Physics for scientists and engineers with modern physics 4th edition douglas c. giancoli all chapter
Physics for Scientists and Engineers with Modern Physics 4th Edition Douglas C. Giancoli
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eTextbook 978-0131495081 Physics for Scientists & Engineers with Modern Physics (4th Edition)
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QUESTlOKS
1 1 INTRODUCTION, MEASUREMENT, ESTIMATING
1-1 The Nature of Science 1-2 Models, Theories, and Laws 1-3 Measurement and Uncertainty; Significant Figures 1-4 Units, Standards, and the SI System 1-5 Converting Units 1-6 Order of Magnitude: Rapid Estimating
9-1 Momentum and Its Relation to Force 215 9-2 Conservation of Momentum 217
9-3 Collisions and Impulse 220
9-4 Conservation of Energy and Momentum in Collisions 222
9-5 Elastic Collisions in One Dimension 222 9-6 Jnelastic Collisioni; 225 9-7 Collisions in Two or Three Dimensions 227
9-8 Center of Mass (C:\-1) 230 9-9 Center of Mass and Translational Motion 234
*9-10 Systems of Variable~; Rocket Propulsion 236 SUMMARY 239 Qt;RSTIONS 239 PROBLEMS 240 GENERAL PROBLEMS 245
10
ROTATIONAL MOTION 248
10-5 Rotational Dynamics; Torque and Rotational Inertia
1()-6 Solving Problems in Rotational Dynamics 26() 10-7 Determining Moments of Inertia 2(i3 10-8 Rotational Kinetic Energy 265 10-9 Rotational Plus Translational Motion; Rolling 267
*10-10 Why Doel$ a Rolling Sphere Slow Down? 273 SUMMARY 274 Ql."ESTIONS 275 PROBLEMS 276 GilNERAL PROBLnMS 281
11 ANGUlAR MOMENrnM; GENERAL ROTATION 284 11-1 Angular Momentum-Objects Rotating About a Fixed Axis 285 11-2 Vector Cross Product;Torque as a Vector 289
11-3 Angular Momentum of a Particle 291 11-4 Angular Momentum and Torque for a System of Particles; General Motion 292 11-5 Angular Momentum and Torque for a Rigid Object 294
11-6 Conservation of Angular Momentum 297
*11-7 The Spinning Top and Gyroscope 299
*11-8 Rotating Frames of Reference; Inertial Forces 300
*11-9 The Coriolis Effect
SUMMARY 302
OUl:iSTIO:,jS 303 PRORI.RMS 303
PROHLl:i.MS 308 301
CONTENTS 9 LINEAR MOMENTUM 214
12 STATIC EQUILIBRIUM; EIASTICfIY AND FRACTIJRE 311
12-1 The Conditions for Equilibrium 312 12-2 Solving Statics Problems 313
12-3 Stability and Balance 317
12-4 Elasticity; Stress and Strain 318
12-5 Fracture 322
*12-6 Trusses and Bridges 324
*12-7 Arches and Domes 327
SUMMARY 329 QUESTIONS 329 PROBI.RM$ 330 GENERAL PRORI.F.MS 334
13 FLums
13-1 Phases of Matter
13-2 Density and Specific Gravity
13-3 Pressure in Fluids
13-4 Atmospheric Pressure and Gauge Pressure 345
13-5 Pascal's Principle 346
l 3-6 Measurement of Pressure; Gauges and the Barometer 346 13-7 Buoyancy and Archimedes' Principle 348
13-8 Fluids in Motion; Flow Rate and the Equation of Continuity 352 13-9 Bernoulli's Equation 354
13-10 Applications of Bernoulli's Principle: Torricelli, Airplanes, Baseballs, TIA 356
*13-11 Viscosity 358
*13-12 Flow in Tubes: Poiseuille's Equation, Blood Flow 358
*13-13 Surface Tension and Capillarity 359
*13-14 Pumps, and the Heart 361
SUMMARY 361 QUESTIONS 362 PROBIBMS 363 GENERAL PROBLEMS 367
14 0SCIU.ATIONS
14-1 Oscillations of a Spring
14-2 Simple Harmonic Motion
14-3 Energy in the Simple Harmonic Oscillator
14-4 Simple Harmonic Motion Related to Uniform Circular Motion
14-5 The Simple Pendulum
*14-6 The Physical Pendulum and the Torsion Pendulum
SUMMARY 490 QUESTIONS 491 PROBLEMS 492 GENERAL PROBLEMS 494
19 H eatandthe F irst Law of T hermod namics 496 19-1 HeatasEnergyTransfer 497 19-2InternalEnergy 498 19-3Specific Heat 499 19-4 alorimetrySolving Problems500 19-5LatentHeat 502
19-6The First LawofThermodynamics505 19-7The First LawofThermodynamics Applied; alculatingtheWork507 19-8MolarSpecific HeatsforGases, andtheEquipartitionofEnergy511
19-9Adiabatic Expansionof a Gas514 19-10 HeatTransfer: onduction, onvection, Radiation 515
SUMMARY 520 QUESTIONS 521 PROBLEMS 522 GENERAL PROBLEMS 526 20 S econd Lawof T hermod namics
20-1TheSecondLawof ThermodynamicsIntroduction529 202HeatEngines 530 203ReversibleandIrreversible Processes; the arnot Engine 533 204Refrigerators,Air onditioners, and HeatPumps 536 205Entropy 539 206EntropyandtheSecondLawof Thermodynamics 541 207 Order toDisorder 544 208UnavailabilityofEnergy; Heat Death545
I wasmotivated from thebeginning towrite a textbookdifferent from othersthat present physics asa sequenceof facts,like a Sears catalog: “herearethefacts and you betterlearnthem.”Insteadof thatapproachin which topics arebegun formally anddogmatically, I havesought tobegineachtopicwithconcrete observations andexperiences studentscan relateto: start with specificsand only then goto thegreat generalizations and themoreformal aspectsofa topic, showing w y webelieve whatwe believe.This approachreflects how scienceisactuallypracticed.
Why FourthEdition?
Tworecenttrendsin physics texbooksaredisturbing: (1) theirrevisioncycles havebecomeshorttheyarebeingrevisedevery3or4years; (2) thebooksare gettinglarger, someover1500pages. I don’tsee how eithertrendcan beof benefittostudents. My response: (1) Ithas been8yearssincetheprevious editionofthisbook.(2) This bookmakesuseofphysics educationresearch, althoughit avoids thedetail a Professor may needtosay in classbut in a bookshuts downthereader. Andthisbookstill remainsamongtheshortest. Thisneweditionintroducessomeimportantnew pedagogictools. Itcontains new physics (such asin cosmology) andmanynewappealingapplications (list on previous page).Pages andpagebreakshavebeencarefully formattedtomakethe physicseasiertofollow: noturninga pagein themiddleof a derivation orExample. Greatefforts weremadeto makethebookattractive sostudents will want to read it. Some of thenewfeaturesarelistedbelow.
Wh t'sNew
C apter-OpeningQuestions :Each hapter begins witha multiple-choice question, whose responses include commonmisconceptions. Students areaskedtoanswer beforestartingthe hapter,togettheminvolved in thematerial andtoget any preconceivednotionsout onthetable. The issuesreappearlaterin the hapter, usually asExercises, afterthematerial has beencovered. The hapterOpening Questionsalsoshowstudentsthepowerandusefulness of Physics.
APPROACHparagrap in workedout numerical Examples.A short introductory paragraphbeforetheSolution, outlininganapproachandthestepswecan taketo getstarted. Brief NOTESaftertheSolution may remarkontheSolution, may give an alternateapproach, ormentionan application.
StepbyStepExamples: Aftermany ProblemSolvingStrategies(more than20in thebook),thenext Exampleisdonestep-by-step following precisely thestepsjust seen.
Exercises within thetext,afteran Exampleorderivation, give studentsachanceto see if theyhaveunderstoodenoughtoanswer a simple questionordo a simple calculation. Manyaremultiplechoice.
Greater clarity: No topic, noparagraphin this bookwas overlookedin thesearch toimprovetheclarity andconciseness ofthepresentation.Phrases andsentences thatmayslowdown theprincipalargument havebeeneliminated:keeptothe essentialsatfirst, give theelaborationslater.
F,y, B Vector notation,arrows: Thesymbols forvectorquantitiesinthetext andFigures now havea tiny arrowoverthem, sotheyaresimilar towhat wewriteby hand. Cosmological Revolution: Withgeneroushelpfromtopexpertsin thefield, readershavethelatestresults.
Page layout :morethanin thepreviousedition, seriousattentionhasbeenpaidto how each pageisformatted. Examples andallimportantderivations and argumentsareon facingpages. Studentsthendon’thavetoturnbackandforth. Throughout, readerssee, ontwo facingpages, an important sliceof physics. NewApplications'. L Ds, digital camerasandelectronicsensors( D, MOS), electrichazards,GF Is, photocopiers, inkjetandlaser printers, metaldetectors, underwatervision, curve balls, airplanewings,DNA, how we actually see images. (Turn backa pagetosee a longer list.)
T isBookisS orter thanothercompletefull-servicebooks atthis level. Shorter explanationsareeasiertounderstandandmorelikelytoberead.
Content ndOrg niz tion lCh nges
• Rot tion l Motion: hapters10and11have beenreorganized. All of angular momentumisnow in hapter11.
• First l w ofthermodyn mics, in hapter19,has beenrewrittenandextended. The full form is given:AK +A U+ AEintQW, where internal energy is E te,and U ispotential energy;the form Q W iskept so that dWPdV.
• KinematicsandDynamics of ircular Motionarenow treatedtogetherin hapter5.
• Work andEnergy, hapters7and8,have beencarefully revised.
• Work donebyfriction isdiscussed now with energy conservation(energy terms duetofriction).
• Graphical Analysis andNumerical Integrationisa new optional Section 29. Problems requiringa computerorgraphingcalculator arefoundattheend of most hapters.
• Lengthofan objectisa script £ ratherthannormal /, which lookslike 1orI (moment ofinertia,current),asin FIIB. apital L isfor angular momentum,latentheat,inductance, dimensions of length [L\.
• Newton’slawofgravitationremainsin hapter6.Why? Becausethe 1/r2 lawistooimportanttorelegatetoa latechapterthatmightnotbecovered atalllatein thesemester; furthermore, itisoneof thebasic forces in nature. In hapter8we can treatrealgravitationalpotentialenergy andhavea fine instanceof using U JF• di.
• New Appendicesincludethedifferential formof Maxwell’sequationsand moreondimensional analysis.
LaszloTakac,University ofMaryland Baltimore o. FranklinD.Trumpy,Des MoinesArea ommunity ollege
RayTurner, lemsonUniversity
SomTyagi,Drexel University
JohnVasut,Baylor University
Robert Webb,TexasA&M
Robert Weidman,MichiganTechnologicalUniversity
EdwardA. Whittaker, StevensInstitute ofTechnology
JohnWolbeck,Orange ounty ommunity ollege
Stanley George Wojcicki,StanfordUniversity
EdwardWright, U LA
ToddYoung,WayneState ollege
WilliamYounger, ollegeofthe Albemarle HsiaoLing Zhou, Georgia State University
Iowespecial thankstoProf. BobDavisformuchvaluableinput,andespeciallyfor working outalltheProblems and producingtheSolutionsManual forallProblems, as well as forprovidingtheanswerstooddnumberedProblemsat theendofthis book. Manythanksalso toJ.ErikHendricksonwhocollaboratedwithBobDavisonthe solutions, andtotheteamtheymanaged(Profs. AnandBatra,MeadeBrooks,David urrott,BlaineNorum,Michael Ottinger,LarryRowan, RayTurner, JohnVasut, William Younger).Iamgrateful toProfs. JohnEssick, BruceBarnett,Robert oakley, BimanDas,MichaelDennin,KathyDimiduk,JohnDiNardo,ScottDudley, DavidHogg, indySchwarz, RayTurner,andSomTyagi, whoinspiredmanyof theExamples,Questions,Problems, andsignificant clarifications. rucialforrootingouterrors,as well asprovidingexcellentsuggestions, were Profs. KathyDimiduk,Ray Turner, andLorraineAllen.Ahugethankyoutothem andtoProf. GiuseppeMolesini forhis suggestions andhis exceptionalphotographs foroptics.
For hapters 43and44onParticlePhysicsand osmologyand Astrophysics, I wasfortunateto receive generous input fromsome of thetopexperts inthefield, to whomIoweadebt of gratitude:George Smoot,Paul Richards, Alex Filippenko, JamesSiegrist,and WilliamHolzapfel (U Berkeley), LymanPage (Princeton and WMAP), EdwardWright(U LAandWMAP), and MichaelStrauss(University ofOklahoma).
I especiallywishto thank Profs.HowardShugart, hair FrancesHeilman, and manyothers at theUniversity of alifornia, Berkeley, PhysicsDepartment for helpfuldiscussions,andfor hospitality. Thanks alsoto Prof.TitoArecchi and others at the IstitutoNazionale diOttica, Florence,Italy.
Finally,I amgratefulto themany peopleat Prentice Hall withwhomI worked on thisproject, especiallyPaul orey, KarenKarlin, hristian Botting, John hristiana, and SeanHogan.
The finalresponsibilityfor allerrors lieswithme.Iwelcomecomments, corrections,and suggestions as soon as possibleto benefit students forthe next reprint.
D. .G.
email:Paul. orey@Pearson.com
Post:Paul orey
OneLake Street
Upper Saddle River, NJ 07458
AbouttheAuthor
Douglas .Giancoli obtained hisBAinphysics(summacumlaude) fromthe University of alifornia, Berkeley,hisMSinphysicsat theMassachusettsInstitute ofTechnology,andhisPhD inelementary particle physicsat theUniversityof ali fornia, Berkeley.He spent 2yearsasapost-doctoral fellowat U Berkeley’sVirus labdevelopingskillsinmolecular biologyand biophysics.Hismentors include Nobel winnersEmilioSegreand Donald Glaser.
He hastaughtawiderange of undergraduatecourses,traditional aswellas innovative ones,and continues toupdatehis texbooksmeticulously,seeking waysto betterprovide an understanding of physicsfor students.
Doug’sfavorite spare-time activityistheoutdoors, especiallyclimbingpeaks (hereonadolomite summit, Italy). Hesaysclimbing peaks islikelearning physics: it takes effort andtherewards are great.
OnlineSupplements(p rti l list)
M steringPhysics™(www.m steringphysics.com) is sophistic tedonlinetutoring ndhomeworksystemdevel opedspeci llyforcoursesusingc lculusb sedphysics. Origin llydevelopedbyD vidPritch rd ndcoll bor tors t MIT, M steringPhysics providesstudents with individu lized online tutoringbyrespondingtotheirwrong nswers ndproviding hints for solving multistep problems when theyget stuck. It gives them immedi te nduptod te ssessment oftheirprogress, nd showswheretheyneedtopr cticemore.M steringPhysics provides instructors with f st nd effective w y to ssign tried nd-testedonlinehomework ssignments th t comprise r nge of problemtypes.Thepowerfulpost ssignmentdi gnostics llow instructors to ssessthe progressoftheir cl ss s whole s well s individu lstudents, ndquicklyidentify re s ofdifficulty.
WebAssign(www.web ssign.com) CAPA ndLONCAPA(www.lonc p .org)
Student Supplements (p rti llist)
StudentStudyGuide&SelectedSolutionsM nu l(VolumeI: 013-2273241, VolumesU&III:013-227325X)by Fr nk Wolfs StudentPocketComp nion(0132273268)byBim nD s
Educ tionGroup t theUniversityofW shington Physlet®Physics(0131019694) byWolfg ngChristi n ndM rioBelloni
R nking T skExercises in Physics, Student Edition (0-13-144851-X) by Thom s L. OKum , D vid P.M loney, ndCurtis J.Hieggelke
E&MTIPERs:Electricity&M gnetismT sks Inspiredby Physics Educ tionRese rch(0-131854992)byCurtisJ.Hieggelke, D vidP.M loney, StephenE. K nim, ndThom sL. OKum
M them ticsforPhysicswithC lculus(0-131913360) byBim nD s
2. Attendallclassmeetings. Listen. Take notes, especially aboutaspects youdo notrememberseeing inthebook. Askquestions(everyoneelse wants to, but maybeyouwillhavethecourage).You willget moreoutof classif youread the hapterfirst.
4. Solve 10to20endof hapterProblems (ormore), especially thoseassigned. IndoingProblemsyoufind outwhatyou learnedandwhat you didn’t.Discuss themwith otherstudents. Problemsolvingisoneofthegreatlearning tools. Don’tjustlookfora formulaitwon’tcutit.
NOTESONTHEFORMATANDPROBLEM SOLVING
1.Sections markedwith a star(*)areconsidered option l. They can beomitted withoutinterruptingthemainflowoftopics. No latermaterialdepends on themexceptpossibly laterstarredSections. They maybefuntoread, though.
2. The customary conventions areused: symbols for quantities(such as m for mass) areitalicized, whereasunits(such asm formeter)arenotitalicized. Symbolsforvectorsareshown in boldface with a small arrowabove: F.
3. Few equationsarevalid in all situations. Wherepractical, the limit tions of importantequationsarestatedin squarebracketsnexttotheequation. The equationsthatrepresentthegreatlawsofphysics aredisplayed with a tan background, asarea fewotherindispensableequations.
4. Attheend of each hapter isa set of Problems whichare ranked asLevelI,II, or III, according to estimateddifficulty.Level I Problems areeasiest, Level IIare standardProblems, and Level IIIare“challenge problems.”These ranked Problems are arrangedbySection, butProblems fora givenSectionmay depend on earlier material too. Therefollowsa group ofGeneral Problems, whichare not arrangedbySection norrankedastodifficulty.Problems thatrelatetooptional Sectionsarestarred(*).Most hapters have 1or2 omputer/Numerical Problems at theend, requiring a computer orgraphing calculator. Answers to odd-numbered Problems are givenattheendofthebook.
5. Being able tosolve Problems isa crucial partoflearning physics, andprovides a powerful means for understandingtheconceptsandprinciples. Thisbook containsmanyaidstoproblemsolving: (a) workedout Ex mples andtheir solutions in thetext,whichshouldbestudied asan integral partofthetext; (b)someof theworked-outExamples are Estim tionEx mples, whichshow howroughorapproximateresultscan beobtainedeven if thegiven dataare sparse (see Section 16);(c)special ProblemSolvingStr tegies placed throughout thetext tosuggest a step-by-stepapproachtoproblemsolving for a particulartopicbutrememberthatthebasicsremainthesame; most ofthese“Strategies”arefollowed by an Examplethatissolved by explicitlyfollowingthesuggested steps; (d) special problem-solvingSections; (e) “ProblemSolving”marginal notes whichrefertohintswithin thetextfor solvingProblems; (f) Exercises withinthetextthatyoushouldwork outimme diately, andthencheck your responseagainst theanswergivenatthebottomof thelast pageofthat hapter; (g)theProblems themselvesat theendofeach hapter(point 4above).
6.Conceptu l Ex mples posea questionwhich hopefullystartsyoutothinkand come upwith a response.Give yourselfa littletimetocome upwith yourown responsebeforereadingtheResponsegiven.
7.M th review,plussome additional topics,arefound inAppendices. Useful data, conversion factors, and math formulas arefound inside the front and back covers.
Vectors
A gener l vector
result nt vector (sum) is slightly thicker components of nyvector red shed
\Westart eac C apter wit aQuestion, like t eone above. Tryto answerit rig t away.Dontworry aboutgetting t erig t answernowt eideaisto getyourpreconceivednotions outont etable.Ift ey are misconceptions, weexpect t em tobe cleared up as youreadt eC apter. Youwill usuallygetanot er c anceatt eQuestion later int eC apter w ent eappropriatematerial asbeen covered. T ese C apter-OpeningQuestions willalso elp youto see t epowerand usefulness ofp ysics. ]
CONTENTS
11The N tureof Science
12Models,Theories, ndL ws
13Me surement nd Uncert inty; Signific nt Figures
14Units, St nd rds, nd theSI System
15ConvertingUnits
16OrderofM gnitude: R pidEstim ting
:17Dimensions ndDimension l An lysis
(b)
FIGURE11( )ThisRom n queductw sbuilt2000ye rs go ndstillst nds,(b)TheH rtford CivicCentercoll psedin1978, just twoye rs fteritw sbuilt.
Physicsisthemost basicofthesciences.Itdealswith thebehavior and structure of matter. The fieldof physicsis usuallydivided into classicalp ysics whichincludesmotion,fluids,heat, sound, light,electricityand magnetism; and modem p ysics whichincludesthetopicsof relativity,atomicstructure, condensedmatter, nuclearphysics,elementaryparticles,and cosmologyand astrophysics. Wewillcoverall these topicsinthisbook, beginningwithmotion (ormechanics, asit is often called)and endingwiththemost recentresults in ourstudy of thecosmos. Anunderstandingof physics iscrucial foranyonemaking acareerin science or technology. Engineers, forexample, must know how to calculate the forces within astructureto design it sothatit remains standing(Fig.1la).Indeed, in hapter12 wewillsee a workedoutExampleofhowasimple physicscalculationoreven intuitionbasedonunderstandingthephysics offorceswouldhave saved hundredsoflives(Fig.1lb).Wewillsee many examples inthisbookofhow physics isuseful in manyfields,andin everyday life.
1—1The N tureofScience
The principal aimofall sciences, including physics, isgenerallyconsidered to be thesearchfor orderinourobservations oftheworld aroundus.Many people thinkthatscience isa mechanical process ofcollecting factsanddevising theories. But it isnotsosimple.Scienceisacreative activitythatinmany respects resem blesothercreative activitiesof thehumanmind.
Oneimportant aspectofscience is observ tion ofevents, whichincludes the designandcarrying outofexperiments. Butobservation and experiment require imagination, for scientists canneverinclude everything inadescriptionofwhat theyobserve. Hence, scientists must makejudgmentsaboutwhat isrelevant in theirobservationsandexperiments. onsider, for example, how twogreat minds, Aristotle(384-322 b .c.) and Galileo (1564-1642), interpretedmotionalongahorizontal surface.Aristotle notedthatobjects givenan initial pushalong theground(oron a tabletop)always slowdown andstop. onsequently, Aristotleargued thatthenaturalstateofan object istobeat rest. Galileo, inhisreexaminationofhorizontalmotion inthe 1600s,imaginedthatif frictioncouldbeeliminated, an object given aninitial pushalong ahorizontalsurfacewouldcontinuetomove indefinitelywithout stopping. Heconcluded thatforan object to beinmotionwasjustasnatural asfor ittobeat rest. Byinventing anewapproach,Galileo founded ourmodernviewof motion( hapters2,3,and4),and hedid sowith a leapof theimagination. Galileo madethisleapconceptually,without actually eliminating friction.
Observation, withcarefulexperimentation and measurement, isone sideofthe scientificprocess.Theothersideistheinvention orcreation oftheoriestoexplain and ordertheobservations. Theories are never derived directlyfrom observations. Observations may help inspire atheory, and theories are accepted or rejectedbased on theresultsof observation and experiment.
The great theories of science maybecompared,ascreative achievements, with greatworks ofartorliterature. Buthow does science differ from theseother creative activities?Oneimportant difference isthatsciencerequires testing of its ideas ortheoriesto see if theirpredictionsareborneoutby experiment.
Althoughthetesting oftheories distinguishes sciencefrom othercreative fields,itshouldnotbeassumed thatatheoryis“proved”by testing. Firstofall,no measuring instrument isperfect, soexact confirmation isnotpossible. Further more, it isnot possible totesta theoryinevery singlepossible circumstance. Hence atheorycannot be absolutely verified.Indeed, thehistory ofsciencetellsusthat long-held theories can bereplacedby newones.
1-2Models,Theories, ndL ws
Whenscientists aretrying tounderstandaparticularset of phenomena, theyoften makeuse of a model. Amodel, in thescientist’ssense,isakindofanalogy or mentalimage ofthephenomenaintermsof something we arefamiliar with. One
The purposeof a model istogiveusanapproximatemental orvisual picture something toholdontowhenwecannot see what actually ishappening.Models oftengiveus a deeperunderstanding:theanalogy toa known system (forinstance, water wavesin theabove example)cansuggest newexperimentstoperformandcan provideideas aboutwhat otherrelatedphenomenamight occur.
Youmaywonder what thedifferenceisbetweenatheoryanda model.Usually a model isrelativelysimple andprovidesastructuralsimilarity tothephenomena beingstudied. A theory isbroader, moredetailed, andcan givequantitativelytestable predictions,oftenwith greatprecision.
Itisimportant,however, nottoconfusea modeloratheorywiththereal system orthephenomenathemselves.
Scientistsgive thetitle l w tocertainconcise butgeneralstatementsabout how naturebehaves(thatenergy isconserved, forexample). Sometimesthestate menttakestheformofarelationshiporequationbetweenquantities(suchas Newton’ssecondlaw, Fma ).
Tobecalled a law, astatementmust befoundexperimentallyvalid overa wide rangeof observedphenomena.Forlessgeneralstatements,theterm principle is oftenused(such asArchimedes’principle).
Scientists normallydotheirresearchasif theacceptedlawsandtheorieswere true. Buttheyareobliged tokeepanopenmindin case newinformationshould alterthevalidity ofany given lawortheory.
1-3Me surement ndUncert
inty; Signific ntFigures
Inthequesttounderstandtheworldaroundus, scientistsseektofind relationships among physicalquantitiesthatcanbemeasured.
Uncert inty
Reliable measurements areanimportant partof physics.But no measurementis absolutelyprecise. There isanuncertainty associated witheverymeasurement. Among the most important sourcesofuncertainty, other thanblunders,arethelimited accuracy of every measuring instrumentandtheinabilitytoreadaninstrument beyond some fractionofthesmallestdivisionshown. For example,ifyou were touse a centimeter ruler tomeasurethewidth of a board(Fig.12), theresult could beclaimed tobe precise toabout 0.1cm(1mm), thesmallestdivisionontheruler, although half of this value might beavalidclaimaswell.The reasonisthat it isdifficult for theobserverto estimate (or interpolate)betweenthesmallest divisions.Furthermore, theruleritself may nothave beenmanufactured to anaccuracy very much better thanthis. Whengiving theresultof ameasurement, it isimportanttostatethe estim ted uncert inty in themeasurement. Forexample,thewidthof a boardmight be writtenas8.8±0.1cm. The ±0.1cm(“plus orminus 0.1cm”)representsthe estimateduncertaintyin themeasurement,so thattheactualwidthmost likely lies between8.7and8.9 cm.The percent uncert inty istheratioof theuncertainty tothemeasuredvalue, multipliedby100.Forexample, if themeasurementis8.8 andtheuncertaintyabout0.1cm, thepercentuncertaintyis
1%, wheremeans“isapproximatelyequalto.’
FIGURE12 Measuringthe width ofaboardwith acentimeter ruler. Theuncertaintyisabout±1 mm.
(a) (b)
FIGURE13 Thesetwocalculators showthewrongnumber ofsignificant figures.In(a),2.0was dividedby3.0. The correctfinalresultwould be0.67. In (b),2.5 was multiplied by 3.2.The correctresultis 8.0.
pPR BLEMS LVING
Significantfigurerule:
Numberofsignificantfiguresinfinal results ouldbesameast eleast significant input value
A AUTION
Calculators errwit significantfigures
IPR BLEIVIS LVING
Reportonlyt epropernumber of significantfiguresint efinalresult. Keepextradigitsduring t ecalculation
FIGURE14 Example1-1. A protractorusedtomeasureanangle.
Often the uncertainty ina measured value isnot specifiedexplicitly.In suchcases, the uncertainty isgenerally assumed tobeone or afew units inthe last digit specified. For example, if a length isgivenas8.8cm,theuncertainty isassumed to beabout 0.1cmor0.2cm.Itisimportant inthiscasethatyou donotwrite 8.80cm,forthis impliesan uncertainty on theorderof 0.01 cm;it assumesthatthelength isprobably between8.79cmand 8.81 cm,when actually you believe it isbetween 8.7and 8.9cm.
Signific ntFigures
The numberofreliablyknown digitsin anumberiscalled thenumberof signific ntfigures. Thus therearefoursignificant figures in thenumber23.21cm andtwo inthenumber0.062cm(thezeros inthelatteraremerelyplaceholders thatshow wherethedecimal pointgoes). Thenumberofsignificant figures may notalwaysbeclear. Take, forexample, thenumber80.Arethereoneortwo significantfigures? Weneedwords here:Ifwesayit is roug ly 80kmbetweentwo cities, thereisonlyonesignificant figure(the8)sincethezero ismerelya place holder. Ifthereisnosuggestionthatthe80isa roughapproximation, thenwe can often assume (as wewillinthis book)thatitis80km within an accuracy ofabout 1or2km, andthenthe80has two significant figures. Ifitisprecisely 80km, to within+0.1km, thenwe write80.0km(threesignificant figures).
When making measurements,orwhen doing calculations, you shouldavoidthe temptationtokeepmoredigitsin thefinal answer thanisjustified. For example, to calculate theareaof a rectangle 11.3 cmby 6.8cm, theresult ofmultiplicationwould be76.84cm2.Butthisanswer isclearlynotaccurateto0.01cm2,since(usingthe outerlimitsof theassumed uncertaintyfor eachmeasurement)theresult couldbe between11.2cm X 6.7cm75.04cm2and 11.4cm X 6.9cm78.66cm2.Atbest, wecanquote theanswer as77cm2, whichimplies an uncertainty ofabout 1or 2 cm2. The other two digits(in the number76.84 cm2)must bedropped because they are not significant.As a roughgeneral rule(i.e.,intheabsence ofa detailedconsideration of uncertainties), we can say that t efinalresultofa multiplication or division s ould aveonlyasmanydigits ast e numberwit t e least numberofsignificant figures used int ecalculation. Inourexample, 6.8cmhas theleastnumberofsignificant figures, namelytwo.Thus theresult76.84cm2needstoberoundedoff to77 cm2.
EXERCISEA Thearea ofarectangle4.5cm by 3.25cm is correctly given by (a) 14.625cm2; (b) 14.63cm2;(c)14.6cm2;(d)15cm2.
Whenadding orsubtractingnumbers,thefinal resultisno moreprecisethan theleast precise numberused. For example, theresultofsubtracting0.57from3.6 is3.0(andnot3.03).
Keep in mind whenyou use a calculatorthatallthedigitsit producesmaynot besignificant. Whenyoudivide 2.0by3.0,theproperanswer is0.67,andnotsome such thingas0.666666666.Digits shouldnotbequotedin a result, unless theyare trulysignificant figures. However, toobtainthemostaccurateresult,youshould normally keep one ormoreextrasignificant figures t roug outacalculation, and roundoffonlyint efinalresult. (With a calculator, you can keepallitsdigitsin intermediateresults.) Notealso thatcalculatorssometimes givetoofewsignificant figures. Forexample, whenyou multiply2.5 X 3.2,a calculatormaygivethe answer assimply 8.But theanswer isaccurate totwosignificant figures,sotheproper answeris8.0.See Fig.13.
C NCEPTUALEXAMPLE11|Significant figures.
Usinga protractor(Fig. 14), you measure an angle tobe 30°. (a) Howmany significant figures should you quotein thismeasurement?(b)Use a calculator tofind the cosine ofthe angleyou measured. RESP NSE (a) Ifyou lookata protractor,you willseethattheprecisionwith which youcan measureanangle isaboutonedegree(certainlynot0.1°). Soyou can quotetwo significant figures, namely, 30°(not30.0°). (b )If youentercos30° inyourcalculator,you willget a numberlike 0.866025403.However, theangle you enteredisknown onlytotwo significant figures, soitscosine iscorrectly given by 0.87;youmustroundyouranswertotwo significant figures.
Wecommonly writenumbers in “powersoften,”or“scientific”notationfor instance36,900as3.69 X 104,or0.0021as2.1 X 103.Oneadvantageofscientific notationisthatitallowsthenumberofsignificant figures tobeclearly expressed. Forexample, it isnotclear whether36,900hasthree, four, orfive significant figures. With powersoftennotationtheambiguity can beavoided: ifthenumberis known tothreesignificant figures, we write3.69 X 104,butif itisknowntofour, we write3.690 X 104.
The significant figures rule isonlyapproximate, and in some cases may underestimate theaccuracy (oruncertainty) of the answer.Supposefor example we divide 97by 92: 97 1.05«1.1. 92
SUGGESTION: Usethesignificant figures rule, butconsiderthe%uncer taintytoo, andaddan extradigitif it gives a morerealisticestimateofuncertainty.
Approxim tions
Muchofphysics involves approximations,oftenbecausewedonothavethe means tosolve a problemprecisely. For example, wemay choosetoignoreair resistanceorfrictionindoinga Problemeventhoughtheyarepresentinthereal world,andthenourcalculationisonly an approximation.IndoingProblems,we shouldbeawareofwhat approximationswearemaking,andbeawarethatthe precisionofouranswer maynotbenearlyasgood asthenumberofsignificant figures given in theresult.
Accur cyversus Precision
There isa technical difference between “precision” and “accuracy.” Predsion in a strict sense refersto therepeatability ofthemeasurementusinga giveninstrument. For example, if you measurethewidthof a boardmany times, getting results like8.81cm, 8.85cm,8.78cm,8.82cm(interpolating betweenthe0.1 cmmarks asbest aspossible eachtime), you couldsaythemeasurementsgivea precision a bit betterthan0.1 cm. Accur cy refers tohowclosea measurement istothetruevalue.For example, if the rulershowninFig.12wasmanufacturedwith a 2%error,the accuracy ofits measurementoftheboard’swidth(about8.8cm)would beabout2%of8.8cmor about +0.2 cm.Estimated uncertainty ismeanttotakebothaccuracyand precision into account.
1—4Units, St nd rds, nd the SISystem
TABLE 1-1 SomeTypical
(orDist nce)
Neutronorproton (di meter)
Atom (di meter)
Virus[seeFig. l5 ]
Sheetofp per (thickness)
Fingerwidth
Footb llfieldlength
Heightof Mt.Everest [seeFig.l5b]
E rthdi meter
E rthtoSun
E rthtone restst r
E rthtone restg l xy
E rthtof rthest
g l xyvisible
( pproxim te)
FIGURE15Somelengths:
( )viruses( bout107 mlong) tt cking cell;(b)Mt. Everests heightisontheorderof104 m (8850 m, tobeprecise).
The measurementof any quantityismaderelativetoa particularstandardor unit, andthis unitmustbespecified along with thenumerical valueofthequantity. For example, wecan measurelengthin Britishunitssuch asinches, feet, ormiles,orin themetricsystem in centimeters, meters,orkilometers. Tospecifythatthelength ofa particularobjectis18.6ismeaningless. The unit must begiven; for clearly, 18.6metersisvery different from18.6inches or18.6millimeters.
Forany unitweuse,such asthemeterfor distanceorthesecond for time, we needtodefinea st nd rd which defines exactly how longonemeteroronesecond is.Itisimportantthatstandardsbechosen thatarereadilyreproduciblesothat anyoneneedingtomakea very accuratemeasurementcan refertothestandardin thelaboratory.
Length
The first trulyinternationalstandardwas the meter (abbreviatedm) establishedas thestandardof length by theFrenchAcademyofSciencesin the1790s.Thestan dardmeterwasoriginally chosentobeoneten-millionthofthedistancefromthe Earth’sequatortoeitherpole,fandaplatinumrodtorepresentthislengthwas made. (Onemeteris, veryroughly, thedistancefromthetipofyournosetothetip ofyour finger, with armandhandstretchedouttotheside.) In1889, themeterwas definedmorepreciselyasthedistancebetweentwofinelyengravedmarksona particularbarofplatinumiridiumalloy.In1960,toprovide greaterprecision and reproducibility, themeter wasredefinedas1,650,763.73wavelengths of a particular orangelight emittedbythegaskrypton-86.In1983themeterwasagain redefined, this timein termsof thespeedof light (whosebestmeasuredvalue intermsof the olderdefinitionofthemeterwas299,792,458m/s,withan uncertaintyoflm/s). The newdefinition reads: “Themeteristhelengthofpathtraveledby light in vacuumduring a timeinterval of1/299,792,458ofa second.”*
Britishunitsoflength(inch,foot,mile) arenowdefinedin termsofthe meter. The inch(in.) isdefinedasprecisely2.54centimeters(cm; 1cm0.01m). Otherconversionfactorsaregiven in theTableontheinside ofthefrontcover of thisbook.Table 11presentssometypical lengths, fromverysmall tovery large,roundedoff tothenearestpoweroften. See also Fig.15.[Notethatthe abbreviationforinches (in.) istheonlyonewitha period,todistinguishitfrom theword“in”.]
Time
The standard unit of time isthe second (s).Formany years, thesecond wasdefined as 1/86,400ofa mean solar day(24h/day X 60min/h X 60s/min86,400s/day). Thestandardsecond isnow defined morepreciselyintermsofthefrequency of radiation emittedby cesiumatomswhen theypassbetween twoparticular states. [Specifically,onesecondisdefined asthetime requiredfor 9,192,631,770periods of thisradiation.] Thereare,by definition, 60sin oneminute (min) and60minutes in onehour(h). Table12presents a range ofmeasuredtime intervals, roundedoffto thenearestpower of ten.
M ss
The standardunitof m ss isthe kilogr m (kg). Thestandardmass isa particular platinumiridiumcylinder, keptat theInternationalBureauof Weightsand MeasuresnearParis,France,whose mass isdefinedasexactly 1kg.Arangeof masses ispresentedinTable13.[Forpracticalpurposes, 1kgweighs about 2.2poundson Earth.]
tModern me surements ofthe E rth’scircumference reve l th t the intended length isoffby bout one-fiftieth of1%.Notb d!
*Thenewdefinition ofthe meter h sthe effectofgiving the speedoflightthe ex ct v lue of 299,792,458m/s.
(a)
TABLE 12SomeTypical TimeIntervals
TimeInterv l Seconds( pproxim te)
Lifetimeofveryunst blesub tomicp rticle
Lifetimeofr dio ctiveelements
Lifetimeofmuon
Timebetweenhum nhe rtbe ts Oned y Oneye r
TABLE 13SomeMasses
Whendealingwithatoms andmolecules, weusuallyusethe unified tomic m ssunit (u). Intermsofthekilogram, lu1.6605 X 1027kg.
Thedefinitions of otherstandardunitsforotherquantitieswillbegiven aswe encountertheminlater hapters. (Precise values of this andothernumbersare giveninsidethefront cover.)
UnitPrefixes
Inthemetric system, thelarger andsmaller units aredefined in multiples of 10from thestandardunit, and thismakes calculation particularly easy.Thus1kilometer (km) is1000m, 1centimeterisifem, 1millimeter(mm) isor ^cm,andsoon. Theprefixes “centi-,”“kilo-,”andothersare listed inTable14and can beapplied notonly tounitsoflengthbuttounitsofvolume, mass,orany othermetricunit. Forexample, a centiliter(cL) is^liter(L)>and a kilogram (kg) is1000grams (g).
Systems ofUnits
Whendealingwith thelawsandequationsofphysics it isveryimportanttouse a consistentset of units.Severalsystems of unitshave beenin useovertheyears. Today themostimportantisthe Systeme Intern tion l (French forInternational System), whichisabbreviatedSI.InSIunits, thestandardoflengthisthemeter, thestandardfortimeisthesecond, andthestandardformass isthekilogram. This system usedtobecalled theMKS(meterkilogramsecond) system.
Asecondmetricsystem isthe cgssystem, in whichthecentimeter,gram, and second arethestandardunitsof length, mass,andtime, asabbreviatedinthetitle. The Britishengineeringsystem hasasitsstandardsthefoot forlength, thepound for force, andthesecond for time.
Weuse SI unitsalmost exclusivelyinthis book.
B se versusDerivedQu ntities
Physicalquantities can bedividedinto twocategories: basequantities and derived quantities. Thecorresponding units for these quantitiesare called baseunits and derived units. A b sequ ntity must bedefined interms of a standard. Scientists, inthe interest of simplicity,want thesmallest number of base quantities possibleconsistent witha fulldescription of thephysicalworld. Thisnumber turns outto beseven,and those used in theSIare given inTable1-5. All other quantities can be defined in terms ofthese sevenbase quantities/and hence arereferredto as derivedqu ntities. An example of a derived quantity is speed, which is defined as distance dividedbythe time it takes to travel thatdistance.ATableinsidethefront cover listsmany derived quantities and their units interms of base units.To define anyquantity,whetherbase or derived, wecan specifya rule or procedure, and thisiscalledan oper tion l definition.
trTheonly exceptions refor ngle(r di ns—see Ch pter8) ndsolid ngle (ster di n). Nogener l greement h s beenre ched stowhether these reb se orderived qu ntities.
TABLE 14Metric (SI)Prefixes
PrefixAbbrevi
V 1(T6 n no n KT9 pico P 1(T12 femto f 1(T15 tto KT18
zepto z 1(T21 yocto y KT24 fju, is theGreek lettermu.
TABLE 15
SIBaseQuantitiesandUnits
Qu ntity
Lengthmeterm
Time seconds
M ss kilogr m kg
Electric current mpere A
Temper turekelvinK
Amount of subst ncemolemol
Luminous intensityc ndel cd
0PHYSICSAPPLIED
T eworld’stallestpeaks
FIGURE 16Theworlds second highestpe k,K2,whosesummitis consideredthemostdifficultof the 8000ers.K2isseenherefrom thenorth(Chin ).
TABLE16
The8000mPeaks
Pe k Height(m)
Mt.Everest 8850
K2 8611
K ngchenjung 8586
Lhotse 8516
M k lu 8462
ChoOyu 8201
Dh ul giri 8167
M n slu 8156
N ng P rb t 8125
Ann purn 8091
G sherbrumI 8068
Bro dPe k 8047
G sherbrumII 8035
Shish P ngm 8013
1—5ConvertingUnits
Anyquantitywemeasure,such as alength,aspeed,oranelectriccurrent,consists ofanumber and aunit. Oftenwe aregivenaquantityinonesetofunits, butwe wantitexpressedinanothersetofunits. Forexample, supposewe measurethata tableis21.5incheswide, andwewanttoexpressthisincentimeters.Wemustusea conversion f ctor, which inthiscaseis(bydefinition)exactly 1in. 2.54 cm or, writtenanotherway, 12.54cm/in.
Since multiplyingbyonedoesnotchangeanything, thewidthofourtable, incm, is 21.5 inches(21.5 X ^2.54^^54.6 cm.
Notehowtheunits(inches in this case) cancelled out. ATable containingmanyunit conversionsis foundinside thefront cover ofthis book.Let’sconsider some Examples.
APPR ACH We needsimply toconvertmeterstofeet,andwecanstartwiththe conversionfactor1in. 2.54 cm, whichisexact.Thatis,1in. 2.5400 cm to anynumberofsignificant figures, becauseitis defined tobe.
S LUTI N Onefootis12in., so we canwrite
cm
1 ft (12is.)(2.54J30.48 cm0.3048m,
whichisexact. Notehowtheunitscancel (coloredslashes). We canrewritethis equationtofindthenumberoffeetin1meter:
lm aUs 3'28084ft
We multiplythisequationby8000.0(tohavefivesignificantfigures):
8000.0m(8000.0l .)^3.28084;^26,247ft.
Anelevationof8000mis26,247ftabovesealevel.
N TE We couldhavedonetheconversionall inoneline: annum 26OT«.
EXERCISEEThere reonly14eightthous ndmeterpe ksintheworld(seeEx mple12), ndtheirn mes ndelev tions regiveninT ble16. They re llintheHim l y moun t inr ngeinIndi ,P kist n,Tibet, ndChin .Determinetheelev tionoftheworlds threehighestpe ksinfeet.
EXAMPLE 13 Apartmentarea. You haveseen a niceapartmentwhose floorareais880squarefeet(ft2).Whatisits areain squaremeters?
APPR ACH Weusethesame conversionfactor, 1in. 2.54 cm,butthis time wehavetouse ittwice.
S LUTI N Because lin. 2.54cm 0.0254m, then lft2(12in.)2(0.0254m/in.)2 0.0929m2.So880ft2(880ft2)(0.0929m2/ft2)«82m2.
N TE Asa ruleofthumb,anareagiven inft2isroughly10times thenumberof squaremeters(moreprecisely, about10.8 X).
APPR ACH Weagainusetheconversionfactor1in.2.54cm, andwe recall thatthereare5280ft in a mile and12inchesin a foot; also,onehourcontains (60min/h) X (60s/min)3600 s/h.
S LUTI N (a) Wecan write1mile as 1mi(5280ir)( 2.54 jGirr1m 'TRv./ \100jGfTf 1609m. We also knowthat1hourcontains3600s,so 'mi. 55 h 55 ir 1609 1JT m mL J V3600 s 25“, s
wherewe roundedofftotwo significant figures. (b) Now weuse 1mi 1609m 1.609 km; then
55 h 55 mi. 1.609 km 'mi km 88 .
N TE Eachconversionfactor isequal toone. You can lookupmost conversion factorsinthe Table inside thefrontcover.
Whenchanging units, youcanavoid makingan errorintheuse ofconversion factorsby checking thatunitscancel outproperly. For example, inourconversion of1 mi to1609m in Example14 (a), if wehadincorrectlyusedthefactor (n^) insteadof(ujoSn),thecentimeterunitswould nothave cancelledout; we would not have endedupwith meters.
1—6OrderofM gnitude:R pidEstim ting
Wearesometimesinterestedonly in anapproximatevaluefora quantity. This mightbebecausean accuratecalculationwouldtakemoretimethanitisworth orwouldrequireadditionaldatathatarenotavailable. Inothercases,we may wanttomakea roughestimatein ordertocheckanaccuratecalculationmade ona calculator,tomakesurethatnoblundersweremadewhenthenumbers wereentered. ^
Aroughestimateismadeby roundingoff all numberstoonesignificant figure \\ PR BLEMS LVING andits powerof10,andafterthecalculationismade, again only onesignificant Howtomakearoug estimate figure iskept.Such anestimateiscalledan order-of-m gnitudeestim te andcan beaccuratewithin a factorof10,andoftenbetter.Infact, thephrase“orderof magnitude”issometimes usedtorefersimplytothepowerof 10.
10 m r500 in (b)
FIGURE 17Ex mple15.( )How muchw terisinthisl ke?(Photoisof oneoftheR eL kesintheSierr Nev d ofC liforni .)(b)Modelof thel ke s cylinder.[Wecouldgoone stepfurther ndestim tethem ssor weightof thisl ke.Wewillseel ter th tw terh s densityof1000 kg/m3, sothisl keh s m ssof bout (I03kg/m3)(l07m3) «1010 kg,whichis bout10billionkgor10millionmetric tons.(Ametrictonis1000kg, bout 2200 lbs,slightlyl rgerth n British ton, 2000lbs.)]
S LUTI N Thevolume V ofacylinderistheproductofits height timesthe areaof its base: V irr2, where r istheradiusof thecircularbase.fTheradius r is \ km500m, so thevolumeisapproximately
V irr2 M(10 m) X (3) X (5 X 102m)2m8 X 106m3«107m3, where tt wasroundedoffto3.So thevolumeisontheorderof107m3,ten millioncubicmeters.Becauseofall theestimatesthatwentintothiscalculation, theorderofmagnitudeestimate(l07m3)isprobablybettertoquotethanthe 8X 106m3figure.
N TE To expressourresultinU.S.gallons,weseeintheTableontheinside frontcoverthat1liter10-3m3« \ gallon.Hence,thelakecontains (8 X 106m3)(lgallon/4 X 103m3) « 2 X 109gallons ofwater.
APPR ACH At first youmight thinkthata special measuringdevice, a micrometer (Fig. 18),isneededtomeasurethethicknessofonepagesinceanordinary rulerclearlywon’tdo. Butwecanuseatrickor, toputitinphysics terms, make useofa symmetry, wecanmakethereasonableassumptionthatall thepagesof thisbookareequalinthickness.
S LUTI N We canusearulertomeasurehundredsofpagesatonce. Ifyou measurethethicknessofthefirst500pagesofthisbook(page1topage500), youmightgetsomethinglike1.5 cm. Notethat500numberedpages, fFormul s like this forvolume, re , etc., refoundinside theb ckcoverofthisbook.
countedfrontandback, is250separatesheetsofpaper. Soonepagemusthave a thickness ofabout
APPR ACH Bystandingyour friend next tothepole, you estimate theheightof thepoletobe3m.Younext step awayfromthepole until thetopof thepoleisin line withthetopofthebuilding,Fig.l9a. You are 5ft6in.tall, soyoureyesare about1.5m abovetheground. Your friend istaller, andwhen she stretches out her arms, onehandtouches you,andtheothertouchesthepole, soyouestimate that distance as2m (Fig.l9a).Youthenpace offthedistance from thepoletothe baseof the building with big, 1-mlongsteps,and you get a total of 16steps or16m.
S LUTI N Nowyou draw,toscale, thediagramshown in Fig.l9busingthese measurements. You can measure, right on thediagram, thelast side ofthe triangletobeabout x 13 m. Alternatively, youcan usesimilar triangles to obtaintheheight x :
EXAMPLE18 ESTIMATEIEstimatingtheradiusofEarth. Believeit or not,youcan estimatetheradius oftheEarthwithout havingtogointospace (see thephotographonpage1).Ifyouhave everbeenontheshoreofa large lake, youmayhavenoticedthatyoucannot see thebeaches, piers, orrocksat water levelacross thelakeontheoppositeshore. The lakeseems tobulgeoutbetween youandtheoppositeshorea good clue thattheEarthisround.Supposeyou climb a stepladderanddiscover thatwhenyoureyes are10ft (3.0m) abovethe water, youcan justsee therocks at waterlevel ontheoppositeshore. From a map, youestimatethedistancetotheoppositeshoreas d 6.1km. Use Fig.110with 3.0m toestimatetheradius R of theEarth.
APPR ACH Weusesimple geometry, including thetheoremofPythagoras, c2a2+b2, where c isthelengthofthehypotenuseofanyright triangle, and a and b arethelengthsoftheothertwo sides.
S LUTI N Fortheright triangleof Fig.110,thetwo sides aretheradius ofthe Earth R andthedistance d 6.1km6100m. Thehypotenuseisapproximatelythelength R+ , where 3.0m.By thePythagoreantheorem,
R2+d2 « (R+ )2 « R2+2 R+ 2.
We solve algebraically for R ,aftercancelling R2 onbothsides: d2 2 (6100m)2(3.0m)2 2 6.0m 6.2 X 106m 6200km.
N TE Precisemeasurementsgive6380km. But lookat yourachievement! Witha fewsimple roughmeasurementsandsimple geometry, youmadea goodestimate of the Earth’s radius.You didnot needto goout inspace,nor didyou need a verylong measuring tape. Nowyou know theanswer to the hapter-Opening Question on p. 1.
FIGURE18Ex mple16.Micrometer used for me suringsm ll thicknesses.
FIGURE19Ex mple17. Di gr ms rere llyuseful!
18m
FIGURE110Ex mple18,but nottosc le.Youc nseesm llrocks tw terlevelontheoppositeshore of l ke6.1kmwideifyoust ndon stepl dder.
