ForceandPotentialEnergy-Njit力势能-新泽西理工学院

Physics430:Lecture8

ForceandPotentialEnergyDaleE.GaryNJITPhysicsDepartmentSeptember23,2021WehaveseenthatthepotentialenergyU(r)correspondingtoaforceF(r)canbeexpressedasanintegralofF(r).Itshouldcomeasnosurprise,then,thatwecanwriteF(r)assomekindofderivativeofU(r),althoughwehavetopreservetheeffectofthedotproductintheintegral,whichturnstheintegralofFintoascalar.Inotherwords,weneedaderivativethatturnsthescalarUintoavector.Aconceptfromvectorcalculusfillsthebill—thegradient.ConsideraparticleactedonbyaconservativeforceF(r),withcorrespondingpotentialenergyU(r).TheworkdonebyF(r)inasmalldisplacementfromrtor+dris: Ontheotherhand,fromthedefinitionofpotentialenergyitisalsoButfromthedefinitionofaderivative4.3ForcesasGradientofPESeptember23,2021Thissuggeststhatwecanwrite wherethederivativesofUarenowpartialderivativeswithrespecttox,y,z.[Forexample,istherateofchangeofUasxchanges,keepingyandzfixed.]Equatingthetwoalternativewaysofwriting

wehavewheretheoperator ispronounced“grad.〞Ittakesascalar“field〞suchasU,andresultsinavectorpointing“uphill.〞Notethattheforce,being,points“downhill.〞GradientofPotentialEnergySeptember23,2021ScalarFieldsAscalarfieldisjustonewhereaquantityin“space〞isrepresentedbynumbers,suchasthistemperaturemap.Hereisanotherscalarfield,heightofamountain.ContoursSideViewContoursclosetogethersteeperContoursfarapartflatterSeptember23,2021VectorFieldsAvectorfieldisonewhereaquantityin“space〞isrepresentedbybothmagnitudeanddirection,i.ebyvectors.Thevectorfieldbearsacloserelationshiptothecontours(linesofconstantpotentialenergy).Thesteeperthegradient,thelargerthevectors.Thegradientvectorspointalongthedirectionofsteepestascent.Theforcevectors(negativeofthegradient)pointalongthedirectionofsteepestdescent,whichisalsoperpendiculartothelinesofconstantpotentialenergy.Imaginerainonthemountain.Thevectorsarealso“streamlines.〞Waterrunningdownthemountainwillfollowthesestreamlines.SideViewSeptember23,2021Surfacevs.VolumeVectorFieldsIntheexampleofthemountain,notethattheseforcevectorsareonlycorrectwhentheobjectisONthesurface.Theactualforcefieldanywhereotherthanthesurfaceiseverywheredownward(towardthecenteroftheEarth.Thesurfacecreatesa“normalforce〞everywherenormal(perpendicular)tothesurface.Thevectorsumofthesetwoforcesiswhatweareshowingonthecontourplot.SideViewSeptember23,2021Statementoftheproblem:ThepotentialenergyofacertainparticleisU=Axy2+BsinCz,whereA,BandCareconstants.Whatisthecorrespondingforce?Solution:Formally,theforceisWhatweneed,then,arethethreepartialderivatives,whichwecanwritedownbyinspection:Pluggingbackintotheequationforforce,wehave:Example4.4:FindingFfromUSeptember23,2021LasttimewesawthatthetwoconditionsforaforcetobeconservativeareItturnsoutthatthereisaneasywaytocheckwhetheraforcehasthesecondproperty,usingaconceptfromvectorcalculus.ItcanbeshownviaatheoremcalledStokes’Theorem(whichyouwillhaveseenifyouhavehadthevectorcalculuscourse)thataforcehasthedesiredproperty,thattheworkitdoesisindependentofthepath,ifandonlyifeverywhere.ThequantityiscalledthecurlofF,orjust“curlF,〞or“delcrossF.〞Itfollowstheusualrulesforthecrossproduct.4.4TheSecondConditionthatFbeConservativeConditionsforaForcetobeConservativeAforceFactingonaparticleisconservativeifandonlyifitsatisfiestwoconditions:1.Fdependsonlyontheparticle’spositionr(andnotonthevelocity v,orthetimet,oranyothervariable);thatis,F=F(r).2.Foranytwopoints1and2,theworkW(1

2)donebyFisthe sameforallpathsbetween1and2.September23,2021Twofindthecurlofavector,youformthematrixandfinditsdeterminant:Itmaynotbeobviousthatthisbeingzeroisequivalenttotheconditionthat ispathindependent,butStokes’Theoremshowsthatitis.Thisgivesahandywaytodeterminethepath-independenceproperty,asthefollowingexampleshows.CurlofFSeptember23,2021Statementoftheproblem:ConsidertheforceFonachargeqduetoafixedchargeQattheorigin.ShowthatitisconservativeandfindthecorrespondingpotentialenergyU.CheckthatSolution:TheCoulombforceiswherewehavesubstitutedgfortheconstantkqQ.Let’sfindthecurlofF,andseeifitiszero.ThexcomponentisButsoTheothertwocomponentsworkexactlythesame,sotheCoulombforceisconservative.Example4.5:IstheCoulombForceConservative?September23,2021Tofindthepotentialenergy,wewritedowntheworkintegral wherewearefreetochoosethepathofintegration(sinceweknowtheanswerispath-independent).Weshouldcertainlychoosearadialpath,sothatdr

isinthedirectionr

,sothat.Theintegralthenbecomestrivial:Asusual,wehavetochooseazeropointforthepotentialenergy.ItiscustomarytochooseU=0at,inwhichcasePleaseseethisexampleinthetexttoseehowtodothelaststep,showingthatExample4.5,cont’dSeptember23,2021Thetextgoesthroughadiscussionoftime-dependentpotentialenergyusingaspecificexample.However,Ithinkitissufficienttogivetheresultsofthatdiscussionwithoutgoingthroughthedetails.ConsideraCoulombforceproblemwherethechargeqisbeingactedonbyachargeQthatischangingwithtime.Intheexample,thechargeQisleakingawayduetointeractionwithairmolecules.Thegistoftheargumentisthatifthepotentialenergyischanginginthisway,thenthetotalmechanicalenergyT+Uisnotconserved.WhileTdoesnotchange,Uslowlydecreasestozero.Inthiscircumstance,U=U(x,y,z,t),sothedifferentialofUisEvenwhenthefirstthreetermscanbeexpressedasthegradientofascalar,thelasttermismanifestlynotzero.Wheredoestheenergygo?Itislostintoheatingtheairmolecules.4.5Time-DependentPotentialEnergySeptember23,2021Theabilitytoexpressforcesasthegradientofpotentialenergyprovidesmanyadvantagesforcertainproblems.Itisperhapseasiesttoseetheseadvantagesbyconsideringone-dimensionalsystems,andinfactone-dimensionalsystemscomeupquiteoften,suchastheproblemofinteractionoftwogravitationalbodies,ortwocharges.Wewillfirstconsider“linear〞systems,wherethesingledimensionis,say,thex-axis,asinacartonatrack.However,wewillseenexttimethatwecanalsoconsideraroller-coasterasaone-dimensionalsystem,eventhoughthetrackcurvesthroughtwo(oreventhree)lineardimensions.Inonedimension(wewillusex),thepotentialenergyisForexample,iftheforceobeysHooke’sLaw,Fx(x)=-kx,andifwechoosexo=0asthereferencepoint,thenthepotentialenergyisthefamiliarGoingthereversedirection,becomes4.6LinearOne-DimensionalSystemsSeptember23,2021Thefactthataconservativeforceisthegradientofthepotentialenergyallowsustouseahelpfulanalogytounderstandtherelationships.Allweneeddoisgraphthepotentialenergy(U(x)inthiscase),andwegeta“rollercoaster〞plotliketheonebelowinwhichwecanuseourintuitiontoseehowanobjectwillbehave.Inthisplot,asonarollercoaster,theforceonanobjectatanypointisequaltothe“downhill〞slope(gradient)ofthepotentialenergy.Youcanseethatatthepointx1,theforceistotheleft(theslopeispositive,soisnegative),whileatthepointx2itistotheright.Atpointsx3andx4theslopeiszero,sotheforceiszero.GraphofPotentialEnergy-1x1x2x3x4xU(x)FxFxSeptember23,2021Thismayseemtooobviousinthecaseofarollercoaster,butitisalsotrueofotherconservativeforcessuchasthespringforceortheelectricforce.Thegraphofpotentialenergyallowsustovisualizethewaytheforcesworkandobjectsbehave.Forinstance,youcancertainlyseethatifyouplaceanobjectnearthepointx3,itwillbestable.Ontheotherhand, ifyouplaceanobjectatthepointx4, eventhoughtheforceiszerothere,a smalldisplacementwillcauseittomove awayfromthatpoint,hencetheposition isunstable.Theconditionforapositiontobestableorunstableisfoundfromthecurvatured2U/dx2atthepoint.Ifthecurvatureisupward(d2U/dx2>0)anddU/dx

=0

(aminimuminU(x)),thenthepositionisstable.Ifthecurvatureisdownward(d2U/dx2<0)anddU/dx

=0(amaximuminU(x)),thenitisunstable.GraphofPotentialEnergy-2x1x2x3x4xU(x)FxFxSeptember23,2021Cantheobjectreachhere?Wecanalsosubtlychangethegraphbyplottingthetotalenergyontheverticalaxis.SincethemechanicalenergyE=T+Uisconserved,Eisrepresentedbyahorizontallineasshownontheplot,whileU(x)isthecurve.Notethatwhentheobjectisatposition

a,wehaveE=T+U=U,sotheKE,T, oftheobjectatthatpointmustbezero.Suchapointiscalledaturningpoint, sincetheobjectmovesuptothatpoint andthenreversesdirection.Whentheobjectmovestopointb,stillwithtotalenergyE=T+U,itfollowsthat,sinceUisaminimumthere,Tmustbeamaximum.Theobjectismovingatitsgreatestspeedatthispoint.Whentheobjectreachespointc,sincethetotalenergyishigherthantheU(x)curve,theobjecthasnon-zeroKEandmovesonpastthispoint.Itactuallyspeedsuppastpointc,andmovesoffthegraphtotheright.GraphofPotentialEnergy-3abcxEenergyHowabouthere?September23,2021ThegraphbelowisthepotentialenergyasafunctionofradiusforanegativelychargedionsuchasCl-(Chlorine).ApositiveionsuchasH+makesabondbybeingtrappedinthe“potentialwell〞formedbytheClion.IftheH+ionhastoomuchkineticenergy, thetotalenergyispositiveandtheH+ion willescape(notbebound).However,iftheH+ionhaslowenough kineticenergy,thetotalenergywillbe negativeandtheH+ionwillbebound.Itwillthenoscillatebetweenturningpoints bandd.Initsminimumenergystate,it willhavetotalenergyequaltothepotential energyatpointc,anditwillremaininequilibriumatpointc.Wewillseeinchapter8thatthisisthesameshapeastheradialpotentialenergyforaplanetorbitingastar.GraphofPotentialEnergy-4abcrE>0energyE<0dSeptember23,2021Althoughitisnotgenerallytrueinthreedimensions,whenwehavea