《物理双语教学课件》Chapter 4 Work and Energy 功和能.docx
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《物理双语教学课件》Chapter4WorkandEnergy功和能
Chapter4WorkandEnergy
Theconceptofenergyisoneofthemostimportantintheworldofscience.Ineverydayusage,thetermenergyhastodowiththecostoffuelfortransportationandheating,electricityforlightsandappliances,andthefoodsweconsume.However,theseideasdonotreallydefineenergy.Theytellusonlythatfuelsareneededtodoajobandthatthosefuelsprovideuswithsomethingwecallenergy.
EnergyispresentintheUniverseinavarietyofforms,includingmechanicalenergy,chemicalenergy,electromagneticenergy,heatenergy,andnuclearenergy.Althoughenergycanbetransformedfromoneformtoanother,thetotalamountofenergyintheUniverseremainsthesame.Ifanisolatedsystemlosesenergyinsomeform,thenbytheprincipleofconservationofenergy,thesystemmustgainanequalamountofenergyinotherform.Thetransformationofenergyfromoneformintoanotherisanessentialpartofthestudyofphysics,chemistry,biology,geology,andastronomy.
Inthischapterweareconcernedonlywithmechanicalenergy.Weintroducetheconceptofkineticenergy,whichisdefinedastheenergyassociatedwithmotion,andtheconceptofpotentialenergy,theenergyassociatedwithposition.WeshallseethattheideasofworkandenergycanbeusedinplaceofNewton’slawtosolvecertainproblems.
4.1WorkandPower
1.WorkWdonebyaconstantforceisdefinedastheproductofthecomponentoftheforcealongthedirectionofdisplacementandthemagnitudeofthedisplacement.
Wheretheforcemakesanangleof
withdisplacement
TheSIunitofworkisthejoule（J）,namedforJamesPrescottJoule,anEnglishscientistofthe1800s.Itisderiveddirectlyfromtheunitsformassandvelocity:
1joule=1J=1（kg）（m/s2）（m）=1kgm2/s2
2.Workdonebyavariableforce
（1）.TheincrementofworkdWdoneontheparticlebyFduringthedisplacementdris
whereForceFisfunctionofitsposition.
（2）.TheworkWdonebyFwhiletheparticlemovesfromaninitialpositionatoafinalpositionbisthen
（3）.Weusethecomponentsof
and
toexpresstheforceanddisplacement,thenwehave
3.Workdonebymultipleforces:
Ifthereareseveralforcesactonaparticle,wecanreplaceFinaboveequationwiththenetforce
where
where
aretheindividualforces.Then
4.Power
（1）.Therateatwhichworkisdonebyaforceissaidtobethepowerduetotheforce.IfanamountofworkWisdoneinatimeinterval
byaforce,thentheaveragepowerduetotheforceis
.
（2）.TheinstantaneouspowerPistheinstantaneousrateofdoingwork,whichcanbewrittenas
.
（3）.TheSIunitofpoweristhejoulepersecond.Thisunitisusedsooftenthatithasaspecialname,thewatt（W）,afterJamesWatt,whogreatlyimprovedtherateatwhichsteamenginescoulddowork.
4.2KineticEnergyandWork-KineticEnergyTheorem
Energyisascalarquantitythatisassociatedwithastateofoneormoreobject.Thetermstateherehasitscommonmeaning:
itistheconditionofanobject.
1.KineticenergyKisassociatedwiththestateofmotionofanobject.Thefastertheobjectmoves,thegreaterisitskineticenergy.Foranobjectofmassmandwhosespeedviswellbelowthespeedoflight,wedefinekineticenergyas
TheSIunitofkineticenergyisthesameaswork—joule.
Aconvenientunitofenergyfordealingwithatomsorwithsubatomicparticlesistheelectron-volt（eV）.
1electron-volt=1eV=1.60x10–19J.
2.Work-kineticenergytheorem:
Ifaforcechangesthespeedofanobject,italsochangesthekineticenergyoftheobject.Ifthekineticenergyistheonlytypeofenergyoftheobjectbeingchangedbytheforce,thenthechangeinkineticenergyisequaltotheworkWdonebytheforce:
Here
istheinitialkineticenergy（=
）and
isthekineticenergy（
）aftertheworkisdone.
3.Wecanprovework-kineticenergytheoremasfollow:
（Numeratoranddenominatorofafraction）
4.3Workdonebyweightandbyaspringforce
1.Workdonebyweight:
Wecanfindthattheworkdonebyweightonaparticlebetweentwopointsdoesnotdependonthepathtakenbytheparticle.Ornomatterwhatpathwechoosetomovetheparticle,theworkdonebyitsweightisthesame.Inotherwordifwemoveaparticlearoundaclosedpath,theworkdonebyweightontheparticleiszero.
2.Workdoneonaparticle-likeobjectbyaparticulartypeofvariableforce,namely,springforce—theforceexertedbyaspring.
3.ConservativeforceandNon-conservativeforce:
Iftheworkdonebytheforceisindependentofthepaththeparticlemoves,theforceisaconservativeforce;otherwiseanon-conservativeforce.Weightandspring-forceareconservativeforces;frictionisanon-conservativeforce.
4.4Potentialenergy
PotentialenergyUisenergythatcanbeassociatedwiththeconfiguration（orarrangement）ofasystemofobjectsthatexertaforceononeanother.Iftheconfigurationofthesystemchanges,thenthepotentialenergyofthesystemalsochanges.
1.Weknowthatworkdonebyaconservativeforcehasnothingtodowiththepaththeparticletaken.Sowecanintroduceaquantitywhichisthefunctionofthestateofthesystemtoindicatethiskindofnatureforaconservativeforce.Wecallitpotential.
2.GravitationalPotentialEnergy
（1）.Theworkdonebyweightcanbeexpressedas:
where
isthechangeinthegravitationalpotentialenergy.Sincetheworkdonebyweighthasdefinitemagnitudefromaninitialpositiontoafinalposition,soonlyachange
ingravitationalpotentialenergyisphysicallyimportant.
（2）.However,tosimplifyacalculationoradiscussion,wecansaythatacertaingravitationalpotentialUisassociatedwithanygivenconfigurationofthesystem,withtheparticleatagivenheighth.Todoso,werewritetheaboveequationas:
Thenwetake
tobethegravitationalpotentialenergyofthesystemwhenitisinareferenceconfiguration,withtheparticleatareferencepoint
.Usually,weset
and
thenwehave
.Sothegravitationalpotentialenergyassociatedwithaparticle-Earthsystemdependsontheheighthoftheparticlerelativetothereferencepositionofhi=0,notthehorizontalposition.
3.ElasticPotentialEnergy
（1）.Similartoaparticle-Earthsystem,theworkdonebythespringforcecanberewrittenas
.
（2）.ToassociatedapotentialenergyUwithanygivenconfigurationofthesystem,withtheblockatpositionx,wesetthereferencepointfortheblockasxi=0,whichisalwaysattheequilibriumpositionoftheblock.AndwesetthecorrespondingelasticpotentialenergyofthesystemasUi=0.Thuswehave
.
Attention:
（1）.Potentialenergybelongstothewholesystem.
（2）.Themagnitudeofpotentialenergydependson
thechoiceofthereferencepoint.
4.5Work-EnergyTheoremandConservationofMechanicalEnergy
1.Work-kineticenergytheoremforoneparticle:
Wehave
2.Work-EnergyTheorem:
SupposewehaveparticlesofNinthesystemwediscussedandweusework-kineticenergytheoremforeachparticle,thenwehavetotalamountofNequationslikethose:
Wecandividetheforcesexertedoneveryparticleintoexternalforcesandinternalforces.Andtheinternalforcescanalsobeclassifiedasconservativeinternalforcesandnon-conservativeforces.Summingthetwosideofaboveequations,wewillhave:
Ortobeexactly,
Wealsoknowthattheworkdonebytheconservativeinternalforcecanbewrittenastheminusdifferenceofpotentialenergy.SowegettheWork-EnergyTheorem:
Theworkdonebyexternalforcesandnon-conservativeinternalforcesinagivensystemisexactlyequaltothedifferenceofitsMechanicalEnergy.
3.ConservationofMechanicalEnergy:
Whenonlyconservativeforcesactwithinasystem,thekineticenergyandpotentialenergycanchange.However,theirsum,themechanicalenergyEofthesystem,doesnotchange.
4.6ReadingaPotentialenergycurve
Consideraparticlethatispartofasysteminwhichaconservativeforceacts.Supposethattheparticleisconstrainedtomovealonganxaxiswhiletheconservativeforcedoesworkonit.
1.FindingtheForceAnalytically:
Forone-dimensionalmotion,theworkWdonebyaconservativeforcethatactsonaparticleastheparticlemoves,andthepotentialenergyhavetherelationasfollow
Socangettheforcefromthepotentialenergy
Wecan,forexample,checkthisresultbyputting
whichistheelasticpotentialenergyfunctionforaspringforce.Aboveequationyields
asexpected.
2.ThePotentialEnergyCurve:
ThefollowingfigureisaplotofapotentialenergyfunctionU（x）forasysteminwhichaparticleisinone-dimensionalmotionwhileaconservativeforceF（x）doesworkonit.WecaneasilyfindF（x）bytakingtheslopeoftheU（x）curveatvariouspoints.Fig.（b）isaplotofF（x）foundinthisway.
3.TuningPoint:
Sincethereisonlyconservativeforceactingontheparticle,thesystemwillremainconservationofitsmechanicalenergy.Sowehave
.Sincekineticenergy
isnotlessthanzero.Astheparticlemovesfrom
to
（Fig.a）,whentheparticlereaches
itskineticenergyiszero,meanwhiletheforceontheparticleispositive.Itmeanstheparticledoesnotremainat
butinsteadbeginstomovebacktotheright.Hence
isatuningpoint,aplacewhereK=0andtheparticlechangesdirectionofitsmotion.
4.EquilibriumPoints:
Neutralequilibrium
Unstableequilibrium
Stableequilibrium
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