . . . . . . . . . 802Exercise 29: Exploring images with acurved mirror. . . . . . . . . . . . 80530Images, quantitatively30.1A real image formed by a convergingmirror . . . . . . . . . . . . . . . 810Location of the image, 810.Magnication,813.30.2Other cases with curved mirrors . 81330.3 Aberrations . . . . . . . . . 817Summary . . . . . . . . . . . . . 821Problems . . . . . . . . . . . . . 823Exercise 30: Object and image distances 82631Refraction31.1Refraction. . . . . . . . . . . 828Refraction, 828.Refractivepropertiesofmedia, 829.Snellslaw, 830.Theindexof refraction is related to the speed oflight., 831.A mechanical model of Snellslaw, 832.A derivation of Snells law,832.Color and refraction, 833.Howmuchlight is reected, andhowmuchistransmitted?,833.31.2Lenses . . . . . . . . . . . . 83531.3 The lensmakers equation . . . 83631.4 The principle of least time forrefraction . . . . . . . . . . . . . 83731.5 Case study: the eye of the jumpingspider . . . . . . . . . . . . . . . 838Summary . . . . . . . . . . . . . 840Problems . . . . . . . . . . . . . 841Exercise 31: How strong are yourglasses? . . . . . . . . . . . . . 84732Wave optics32.1Diffraction. . . . . . . . . . . 85032.2Scaling of diffraction . . . . . . 85132.3The correspondence principle . . 85232.4Huygens principle. . . . . . . 85332.5Double-slit diffraction. . . . . . 85432.6Repetition. . . . . . . . . . . 85832.7Single-slit diffraction . . . . . . 85932.8 The principle of least time . . 860Summary . . . . . . . . . . . . . 863Problems . . . . . . . . . . . . . 86513Exercise 32A: Double-source interference870Exercise 32B: Single-slit diffraction . . 872Exercise 32C: Diffraction of light . . . . 874The modern revolution inphysics33Rules of randomness33.1Randomness isnt random. . . . 87933.2Calculating randomness. . . . . 880Statistical independence, 880.Additionof probabilities, 881.Normalization,882.Averages,882.33.3Probability distributions. . . . . 884Average and width of a probabilitydistribution,885.33.4Exponential decay and half-life . . 886Rateofdecay,889.33.5 Applications of calculus. . . . 891Summary . . . . . . . . . . . . . 894Problems . . . . . . . . . . . . . 89634Light as a particle34.1Evidence for light as a particle. . 90234.2How much light is one photon? . . 904The photoelectric eect, 904.An un-expected dependence on frequency,905.Numerical relationshipbetweenen-ergyandfrequency,906.34.3Wave-particle duality. . . . . . 909Awrong interpretation: photons inter-fering with each other, 910.The con-cept of a photons path is undened.,910.Another wrong interpretation: thepilot wave hypothesis, 911.The probabil-ityinterpretation,911.34.4Photons in three dimensions . . . 914Summary . . . . . . . . . . . . . 915Problems . . . . . . . . . . . . . 91635Matter as a wave35.1Electrons as waves . . . . . . . 920Whatkindofwaveisit?,923.35.2 Dispersive waves. . . . . . 92535.3Bound states . . . . . . . . . 92735.4The uncertainty principle . . . . 930The uncertainty principle, 930.MeasurementandSchrodingerscat,934.35.5Electrons in electric elds. . . . 935Tunneling,936.35.6 The Schr odinger equation . . 936Useofcomplexnumbers,939.Summary . . . . . . . . . . . . . 940Problems . . . . . . . . . . . . . 94236The atom36.1Classifying states . . . . . . . 94636.2Angular momentum in threedimensions . . . . . . . . . . . . 947Three-dimensional angular momentuminclassical physics, 947.Three-dimensionalangular momentumin quantumphysics,948.36.3The hydrogen atom. . . . . . . 94936.4 Energies of states in hydrogen . 952History, 952.Approximate treatment,952.36.5Electron spin . . . . . . . . . 95436.6Atoms with more than one electron955Derivingtheperiodictable,957.Summary . . . . . . . . . . . . . 959Problems . . . . . . . . . . . . . 961Exercise36: Quantumversusclassicalrandomness. . . . . . . . . . . . 964Hints for volume 2. . . . . . . . . . 965Answers for volume 2 . . . . . . . . 965Photo credits for volume 2 . . . . . . 97414The 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 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 The scientic methodUntil veryrecentlyinhistory, noprogresswasmadeinansweringquestionslikethese. Worsethanthat, thewrong answerswrittenbythinkersliketheancientGreekphysicistAristotlewereacceptedwithoutquestionforthousandsof years. WhyisitthatscienticknowledgehasprogressedmoresincetheRenaissancethanithadinall theprecedingmillenniasincethebeginningof recordedhis-tory? Undoubtedlytheindustrial revolutionispartoftheanswer.Buildingitscenterpiece, thesteamengine, requiredimprovedtech-15a / 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).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-ories1arecreatedtoexplaintheresultsof experimentsthatwerecreated 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 in1The term theory in science does not just mean what someone thinks, orevenwhatalotofscientiststhink.Itmeansaninterrelatedsetofstatementsthat have predictive value, andthat have surviveda broadset of empiricaltests. Thus, both Newtons law of gravity and Darwinian evolution are scientictheories. Ahypothesis,incontrasttoatheory, isanystatementof interestthat can be empirically tested. That the moon is made of cheese is a hypothesis,whichwasempiricallytested,forexample,bytheApolloastronauts.16 Chapter 0 Introduction and reviewdierentcombinations, withoutanychangeintheidentitiesoftheelements themselves. The theory worked for hundreds ofyears,andwas conrmedexperimentallyover awiderangeof pressures andtemperatures andwithmanycombinations of elements. Onlyinthe twentieth century did we learn that one element could be trans-formed into one another under the conditions of extremely high pres-sureandtemperatureexistinginanuclearbomborinsideastar.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. 522Thescienticmethodasdescribedhereisanidealization, andshouldnotbeunderstoodasasetprocedurefordoingscience. Sci-entistshaveasmanyweaknessesandcharacterawsasanyothergroup,anditisverycommonforscientiststotrytodiscreditotherpeoplesexperimentswhentheresultsruncontrarytotheirownfa-voredpoint of view. Successful sciencealsohas moretodowithluck, intuition, andcreativitythanmost people realize, andtherestrictionsof thescienticmethoddonotstieindividualityandself-expressionanymorethanthefugue andsonataforms 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 The scientic method 17B 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?0.2 What is physics?Given for one instant an intelligence which could comprehendall the forces by which nature is animated and the respectivepositions of the things which compose it...nothing would beuncertain, and the future as the past would be laid out beforeits eyes.Pierre Simon de LaplacePhysicsistheuseofthescienticmethodtondoutthebasicprinciplesgoverninglightandmatter, andtodiscovertheimplica-tionsofthoselaws. Partofwhatdistinguishesthemodernoutlookfromtheancientmind-setistheassumptionthattherearerulesbywhich the universe functions, and that those laws can be at least par-tiallyunderstoodbyhumans. FromtheAgeofReasonthroughthenineteenthcentury,manyscientistsbegantobeconvincedthatthelaws ofnaturenotonlycouldbeknownbut,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 phenomena18 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 knownby its catalognumber, MG1131+0456, or moreinformally as Einsteins Ring.thatarenotvisibletotheeye, includingradiowaves, microwaves,x-rays, and gamma rays. These are the colors of light that do nothappentofallwithinthenarrowviolet-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. 522Manyphysical phenomenaarenot themselveslight ormatter,butarepropertiesof lightormatterorinteractionsbetweenlightand matter. For instance, motion is a property of all light and somematter,butitisnotitselflightormatter. Thepressurethatkeepsabicycletireblownupisaninteractionbetweentheair andthetire. Pressure is not aformof matter inandof itself. It is asmuchapropertyof thetireasof theair. Analogously, sisterhoodandemploymentare relationshipsamongpeople butarenotpeoplethemselves.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 lightSection 0.2 What is physics? 19d / Reductionism.beam 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 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.Isolated systems and reductionismTo 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 apart20 Chapter 0 Introduction and reviewintotheirconstituentneutrons,protonsandelectrons. Protonsandneutrons appear tobe made out of evensmaller particles calledquarks, andtherehaveevenbeensomeclaimsof experimental ev-idencethat quarks havesmaller parts insidethem. This methodofsplittingthingsintosmallerandsmallerpartsandstudyinghowthosepartsinuenceeachotheriscalledreductionism. 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?0.3 How to learn physicsFor as knowledges are now delivered, there is a kind of con-tract of error between the deliverer and the receiver; for hethat delivereth knowledge desireth to deliver it in such a formas may be best believed, and not as may be best examined;and he that receiveth knowledge desireth rather present sat-isfaction than expectant inquiry.Francis BaconMany students approach a science course with the idea that theycansucceedbymemorizingtheformulas, sothatwhenaproblemSection 0.3 How to learn physics 21isassignedonthehomeworkoranexam,theywillbeabletoplugnumbersintotheformulaandgetanumerical resultontheircal-culator. Wrong! Thatsnotwhatlearningscienceisabout! Thereisabigdierencebetweenmemorizingformulasandunderstandingconcepts. Tostartwith, dierentformulasmayapplyindierentsituations. Oneequationmightrepresentadenition, whichisal-waystrue. Anothermightbeaveryspecicequationforthespeedof an object sliding down an inclined plane, which would not be trueiftheobjectwasarockdriftingdowntothebottomoftheocean.Ifyoudontworktounderstandphysicsonaconceptuallevel,youwontknowwhichformulascanbeusedwhen.Most students takingcollegesciencecourses for therst timealsohaveverylittleexperiencewithinterpretingthemeaningofanequation. Considertheequationw=A/hrelatingthewidthof arectangletoitsheightandarea. Astudentwhohasnotdevelopedskill at interpretationmight viewthis as yet another equationtomemorizeandplugintowhenneeded. Aslightlymoresavvystu-dent might realizethat it issimplythefamiliarformulaA=whinadierent form. Whenaskedwhether arectanglewouldhaveagreater or smaller widththananother withthe same areabutasmaller height, theunsophisticatedstudent might beat aloss,nothavinganynumberstopluginonacalculator. Themoreex-periencedstudent wouldknowhowtoreasonabout anequationinvolvingdivisionifhissmaller, andAstaysthesame, thenwmust 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 metheykeeptryingtounderstanda22 Chapter 0 Introduction and reviewcertain 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.Finally, whenreviewingfor anexam, dont simplyreadbackoverthetextandyourlecturenotes. Instead, trytouseanactivemethodofreviewing,forinstancebydiscussingsomeofthediscus-sionquestionswithanotherstudent, ordoinghomeworkproblemsyouhadntdonethersttime.0.4 Self-evaluationTheintroductorypartofabooklikethisishardtowrite, becauseevery student arrives at this starting point with a dierent prepara-tion. OnestudentmayhavegrownupoutsidetheU.S.andsomaybecompletelycomfortablewiththemetricsystem, butmayhavehadanalgebracourseinwhichtheinstructor passedtooquicklyoverscienticnotation. Anotherstudentmayhavealreadytakencalculus, butmayhaveneverlearnedthemetricsystem. Thefol-lowingself-evaluationisachecklisttohelpyougureoutwhatyouneedtostudytobepreparedfortherestofthecourse.If youdisagreewiththisstate-ment. . .youshouldstudythissection:Iamfamiliarwiththebasicmetricunitsofmeters, kilograms, andsec-onds,andthemostcommonmetricprexes: milli- (m), kilo- (k), andcenti-(c).section0.5BasicoftheMetricSys-temIknowaboutthenewton, aunitofforcesection0.6Thenewton, theMetricUnitofForceI amfamiliar withthese less com-mon metric prexes: mega- (M),micro-(),andnano-(n).section 0.7 Less Common MetricPrexesIamcomfortablewithscienticno-tation.section0.8ScienticNotationI cancondentlydometricconver-sions.section0.9ConversionsI understand the purpose and use ofsignicantgures.section0.10SignicantFiguresIt wouldnt hurt youtoskimthe sections youthinkyoualreadyknowabout,andtodotheself-checksinthosesections.Section 0.4 Self-evaluation 230.5 Basics of the metric systemThe metric systemUnitswerenotstandardizeduntil 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 of units known inEnglish 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;24 Chapter 0 Introduction and reviewahundredthofagramwouldnotbewrittenas1cgbutas10mg.Thecenti-prexcanbeeasilyrememberedbecauseacentis102dollars. Theocial SIabbreviationforsecondsiss(notsec)andgramsareg(notgm).The secondThe sun stood still and the moon halted until the nation hadtaken vengeance on its enemies. . .Joshua 10:12-14Absolute, true, and mathematical time, of itself, and from itsown nature, ows equably without relation to anything exter-nal. . .Isaac NewtonWhenIstatedbrieyabovethatthesecondwasaunitoftime,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.Section 0.5 Basics of the metric system 25e / The original denition ofthe meter.f / A duplicate of the Pariskilogram, maintained at the Dan-ishNational Metrology Institute.self-check CWhat is a possible operational denition of how strong a person is? Answer, p. 522The meterTheFrenchoriginallydenedthemeteras107timesthedis-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.The kilogramThethirdbaseunit of theSI isthekilogram, aunit of mass.Mass is intendedtobeameasureof theamount 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.Combinations of metric unitsJust 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 like26 Chapter 0 Introduction and reviewacordof wood, abolt of cloth, or ajigger of whiskey. Nomoreliquidanddrymeasure. Justasimple,consistentsetofunits. TheSI measures put together from meters, kilograms, and seconds makeupthemkssystem. Forexample,themksunitofspeedism/s,notkm/hr.Checking unitsAuseful technique for ndingmistakes inones algebrais toanalyzetheunitsassociatedwiththevariables.Checking units example 1 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 aSection 0.5 Basics of the metric system 27pendulum 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?0.6 The newton, the metric unit of forceAforceis apushor apull, or moregenerallyanythingthat canchangeanobjectsspeedordirectionofmotion. Aforceisrequiredtostartacarmoving, toslowdownabaseball playerslidingintohome base,ortomake anairplaneturn. (Forces mayfailtochangeanobjects motionif theyarecanceledbyother forces, e.g., theforce of gravity pulling you down right now is being canceled by theforceof thechairpushinguponyou.) Themetricunitof forceistheNewton, denedastheforcewhich, if appliedforonesecond,will cause a 1-kilogram object starting from rest to reach a speed of1m/s. Laterchapterswilldiscusstheforceconceptinmoredetail.In fact, this entire book is about the relationship between force andmotion.Insection0.5, Igaveagravitational denitionofmass, butbydeninganumericalscaleofforce,wecanalsoturnaroundandde-neascaleofmasswithoutreferencetogravity. Forinstance, ifaforceoftwoNewtonsisrequiredtoaccelerateacertainobjectfromrest to1m/sin1s, thenthat object must haveamassof 2kg.Fromthispointof view, masscharacterizesanobjectsresistancetoachangeinits motion, whichwecall inertiaor inertial mass.Although there is no fundamental reason why an objects resistancetoachangeinitsmotionmustberelatedtohowstronglygravityaectsit, careful andpreciseexperimentshaveshownthatthein-ertial denitionandthegravitational denitionofmassarehighlyconsistent for a variety of objects. It therefore doesnt really matterforanypracticalpurposewhichdenitiononeadopts.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.)28 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.0.7 Less common metric prexesThe following are three metric prexes which, while less commonthantheonesdiscussedpreviously,arewellworthmemorizing.prex meaning examplemega- M 1066.4Mm =radiusoftheearthmicro- 10610m =sizeofawhitebloodcellnano- n 1090.154nm =distance between carbonnucleiinanethanemoleculeNotethattheabbreviationformicroistheGreeklettermu, acommonmistakeistoconfuseitwithm(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. 5220.8 Scientic notationMostof theinterestingphenomenainouruniversearenotonthehumanscale. It wouldtake 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.Section 0.7 Less common metric prexes 29Since101istentimessmallerthan102, itmakessensetousethenotation100tostandforone, 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 3.26reallystandsforthenumber3.2 3.2 3.2 3.2 3.2 3.2=1074, atotallydierentnumberfrom3.2 106=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. 5220.9 ConversionsIsuggestyouavoidmemorizinglotsof conversionfactorsbetweenSIunitsandU.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.units of force and mass are confusing, so its a good thing theyre notusedinscience. InU.S. units, theunitofforceisthepound-force,30 Chapter 0 Introduction and reviewandthebestunittouseformassistheslug, whichisabout14.6kg.)More 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$$$year 365days1$$$year24 $$$hours1day60min1 $$$hour 60s1min== 3.15 107s .Should that exponent be positive, or negative?A common mistake is to write the conversion fraction incorrectly.Forinstancethefraction103kg1g(incorrect)doesnotequalone, because103kgisthemassofacar, and1gisthemassofaraisin. Onecorrectwayofsettinguptheconversionfactorwouldbe103kg1g(correct) .Section 0.9 Conversions 31Youcanusuallydetectsuchamistakeifyoutakethetimetocheckyouranswerandseeifitisreasonable.Ifcommonsensedoesntruleouteitherapositiveoranegativeexponent, heresanotherwaytomakesureyougetitright. Therearebigprexesandsmallprexes:bigprexes: k Msmallprexes: m n(Itsnothardtokeepstraightwhicharewhich, sincemegaandmicroareevocative, anditseasytorememberthatakilometerisbiggerthanameterandamillimeterissmaller.) Intheexampleabove, we want the top of the fraction to be the same as the bottom.Sincekisabigprex, weneedtocompensate byputtingasmallnumberlike103infrontofit,notabignumberlike103. Solved problem: a simple conversion page 37, problem 6 Solved problem: the geometric mean page 38, 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.0.10 Signicant guresAnengineerisdesigningacarengine, andhasbeentoldthatthediameterofthepistons(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. Asinthe32 Chapter 0 Introduction and reviewexampleabove,siggsprovideawayofshowingtheaccuracyofanumber. Inmostcases,theresultofacalculationinvolvingseveralpieces 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, inwhich5 101cmwouldimplyonesiggofaccuracy,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. 522Dealingcorrectlywithsignicantgurescansaveyoutime! 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.Section 0.10 Signicant gures 33Whenindoubt, dontusethesiggrulesatall. Instead, in-tentionallychangeonepieceof 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 kgAnswer, p. 52234 Chapter 0 Introduction and reviewSummarySelected vocabularymatter . . . . . . Anythingthatisaectedbygravity.light. . . . . . . . Anything that can travel from one place to an-otherthroughemptyspaceandcaninuencematter,butisnotaectedbygravity.operational de-nition. . . . . . .A denition that states what operationsshould be carried out to measure the thing be-ingdened.Syst`eme Interna-tional . . . . . . .Afancynameforthemetricsystem.mkssystem . . . Theuseof metricunits basedonthemeter,kilogram, andsecond. Example: meters persecondisthemksunitof speed, notcm/sorkm/hr.mass . . . . . . . Anumerical measureof howdicultitistochangeanobjectsmotion.signicantgures Digits that contribute to the accuracy of ameasurement.Notationm . . . . . . . . . meter,themetricdistanceunitkg . . . . . . . . . kilogram,themetricunitofmasss . . . . . . . . . . second,themetricunitoftimeM-. . . . . . . . . themetricprexmega-,106k- . . . . . . . . . themetricprexkilo-,103m- . . . . . . . . . themetricprexmilli-,103- . . . . . . . . . themetricprexmicro-,106n- . . . . . . . . . themetricprexnano-,109SummaryPhysics is the use of the scientic method to study the behaviorof lightandmatter. Thescienticmethodrequiresacycleof the-oryandexperiment,theorieswithbothpredictiveandexplanatoryvalue,andreproducibleexperiments.The metric system is a simple, consistent framework for measure-ment built out of the meter, the kilogram, and the second plus a setof prexes denoting powers of ten. The most systematic method fordoingconversionsisshowninthefollowingexample:370ms 103s1ms= 0.37sMassisameasureof theamountof asubstance. Masscanbedenedgravitationally,bycomparinganobjecttoastandardmassonadouble-panbalance, orintermsof inertia, bycomparingtheeectof aforceonanobjecttotheeectof thesameforceonastandardmass. The twodenitions are foundexperimentallytobeproportional toeachothertoahighdegreeof precision, soweSummary 35usuallyrefersimplytomass,withoutbotheringtospecifywhichtype.Aforceisthatwhichcanchangethemotionofanobject. ThemetricunitofforceistheNewton,denedastheforcerequiredtoaccelerateastandard1-kgmassfromresttoaspeedof1m/sin1s.Scienticnotationmeans,forexample,writing3.2 105ratherthan320000.Writingnumberswiththecorrectnumberof signicantgurescorrectlycommunicateshowaccuratetheyare. Asaruleofthumb,the nal result of a calculation is no more accurate than, and shouldhavenomoresignicant gures than, theleast accuratepieceofdata.36 Chapter 0 Introduction and reviewProblemsKeyAcomputerizedanswercheckisavailableonline.

Aproblemthatrequirescalculus. Adicultproblem.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 Thespeedoflightis3.0 108m/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. 5087 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?Problems 37Problem 10.8 Theusualdenitionofthemean(average)oftwonumbersaand b is (a+b)/2. This is called the arithmetic mean. The geometricmean, however, is dened as (ab)1/2(i.e., the square root of ab). Forthesakeofdeniteness,letssaybothnumbershaveunitsofmass.(a)Computethearithmeticmeanof twonumbersthathaveunitsof grams. Thenconvert the numbers to units of kilograms andrecomputetheirmean. Istheanswerconsistent? (b)Dothesameforthegeometricmean. (c)If aandbbothhaveunitsof grams,whatshouldwecalltheunitsofab? Doesyouranswermakesensewhenyoutakethesquareroot? (d)Supposesomeoneproposestoyouathirdkindof mean, calledthesuperdupermean, denedas(ab)1/3. Isthisreasonable? Solution,p. 5089 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 expression2rh,whereristheradiusof theEarth, andhistheobserversheightabove the Earths surface. (This can be proved using the Pythagoreantheorem.) Showthattheunitsofthisexpressionmakesense. (Seeexample1onp. 27foranexampleofhowtodothis.) Donttrytoprovetheresult,justcheckitsunits.38 Chapter 0 Introduction and reviewExercise 0: Models and idealizationEquipment:coeeltersramps(onepergroup)ballsofvarioussizesstickytapevacuumpumpandguineaandfeatherapparatus(one)Themotionoffallingobjectshasbeenrecognizedsinceancienttimesasanimportantpieceofphysics,butthemotionisinconvenientlyfast, soinoureverydayexperienceitcanbehardtotell exactly what objects are doing when they fall. In this exercise you will use several techniquesto get around this problem and study the motion. Your goal is to construct a scientic model offalling. Amodelmeansanexplanationthatmakestestablepredictions. Oftenmodelscontainsimplicationsoridealizationsthatmakethemeasiertoworkwith, eventhoughtheyarenotstrictlyrealistic.1. Onemethodofmakingfallingeasiertoobserveistouseobjectslikefeathersthatweknowfromeverydayexperiencewill notfall asfast. Youwill usecoeelters, instacksof varioussizes,totestthefollowingtwohypothesesandseewhichoneistrue,orwhetherneitheristrue:Hypothesis1A: Whenanobjectisdropped, itrapidlyspeedsuptoacertainnatural fallingspeed, and then continues to fall at that speed. The falling speed is proportional to the objectsweight. (Aproportionalityisnotjustastatementthatifonethinggetsbigger,theotherdoestoo. Itsaysthatifonebecomesthreetimesbigger,theotheralsogetsthreetimesbigger,etc.)Hypothesis1B:Dierentobjectsfallthesameway,regardlessofweight.Testthesehypothesesanddiscussyourresultswithyourinstructor.2. Asecondwaytoslowdowntheactionistoletaball roll downaramp. Thesteepertheramp,the closer to free fall. Based on your experience in part 1,write a hypothesis about whatwill happenwhenyouraceaheavierball againstalighterball downthesameramp, startingthembothfromrest.Hypothesis:Showyourhypothesistoyourinstructor,andthentestit.Youhaveprobablyfoundthatfallingwasmorecomplicatedthanyouthought! Istheremorethanonefactorthataectsthemotionofafallingobject? Canyouimaginecertainidealizedsituationsthataresimpler? Trytoagreeverballywithyourgrouponaninformal model offallingthatcanmakepredictionsabouttheexperimentsdescribedinparts3and4.3. Youhavethreeballs: astandardcomparisonballof mediumweight, alightball, andaheavyball. Supposeyoustandonachair and(a) dropthelight ball sidebysidewiththecomparisonball, then(b)droptheheavyball sidebysidewiththecomparisonball, then(c)jointhelightandheavyballstogetherwithstickytapeanddropthemsidebysidewiththecomparisonball.Useyourmodeltomakeaprediction:Testyourprediction.Exercise 0: Models and idealization 394. Yourinstructorwill pumpnearlyall theairoutof achambercontainingafeatherandaheavierobject,thenletthemfallsidebysideinthechamber.Useyourmodeltomakeaprediction:40 Chapter 0 Introduction and reviewa / Amoebas this size areseldom encountered.Life would be very different if youwere the size of an insect.Chapter 1Scaling and estimation1.1 IntroductionWhy cant an insect be the size of a dog?Some skinny stretched-outcellsinyourspinal cordareametertall whydoesnaturedisplaynosinglecellsthatarenotjustametertall, butameterwide, andameterthickaswell? Believeitornot, theseareques-tions that can be answered fairly easily without knowing much moreaboutphysicsthanyoualreadydo. Theonlymathematical tech-niqueyoureallyneedisthehumbleconversion,appliedtoareaandvolume.Area and volumeAreacanbedenedbysayingthat wecancopytheshapeofinterestontographpaperwith1cm 1cmsquaresandcountthenumberofsquaresinside. Fractionsofsquarescanbeestimatedbyeye. Wethensaytheareaequalsthenumberofsquares,inunitsofsquarecm. AlthoughthismightseemlesspurethancomputingareasusingformulaelikeA=r2foracircleorA=wh/2foratriangle, those formulae are not useful as denitions of area becausetheycannotbeappliedtoirregularlyshapedareas.Units of square cm are more commonly written as cm2in science.41Of 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,soweshouldalsosaythatonecanberepresentedby100cm1m100cm1m,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 amillion 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 ABased 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.52242 Chapter 1 Scaling and estimation Solved problem: converting mm2to cm2page 59, problem 10 Solved problem: scaling a liter page 60, problem 19Discussion 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.c / Galileo Galilei (1564-1642) was a Renaissance Italian who brought thescientic method to bear on physics, creating the modern version of thescience. Coming from a noble but very poor family, Galileo had to dropout of medical school at the University of Pisa when he ran out of money.Eventuallybecomingalecturerinmathematicsatthesameschool, hebegan a career as a notorious troublemaker by writing a burlesque ridi-culing the universitys regulations he was forced to resign, but found anew teaching position at Padua. He invented the pendulum clock, inves-tigated the motion of falling bodies, and discovered the moons of Jupiter.ThethrustofhislifesworkwastodiscreditAristotlesphysicsbycon-fronting it with contradictory experiments, a program that paved the wayfor Newtons discovery of the relationship between force and motion. Inchapter 3 well come to the story of Galileos ultimate fate at the hands ofthe Church.1.2 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 SwiftNowhowdotheseconversionsofareaandvolumerelatetothequestionsIposedaboutsizesof 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, andjumptomanytimesyourownheight. 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 lawsSection 1.2 Scaling of area and volume 43d / 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.of natureasamachine. Wewill followhisleadbyrstdiscussingmachinesandthenlivingthings.Galileo on the behavior of nature on large and small scalesOne 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 beraised, 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.44 Chapter 1 Scaling and estimationg / 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.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.h / 1. Thisplankisaslongasitcanbewithout collapsingunderitsownweight. If it wasahun-dredth of an inch longer, it wouldcollapse. 2. Thisplankismadeout of the same kind of wood. It istwice as thick, twice as long, andtwice as wide. It will collapse un-der its own weight.Section 1.2 Scaling of area and volume 45Also, 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-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: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, andthetotal1Galileomakes aslightlymorecomplicatedargument, takingintoaccounttheeectofleverage(torque). TheresultImreferringtocomesoutthesameregardlessofthiseect.46 Chapter 1 Scaling and estimationi / The area of a shape isproportional tothesquareof itslinear dimensions, even if theshape is irregular.surfaceof 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.Scaling of area and volume for irregularly shaped objectsYouprobablyarenotgoingtobelieveGalileosclaimthatthishasdeepimplicationsforallofnatureunlessyoucanbeconvincedthatthesameistrueforanyshape. 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,thecorrespondingsquareonthesmallestviolinhashalftheheight