Shell theorem.docx
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Shell theorem.docx
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Shelltheorem
Shelltheorem
Inclassicalmechanics,theshelltheoremgivesgravitationalsimplificationswhichcanbeappliedtoobjectsinsideoroutsideasphericallysymmetricalbody.Thistheoremhasparticularapplicationtoastronomy.
IsaacNewtonshowedthatasphericallysymmetricbodyaffectsotherobjectsgravitationallyasthoughallofitsmasswereconcentratedatapointatitscenter.Ifthebodyisasphericallysymmetricshell(i.e.ahollowball),nogravitationalforceisexertedbytheshellonanyobjectinside.Finally,insideasolidspherethegravitationalforcevarieslinearlywithdistancefromthecenter,becomingzeroatthecenterofmass.
Theseresultsarenotimmediatelyobvious,buttheycanbeprovenwithcalculus.
Contents
∙1Derivation
∙2Thickshells
∙3Solidspheres
∙4Seealso
Derivation
Asphericallysymmetricbodycanbeconsideredasaninfinitenumberofconcentric,infinitesimallythinsphericalshells.Consideronesuchshell:
Theforceduetotheshadedbandis
Thesurfacedensityoftheentireshellis
andtheareaofthebandis
makingthemassoftheband
Theforcecanthenbewritten
Bythelawofcosines,
thus
Togetthetotalforce,weintegrateoversastheshadedbandsweepsfromthepointonthesphereclosesttomtothefarthest(i.e.asθgoesfrom0toπ).Assumingr>R:
Aninterestingresultoccurswhenweconsiderthecaseinwhichr Thelowerconstantofintegrationisreversedinthiscase,giving: Therefore,theshellexertsnonetforceonparticlesanywherewithinitsvolume.Ingeneral,wewrite: Thickshells Nowconsiderasphericallysymmetricshelloffinitethickness,withinnerradiusRaandouterradiusRb.Thebehaviorentirelyinsideoroutsidetheshellisnodifferentthanforathinshell,butwhatistheforcefeltbyanobserversomewherewithintheshell(i.e.Ra Again,thisthickshellbodymaybeconsideredasmanyconcentricthinshells.Theforcecontributionfromeachthinshellis: ThemassofathinshellwithradiusRandthicknessdRis: Therefore, SincealloftheshellswithR>rhavenoeffectontheobserver,thesecondtermdropsout: Ifthedensityisconstantthroughoutthebody,ρ(R)=ρand Ingeneral,forconstantρ: Thefactors and aresimplythemassMofeachthickshell.ThusthefirsttwocasesreducetoNewton'slawofuniversalgravitation. Solidspheres AsolidspheremaybetreatedasaspecialcaseofathickshellwhereRa=0: Therefore,forr Andforconstantρ(renamingRbtoR): Manysufficientlylargecelestialbodiesareagoodapproximationofsphericallysymmetricalsolids.However,thedensityfunctionρ(R)isgenerallynotconstant,buttendstobeinverselyrelatedtoR.Thiscancausesomecounterintuitivebehaviors.Forexample,onemightexpectgravitytodecreasewhendescendingintoadeepmineshaftonEarth.However,sincedensityincreaseswithdepth,thegravityinitiallyincreasesslightly.ThiseffectwouldbeevenmorepronouncedonagasgiantplanetsuchasJupiter.
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