Proof of F = mv²/R - IOPSpark - Institute of Physics

Using F = ma then F = mv 2R. The same holds for the motion at all places round the circle. The vertical is always taken to mean the direction from the ... Skiptomaincontent F=m(v^2)/R ForcesandMotion ProofofF=mv²/R TeachingGuidance for14-16 MethodA Thisfollowsdirectlyfromtheexperiment Sketchingasatelliteorbitandpredictingitsperiod Themathematicsfollowsdirectlyfromthesketchproducedinthatexperiment,reproducedbelow.ThisisNewton’smethod. Itreliescruciallyonthecrossedchordstheoremforacircle,whichshouldbegiven. ThecirclerepresentstheorbitofasatelliteofradiusR,movingwithspeedv.ThesatellitemovesfromAtoBinatimet.WithoutaforcethesatellitewouldhavemovedtoKatconstantspeed. Now'switch-on'gravityandthesatellitewillfalladistancehinthesametimefromthetangentfromAtothepointB.Itdoesn’tmatterwhetheryouletitfallfromAfirst,andthencontinueinthetangentialdirection,orviceversa.AnyonewhoobjectsthatthefallfromKtoBisnotalongtheradiusshouldlookattheirscalediagramagain:itisalmostimpossibletoseethedifferencebetweenhandtheradialdrop.[Youmayneedtotalkaboutinthelimit.] Fromthecrossedchordsproperty,h(2R-h)=x 2 but2R>>htherefore2v=x 2andsoh=x 22R(equation1) nowx=AKwhichisalmostthearcAB=vt(equation2) Combining1and2,h=(vt)22R(equation3) histheverticalfallandsousings=½at 2=h(equation4) Thenfrom(equation3)and(equation4) ½at 2=(vt)22R leadingtoa=v 2R UsingF=mathenF=mv 2R Thesameholdsforthemotionatallplacesroundthecircle.Theverticalisalwaystakentomeanthedirectionfromthesatellitetothecentreoftheattractingbody. MethodB Thismethodreliesonanunderstandingofvectors. ThecirclerepresentstheorbitofasatelliteofradiusR,movingwithspeedv. ThesatellitemovesfromAtoBinatimet. DrawvectorAPtorepresenttheinitialvelocityofthesatelliteatA,whichisalongatangentatA.Drawasecondvectorofthesamelength,BQ,torepresentthelatervelocityatB. Redrawtheinitialandlatervectors,bothstartingfromthesamepointD.Bothhavemagnitudeequaltov.FG,representingthechangeinvelocity,mustbeaddedtotheoldvelocitytomakethenewvelocity. AOBandFDGaresimilartriangles. changeinvelocityv = ABR acceleration=changeinvelocitytimetakenAtoB=ABxvRxtimetoAtoB=v 2R UsingF=mathenF=mv 2R Theequation,F=mv 2R,illustratestheserelationships: thehigherthespeedv,thebiggertheforceneededtoholdobjectsinorbit,sothebiggerthecentralacceleration forthesamespeed,thesmallertheradius,orthesharperthecurve,thebiggertheforceandthereforethebiggertheaccelerationmustbe. Theaccelerationgoesupwithorbitalspeed,v,butdecreasesastheradius,R,increases. AppearsintheseCollections CuratedCollection Circularmotionandcentripetalforce Commonsensesuggeststhatanyobjectincircularmotionstrivesconstantlyto... For14-16 17Resources OtherresourcesonF=m(v^2)/R CircularMotion ForcesandMotion Episode225:Quantitativecircularmotion Thisepisodediscusses:linearandangularvelocity;degreesandradians;andangularacceleration.Andisaccompaniedby... Lesson 16-19 CentripetalForce ForcesandMotion Furthertestofmv²/R Classpractical:Thisisamodificationofthepractical,ExperimentaltestofF=mv²/R. PracticalActivity 14-16 CircularMotion ForcesandMotion Episode224:Describingcircularmotion Inthisepisode,youwillintroducetheimportanceofcircularmotionandexplaintheneedforacentripetalforcetokeepan... PracticalActivity 16-19 Misconceptions ForcesandMotion Manystudentsthinkthatanobjectmovinginacircleexperiencesaforceawayfromthecentreofthecircle(acentrifugalforce) NumberofResources3NumberofReferences2NumberofDiagnosticResources2 ForcesandMotion Manystudentsthinkthatanobjectthatismovingalongacurvedpathwillcontinuetofollowacurvedpathwhenitsmotionbecomesunconstrained NumberofResources2NumberofReferences2NumberofDiagnosticResources2 F=m(v^2)/R involvesthequantity Force describes CircularMotion LimitLessCampaign Supportourmanifestoforchange TheIOPwantstosupportyoungpeopletofulfiltheirpotentialbydoingphysics.Pleasesignthemanifestotodaysothatwecanshowourpoliticiansthereiswidespreadsupportforimprovingequityandinclusionacrosstheeducationsector. Signtoday Close PhysicsLinksExplorer Explorethelinksbetweendifferentconceptsinthephysicscurriculum Domains EarthandSpaceElectricityandMagnetismEnergyandThermalPhysicsForcesandMotionLight,SoundandWavesPropertiesofMatterQuantumandNuclear Manageprivacyandcookies