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ClassicalDynamicsP1 CLASSICALDYNAMICS
20LecturesProf.S.F.Gull
â˘HANDOUTâcomprehensivesetofnotescontainingallrelevantderivations.
Pleasereportallerrorsandtypos.
â˘NOTESâProvisionalhardcopyavailableinadvance.
Definitivecopiesofoverheadsavailableonweb.
Pleasereportallerrorsandtypos.
â˘SUMMARYSHEETSâ1pagesummaryofeachlecture.
â˘EXAMPLESâ2examplesheetsâ2examplesperlecture.
â˘WORKEDEXAMPLESâWillbeavailableontheweblater.
â˘WEBPAGEâForfeedback,additionalpictures,moviesetc.
http://www.mrao.cam.ac.uk/âźsteve/part1bdyn/
ThereisalinktoitfromtheCavendishteachingpages.
ClassicalDynamicsP2 NEWTONIANMECHANICS
â˘NewtonianMechanicsis:
-Non-relativistici.e.velocitiesvc(speedoflight=3Ă108
msâ1
.)
-Classicali.e.Eth(Planckâsconstant=1.05Ă10â34
Js.)
â˘Assumptions:
-massindependentofvelocity,timeorframeofreference;
-measurementsoflengthandtimeareindependentoftheframeofreference;
-allparameterscanbeknownprecisely.
â˘Mechanics:
=Statics(absenceofmotion);
+Kinematics(descriptionofmotion,usingvectorsforpositionandvelocity);
+Dynamics(predictionofmotion,andinvolvesforcesand/orenergy).
ClassicalDynamicsP3
â˘ThiscoursecontainsmanyapplicationsofNewtonâsSecondLaw.
BASICPRINCIPLESOFNEWTONIANDYNAMICS
â˘Massesaccelerateifaforceisapplied...
â˘Therateofchangeofmomentum(massĂvelocity)isequaltotheappliedforce.
â˘Vectorially(p=mv):dp
dt=
d(mv)
dt=F
â˘Usuallymisaconstantsomdv
dt=F
â˘Generalcasemdv
dt+
dm
dtv=Fenablesyoutodorocketscience.
v
u
m
dm
0
â˘Rocketofmassm(t)movingwithvelocityv(t)expelsamassdm
ofexhaustgasesbackwardsatvelocityâu0relativetotherocket.
â˘Intheabsenceofgravityorotherexternalforcesdmu0+mdv=0.
â˘Integrating,wefindv=u0log(mi/mf),
wheremi,faretheinitialandfinalmasses.
â˘Forarocketacceleratingupwardsagainstgravitymdv
dt+
dm
dtu0+mg=0.
ClassicalDynamicsP4
â˘Thesimpleharmonicoscillator(SHO)occursmanytimesduringthecourse.
SIMPLEHARMONICOSCILLATOR
â˘Massmmovinginonedimensionwithcoordinatexonaspringwithrestoringforce
F=âkx.Theconstantkisknownasthespringconstant.
â˘Newtonianequationofmotion:mx=âkx,wherexdenotesdx
dtetc.
â˘Generalsolution:x=AcosĎt+BsinĎtwhereĎ2
=k/m.
â˘Canalsowritesolutionasx=<(AeiĎt
),whereAiscomplex.
â˘Wecanintegratetheequationofmotiontogetaconservedquantityâtheenergy.
â˘Multiplyingtheequationofmotionbyx(agoodgeneraltrick)weget
mxx+kxx=0â12mx
2+
12kx
2=E=constant
â˘HerethequantityTâĄ12mx
2isthekineticenergyofthemassandVâĄ
12kx
2isthe
potentialenergystoredinthespring.
â˘Formanydynamicalsystems(suchastheSHO)thetimetdoesnotappearexplicitlyinthe
equationsofmotionandthetotalenergyE=T+Visconserved.Thisconserved
quantityisalsoknownastheHamiltonian.
ClassicalDynamicsP5 THEENERGYMETHOD
â˘Ifweknowfromphysicalgroundsthattheenergyisconserved,wecanalwaysderivethe
equationsofmotionofsystemsthatonlyhaveonedegreeoffreedom(suchastheSimple
HarmonicOscillator):12mx
2+
12kx
2=Eâx(mx+kx)=0âmx=âkx.
â˘Thisworksbecausexisnotalwayszero.Wewillcallthistheenergymethod.
â˘Wecansometimesderivetheequationsofmotionofmuchmorecomplicatedsystemswith
ndegreesoffreedomisasimilarway.Itâscertainlynotrigorous,butworksformostofthe
systemsstudiedinthiscourse.ThetheoreticallymoreadvancedmethodsofLagrangian
andHamiltonianmechanicsderivetheequationsofmotionfromavariationalprinciple.They
arerigorous,butstillusethekineticenergyTandpotentialenergyV(actuallyinthe
combinationL=TâV).
â˘Wewillseelaterinthecoursetheuseoftheenergymethodtoderivetheequationofmotion
ofaparticleatradiusrinacentralforce:12mr
2+Veff(r)=E.
Differentiatingwithrespecttotimeweget:r
(
mr+dVeff
dr
)
=0
â˘Westrikeoutthertoobtaintheequationofmotion.
ClassicalDynamicsP6 THEENERGYMETHOD:EXAMPLE
â˘Ladderleaningagainstasmoothwall,restingonasmoothfloor
(notrecommendedpractice).
â˘NewtonianmethodneedsreactionforcesNandR.
Takemomentstogettheangularacceleration.Tryit...
â˘Theenergymethodiseasier:
(1)Potentialenergy.Thisiseasy:V=12mglcosθ.
(2)Kineticenergy.Writethisasthesumofthekineticenergy
ofthecentreofmassplustheenergyofrotationabout
thecentreofmass:T=12m(x
2+y
2)+
12Iθ
2,
whereI=112ml
2isthemomentofinertiaofauniformrodaboutitscentre.
Coordinatesofcentreofmass:x=12lsinθ;y=
12lcosθ
Workoutvelocities:x=12lcosθθ;y=â
12lsinθθ
T=12m(x
2+y
2)+
12Iθ
2=
18ml
2θ2
+124ml
2θ2
=16ml
2θ2
(3)Energymethod:d(T+V)
dt=0âθ
(13θml
2â
12mglsinθ
)
=0.
(4)Equationofmotion:θ=3g
2lsinθ.
ClassicalDynamicsP7 REVISION:VECTORCALCULUS
(SeeSection1.2ofHandout)
â˘Indynamicsweusevectorstodescribethepositions,velocitiesandaccelerationsof
particlesandotherbodies,aswellastheforcesandcouplesthatactonthem
â˘Weneedtorevisevectors,vectorfunctions,vectoridentitiesandintegraltheorems.
â˘Revisescalarproducta¡bandvectorproductaĂĂĂĂĂb(preferabletoaâ§b).
â˘Thereareonlyacoupleofvectoridentities,butyouMUSTLEARNTHEM.
(1)Scalartripleproduct
a¡(bĂĂĂĂĂc)=(aĂĂĂĂĂb)¡c(Interchangeofdotandcross)
a¡(bĂĂĂĂĂc)=b¡cĂĂĂĂĂa=âb¡aĂĂĂĂĂc(Permutationschangesign)
(2)Vectortripleproduct
Thisisthemostimportantidentity:aĂĂĂĂĂ(bĂĂĂĂĂc)=a¡cbâa¡bc
Rule:Vectoroutsidebracketappearsinboth
scalarproducts.
Rule:Outerpairtakestheplussign.(aĂĂĂĂĂb)ĂĂĂĂĂc=a¡cbâb¡ca
ClassicalDynamicsP8 REVISION:VECTORCALCULUSII
(SeeSection1.2ofHandout)
Gradientoperatorâ(Vectordifferentialoperator).
â˘grad(ÎŚ);âÎŚVectorgradientofascalarfieldÎŚ.
â˘div(E);â¡EScalardivergenceofavectorfieldE.
â˘curl(E);âĂĂĂĂĂE.VectorcurlofavectorfieldE.
DivergenceTheorem(GaussâTheorem)
d=d| | SnS
â˘RelatesintegraloffluxvectorEthroughclosed
surfaceS(noutwards)to
volumeintegralofâ¡E.
âŽ
dS¡E=
âŤ
dVâ¡E
StokesâTheorem
â˘RelateslineintegralvectorEaroundclosedlooplto
surfaceintegralofâĂĂĂĂĂE.âŽ
dl¡E=
âŤ
dS¡âĂĂĂĂĂE
Furtheridentities:
âĂĂĂĂĂ(âÎŚ)=0;â¡(âĂĂĂĂĂE)=0;âĂĂĂĂĂ(âĂĂĂĂĂE)=â(â¡E)ââ2E
ClassicalDynamicsP9 MECHANICSâREVIEWOFPARTIA
Revisionofdynamicsofmany-particlesystem.(DetailedderivationsinSection3ofHandout.)
â˘SystemofNparticles.Theathparticleofmassmaisatpositionraandvelocityva.
ActedonbyexternalforceFa0andinternalforcesFabfromotherparticles.
â˘Importantdefinition:CentreofmassR.DefineMâĄâ
a
maandMRâĄâ
a
mara.
â˘Otherconcepts:TotalmomentumP.TotalangularmomentumJ.TotalexternalforceF0
andcoupleG0.KineticenergyT,potentialenergyU,totalenergyE.
â˘Totalmomentumactsasifitwereacteduponbythetotalexternalforce.
â˘Totalangularmomentumactsasifitwereacteduponbythetotalexternalcouple.
â˘Intrinsicangularmomentum:Jâ˛
intheframeSâ˛
inwhichPâ˛=0(zero-momentum,or
CentreofMass(CoM)frame).Theintrinsicangularmomentumisindependentoforigin.
â˘TheCentreofMassframeisthusspecial,andshouldbeusedwhereverpossible.
â˘GalileantransformationfromCoMframeSâ˛
toframeSmovingatvelocityV
(CoMatRâ˛=Vt).Momentum:P=P
â˛+MV
AngularMomentum:J=Jâ˛+MR
â˛ĂĂĂĂĂV.Kineticenergy:T=T
â˛+
12MV
2
ClassicalDynamicsP10 OVERALLMOTION
(SeeSection3.1ofHandout)
â˘DevelopmentfromNewtonâsLaws:mara=Fa,
wherea=1,NfortheathofNparticles.Arigidbodyisaspecialcase.
â˘Overallmotionâ
a
mara=â
a
Fa=â
a
Fa0+â
a
â
b
Fab,
whereFa0istheexternalforceonparticleaandFabistheforceonaduetob.
SinceFab=âFbabyNewtonâs3rdLaw,theâ
a
â
b
termabovesumstozero.
â˘DefineMâĄâ
a
maandMRâĄâ
a
mara.RisthepositionoftheCentreofMass.
â˘WiththesedefinitionsMR=â
a
Fa0âĄF0
â˘TheCentreofMassmovesasifitwereaparticleofmassMacteduponbythetotal
externalforceF0.
â˘Intermsofmomentumpa=Fa;P=F0,wherePisthetotalmomentum.
ClassicalDynamicsP11 MOMENTS
(SeeSection3.2ofHandout)
â˘Couple,torque:GâĄrĂĂĂĂĂF.Angularmomentum:JâĄrĂĂĂĂĂp.
â˘Sincepa=Fa,wehaveâ
a
raĂĂĂĂĂpa=â
a
raĂĂĂĂĂFa
â˘ExpandRHS:RHS=â
a
raĂĂĂĂĂFa0+â
a
â
b
raĂĂĂĂĂFab
︸︡︡︸â
b
â
a<b
(raârb)ĂĂĂĂĂFab︸︡︡︸
=0â˘ThelattertermiszerosinceFabisassumedtobealongthelinebetweenaandb.
WehaveagainusedNewtonâsThirdLaw.
â˘TheLHSforoneparticleisJa=d
dt(raĂĂĂĂĂpa)=raĂĂĂĂĂpa
︸︡︡︸+raĂĂĂĂĂpa
zero,sincemr=pâ˘Forthesystemofparticles
JâĄâ
a
Ja=â
a
raĂĂĂĂĂpa=RHS=â
a
raĂĂĂĂĂFa0âĄG0
â˘G0istheresultantcoupleGfromallexternalforces.
ClassicalDynamicsP12 CHOICEOFORIGIN
(SeeSection3.3ofHandout)
â˘Supposewemovetheoriginbyaconstanta,givingnewcoordinatesrâ˛
withr=râ˛+a.
â˘Thenr=râ˛
andtheoverallmotionisunaffected.
â˘WhatabouttheangularmomentumJ?ForoneparticleJa=Jâ˛
a+aĂĂĂĂĂpa,orforthe
system
J=Jâ˛+
â
a
aĂĂĂĂĂpa=Jâ˛+aĂĂĂĂĂP
i.e.JdependsonthechoiceoforiginunlessP=0.
â˘Intrinsicangularmomentum:JintheframeinwhichP=0(zero-momentum,orCentre
ofMassframe).TheIntrinsicangularmomentumisindependentoforigin.
â˘TheCentreofMassframeisthusspecial,andshouldbeusedwhereverpossible.
â˘SimilarlyG=Gâ˛+aĂĂĂĂĂF.
ClassicalDynamicsP13 KINETICENERGY
(SeeSection3.4ofHandout)
â˘Workdone:forceĂdistancemovedâforce=changeinenergy.
â˘ForasingleparticleF¡dr=mr¡dr=mr¡r ︸︡︡︸dt
d
dt(12r¡r)
or
F¡dr=d(12mv
2)
â˘Kineticenergy:TâĄ12mv
2.
â˘Workdoneonparticle=changeinkineticenergy.
â˘Forasystemofparticles
dT=â
a
dTa=â
a
Fa¡dra
=â
a
Fa0¡dra+ââ
a<b
Fab¡(draâdrb)
wherewehaveusedFab=âFba.Wecanwritetheab-termasâFabd|raârb|,where
Fabhasmagnitude=|Fab|andispositiveifforceisattractive,negativeifrepulsive.
ClassicalDynamicsP14 POTENTIALANDTOTALENERGY
(SeeSection3.4ofHandout)
â˘Potentialenergy:Uisdefinedas
dU=ââ
a<b
Fabd|raârb|
â˘NotethezeroofU=
âŤ
dUisundefined.ItisoftentakenwithU=0withparticlesat
infiniteseparation,givingnegativeUforasystemofparticleswithattractiveforces.
â˘ForarigidbodydU=0since|raârb|isfixed.
â˘TotalEnergy:E=T+U.
â˘Asdefinedabove
dE=dT+dU=â
a
Fa0¡dra
â˘TheRHStermistheworkdonebyexternalforces;itcanbeincorporatedintoUifdesired.
ClassicalDynamicsP15 GALILEANTRANSFORMATION
(SeeSection3.5ofHandout)
â˘GofromframeSâ˛
toSwithr=râ˛+Vt;Vsteady;t=t
â˛.
â˘Momentump=pâ˛+mV;P=P
â˛+MV
i.e.PinSandPâ˛
inSâ˛
changetogether(orremainsteadytogetherifthereifnoexternal
force).IfPâ˛=0,thenS
â˛isthezero-momentumorCentreofMassframe.
â˘AngularmomentumJ=â
a
(râ˛
a+Vt)ĂĂĂĂĂ(pâ˛
a+maV)
â˘Thereare4terms.The4thisVĂĂĂĂĂV=0.Theothersgive
J=Jâ˛+VtĂĂĂĂĂP
â˛+
â
a
râ˛
aĂĂĂĂĂmaV
︸︡︡︸â
a
(marâ˛
a)ĂĂĂĂĂV=MRâ˛ĂĂĂĂĂV
â˘ThusifSâ˛
isthezero-momentumframe,Pâ˛=0and
J=Jâ˛
+MRâ˛ĂĂĂĂĂV
︸︡︡︸.
inSintrinsicmotionofCofMinS
ClassicalDynamicsP16 GALILEANTRANSFORMATIONâENERGY
(SeeSection3.5.3ofHandout)
â˘Energy
T=â
a
12mav
2a=
12ma(v
â˛
a+V)¡(vâ˛
a+V)
=Tâ˛+
â
a
mavâ˛
a
︸︡︡︸
¡V+12MV
2.
(=0,ifSâ˛
=zero-momentumframe)
or
T=KEinzero-momentumframe+12MV
2
ClassicalDynamicsP17 COORDINATESYSTEMS
(SeeSection1.1ofHandout)
â˘PositionvectorrhasCartesiancoordinates(x,y,z),
cylindricalpolarcoordinates(Ď,Ď,z)
andsphericalpolarcoordinates(r,θ,Ď).
â˘Relationbetweencoordinatesystems:
x=ĎcosĎ=rsinθcosĎ
y=ĎsinĎ=rsinθsinĎ
z=z=rcosθ
Ď=â
x2+y2
r=â
x2+y2+z2
â˘Wedefineunitvectors(ex,ey,ez)alongx,y,zaxes.
â˘Similarlywedefineunitvectors(eĎ,eĎ,eθ)alongdirectionsofincreasing(Ď,Ď,θ)
(unambiguousbecausetheseareorthogonalcoordinatesystems).
â˘Positionvectorr=xex+yey+zez=ĎeĎ+zez=rer
ClassicalDynamicsP18 DYNAMICSINCYLINDRICALPOLARS
(SeeSection2.1ofHandout)
â˘InCartesians,theequationofmotionofaparticleismr=F;ormx=Fxetc,
wherewedenotetimederivativesdr
dtâĄr,d
2r
dt2âĄretc.
eĎ
eĎ â˘Considercylindricalpolars;ignorez-motionforthemoment
r=ĎeĎwhereeĎ,eĎandezareunitvectorsinthe
directionsofincreasingĎ,Ď,z.
â˘NotethatthedirectionofthevectorseĎandeĎ
changeastheparticlemoves:r=ĎeĎ+ĎËeĎ
â˘AstheparticlemovesfromsayPtoPâ˛
indt,eĎandeĎrotatebydĎ.
â˘ElementarygeometrygivesdeĎ=dĎeĎorËeĎ=ĎeĎandsimilarlyËeĎ=âĎeĎ
givingr=ĎeĎ︸︡︡︸
+ĎĎeĎ︸︡︡︸
radialtransverse
â˘IncylindricalpolarcoordinatestheradialvelocityisĎandthetransversevelocityisĎĎ.
ClassicalDynamicsP19 ACCELERATIONINPOLARCOORDINATES
(SeeSection2.1ofHandout)
â˘Similarly,wecanworkouttherateofchangeofvelocity:
r=ĎeĎ+ĎËeĎ︸︡︡︸
+ĎĎeĎ+ĎĎeĎ+ĎĎËeĎ︸︡︡︸
ĎeĎâĎeĎ
=(ĎâĎĎ2)
︸︡︡︸eĎ+(2ĎĎ+ĎĎ)
︸︡︡︸eĎ
radialtransverseâ˘Thez-motionisindependent:(r)zisjustzezsinceËez=0.
â˘TheradialaccelerationisĎâĎĎ2,thesecondtermbeingthecentripetalacceleration
requiredtokeepaparticleinanorbitofconstantradius.
â˘Thetransverseaccelerationis2ĎĎ+ĎĎ=1
Ď
d
dt
(
Ď2Ď)
andshowsthatitisrelatedtothe
angularmomentumperunitmassĎ2Ď.
â˘Sphericalpolarscanbetreatedbyputtingr=rer,andexpandingretc.withËer
expressedintermsofer,eθandeĎ.Weshallnotneedthishere,asitâsslightly
complicated,butifyouhavecomputeralgebraavailableitâsveryuseful...
ClassicalDynamicsP20 POLARSANDTHEARGANDDIAGRAM
eiĎ
ieiĎ
(SeeHandoutSection2.1)
â˘ThecomplexplanezâĄx+iy=ĎeiĎ
hasthe
samestructureasthetwo-dimensionalplane
r=xex+yey=ĎeĎ.
â˘Theunitvectorscorrespondtocomplexnumbers:
eĎâeiĎ
eĎâieiĎ
â˘WecanthereforederivetheradialandtransversecomponentseasilyusingtheArgand
diagram.
â˘Velocity:d
dt(Ďe
iĎ)=Ďe
iĎ+ĎĎie
iĎ.
â˘Acceleration:
d2
dt2(ĎeiĎ
)=ĎeiĎ
+2ĎĎieiĎ
+ĎĎieiĎ
âĎĎ2eiĎ
=(ĎâĎĎ2
︸︡︡︸)e
iĎ+(ĎĎ+2ĎĎ
︸︡︡︸)ie
iĎ
radialtransverse
ClassicalDynamicsP21 FRAMESINRELATIVEMOTION
(SeeHandoutSection2.2)
â˘SupposewehaveaframeS0inwhichmr0=F,withFascribedtoknownphysical
causes.WhatistheapparentequationofmotioninamovingframeS?
â˘Case1:Supposer=r0âR(t).SupposetheaxesinS0andSremainparalleland
t=t0(asalwaysinclassicalphysics):r=r0âR
â˘ForthespecialcaseR=0(i.e.steadymotionbetweenframes),mr=mr0=F,
i.e.thesameequationofmotion(Galileantransformation).
â˘ForgeneralR(t)mr=mr0âmR=FâmR
â˘TheapparentforceinSincludesboththeactualforcemr0andafictitiousforceâmR.
â˘Fictitiousforcesare:(a)associatedwithacceleratedframes;(b)proportionaltomass.
â˘Question:Isgravityafictitiousforce?
â˘Answer:(accordingtogeneralrelativity).Yes!Gravityisequivalenttoacceleration.
ClassicalDynamicsP22 ROTATINGFRAMES
ĎĂĂĂĂĂez
ez
ey
ex
(SeeSection2.3ofHandout)â˘Case2:FrameSrotateswithangularvelocityĎ,sothat
theunitvectorsrotatewithrespecttotheinertialframeS0.
â˘TherateofchangeisgivenbyËez=ĎĂĂĂĂĂezetc.
Lettheframescoincideatt=0:
r0=xex+yey+zez=r
r0=xex+xËex+yandzterms
=v+ĎĂĂĂĂĂr
,
vâĄxex+yey+zezistheapparentvelocityinS.
â˘TheaccelerationinS0isr0=xex+2xËex+x¨ex+yandzterms
=xex+2(ĎĂĂĂĂĂex)x+ĎĂĂĂĂĂ(ĎĂĂĂĂĂex)x+yandzterms
=a+2ĎĂĂĂĂĂv+ĎĂĂĂĂĂ(ĎĂĂĂĂĂr)
whereaâĄxex+yey+zezistheapparentaccelerationinS.Werewritethe
momentumequationmr0=Fintermsoftheapparentquantitiesr,vanda:
ma=Fâ2m(ĎĂĂĂĂĂv)âmĎĂĂĂĂĂ(ĎĂĂĂĂĂr)
â˘TheobserverinSaddsCoriolisandCentrifugalforces(inertialorfictitiousforces).
ClassicalDynamicsP23 ROTATINGFRAMES
(SeeSection2.3ofHandout)
â˘Thereisanoperatorapproachtorotatingframesthatisagoodaidtomemory(andisrigorous).
â˘ForanyvectorAtheratesofchangeinframeS0andinframeS
arerelatedby
[dA
dt
]
S0
=
[dA
dt
]
S
+ĎĂĂĂĂĂA
â˘Applythisoperatorrelationtwicetor(r=r0att=0):[d
2r0
dt2
]
S0
=
([d
dt
]
S
+ĎĂĂĂĂĂ
)([dr
dt
]
S
+ĎĂĂĂĂĂr
)
â˘Expandingandsetting
[dr
dt
]
S
=vand
[dv
dt
]
S
=awerecover
mr0=F=ma+2m(ĎĂĂĂĂĂv)+mĎĂĂĂĂĂ(ĎĂĂĂĂĂr)â˘Generalcase:ObservermovesonapathR(t)andusesaframerotatingatangular
velocityĎ(t)whichisalsochanging.Frompreviousresultsand,becausethetime
derivativenowoperatesonĎwegetthegeneralformula:
ma=Fâ2m(ĎĂĂĂĂĂv)âmĎĂĂĂĂĂ(ĎĂĂĂĂĂr)âmRâmĎĂĂĂĂĂr
â˘ThemĎĂĂĂĂĂrtermiscalledtheEulerforce.
ClassicalDynamicsP24 CENTRIFUGALANDCORIOLISFORCES
â˘CentrifugalForce:âmĎĂĂĂĂĂ(ĎĂĂĂĂĂr).
âmĎĂĂĂĂĂ(ĎĂĂĂĂĂr)=m(Ď2râr¡ĎĎ)
â˘CentrifugalForcemĎ2Ďoutwards.
â˘CoriolisForce:â2m(ĎĂĂĂĂĂv)appearsifabodyismoving
withrespecttoarotatingframe.
â˘CoriolisForceisasidewaysforce,perpendicularbothto
therotationaxisandtothevelocity.
â˘ProblemsinvolvingCoriolisForcecanoftenbedonebyconsideringangularmomentum.
â˘Advice:Donotmeddlewiththesignsortheorderingoftheterms.Theminussignreminds
usthatthesetermscamefromtheothersideoftheequation,andĎĂĂĂĂĂ(ĎĂĂĂĂĂr)construction
remindsusoftheoperatorrelation
[d
dt
]
S0
=
[d
dt
]
S
+ĎĂĂĂĂĂ.
ClassicalDynamicsP25 FICTITIOUSFORCESâAPPLICATIONS
â˘CentrifugalforcegivesrisetotheEarthâsequatorialbulge:âźÎŠ
2R
gâ1
300.
l
N
You are here Wâ˘CoriolisforceduetomotiononEarthâssurface:F=2mΊvsinÎť
DirectionissidewaysandtotherightintheNorthernHemisphere.
Independentofdirectionoftravel[NSEW].F=2mvWl cos
zx
East
â˘Coriolisforceonafallingbody.
Startsfromrestattimet=0.v=gt
mx=2mgtΊcosÎťâx=13gΊt
3cosÎť
â˘Foucaultpendulum.PrecessesatΊsinÎť.
Whichway?
â˘Roundaboutsandotherfairgroundrides.Rollercoasters.
Low
High
High
High
N
w
Low
High â˘Weatherpatterns,tradewinds,jetstreams,
tornados,bathtubs(??).
ClassicalDynamicsP26 ORBITSâCENTRALFORCEFIELD
(SeeSection4.1ofHandout)
F
O
P
r
v â˘Particlemovingincentralforcefield.PotentialU(r)yields
radialforceF=ââU=âdU
drer.
â˘Motionremainsintheplanedefinedbypositionvectorr
andvelocityv.
â˘Nocouplefromcentralforceâangularmomentumisconserved:
mr2Ď=J=constant(KeplerII)
â˘Totalenergyisconserved:
E=U(r)+12m(r
2+r
2Ď
2)=
12mr
2+U(r)+
J2
2mr2
â˘TheeffectivepotentialUeff(r)hasacontributionfromtheangularvelocity.
Ueff(r)âĄU(r)+J
2
2mr2
â˘Theeffectivepotentialhasacentrifugalrepulsivetermâ1
r2.
ClassicalDynamicsP27 ORBITSINPOWER-LAWFORCE
(SeeSection4.2ofHandout)
â˘WecangainalotofinsightintoorbitsbystudyingtheforcelawF=âArn
withApositive,
soforceisattractive.
â˘TheeffectivepotentialisthenUeff(r)=Ar
n+1
n+1+
J2
2mr2
theonlyexceptionbeingn=â1(Ueffthencontainsalogrterm).
â˘Thecentrifugalpotentialisrepulsiveandârâ2
.AplotofUeff(r)showswhichvaluesof
theindexnleadtoboundorunboundorbits,andwhichleadtostableorunstableorbits.
â˘FornâĽâ1(includingthelogrpotential),thepotentialincreasesasrââandthe
orbitsareboundandstable.
â˘Forâ3<n<â1thepotentialgoestozeroatr=âandtheorbitscaneitherbebound
orunbound.
â˘Forn<â3theattractionatrâ0overcomesthecentrifugalrepulsionandtheorbitsare
notstable(thisisthecaseforthecentralregionofblackholesinGR).
ClassicalDynamicsP28 ORBITSINPOWER-LAWFORCE
(SeeSection4.2ofHandout)
r
eff U
r0
E0
r r
eff U
0
r
eff U
r0
U0
nâĽâ1
â3<n<â1
n<â3
â˘Thepotentialisqualitativelydifferentfordifferentvaluesofn:
nâĽâ1:Orbitatr0stable.Allorbitsbound.
â3<n<â1:Orbitatr0stable.UnboundorbitsforE>0.
n<â3:Orbitatr0unstable.Willgotor=0orr=â
ClassicalDynamicsP29 NEARLYCIRCULARORBITSINPOWER-LAWFORCE
r
eff U
r0
U0
â˘LetF=âArn
,n=index,withcommoncases
n=+1(2DSHM)andn=â2(gravity,electrostatics).
Ueff=Ar
n+1
n+1+
J2
2mr2
â˘Nearlycircularorbitsareoscillations/perturbationsaboutr0.
TaylorexpansionofUeffgives
Ueff=U0+(râr0)dUeff
dr
âŁâŁâŁâŁr0
+12(râr0)
2d2Ueff
dr2
âŁâŁâŁâŁr0
+¡¡¡
â˘Atr=r0
dUeff
driszero,giving
dUeff
dr=Ar
nâ
J2
mr3=0atr0.
â˘ThesecondderivativeofUeffisd
2Ueff
dr2=nArnâ1
+3J
2
mr4=(n+3)J
2
mr40
atr0.
â˘Usingtheenergymethodd
dt
(12mr
2+Ueff
)=r
(
mr+dUeff
dr
)
=0
wegettheSHMequationmr+(n+3)J
2
mr40
(râr0)=0,
i.e.simpleharmonicmotionaboutr0withangularfrequencyĎp=â
n+3J
mr20
.
ClassicalDynamicsP30 NEARLYCIRCULARORBITSII
â˘HowdoesĎpoftheperturbationcomparewithĎcofthecircularorbitatr0?
Ďc=Ď=J/mr20.ThereforeĎp=
ân+3Ďc.
â˘Thesimplecasesare
1.n=1.Forceproportionaltor,i.e.simpleharmonicmotion.
Ďp=2Ďc,givingacentralellipse(Lissajousfigure).
2.n=â2.Inversesquareforce.Ďp=Ďc,
givinganellipsewithafocusatr=0
(planetaryorbit).
â˘Generalngivesnon-commensurateĎpandĎc,
withnon-repeatingorbits(e.g.galacticorbits).
Caseillustratedisn=â1.
ClassicalDynamicsP31 KEPLERâSLAWS
â˘150ADPtolemy-Earthatcentreofsolarsystem,withSunrotatingaroundit,andthe
planetaryorbitsdescribedbyacombinationofcircles(epicycles).
â˘TychoBrahe(1546-1601)madeobservationsofplanetaryandstellarpositionstoan
accuracyof10arcsec(theresolutionoftheeyeis1arcmin).
â˘JohannesKepler(1571-1630)spent5yearsfittingcirclestotheBraheâsdataforMarsâsorbit
andfounddifferencesoftheorderof8arcmin.Rejectedmodelbecauseofknownaccuracy
ofBraheâsmeasurements.
â˘EventuallyKeplerconcludedthattheorbitswereellipses.
â˘KeplerâsLaws
â˘FirstLaw:PlanetaryorbitsareellipseswiththeSunatonefocus.
â˘SecondLaw:ThelinejoiningtheplanetstotheSunsweepsoutequalareasinequal
times.(Impliesconservationofangularmomentum.)
â˘ThirdLaw:Thesquareoftheperiodofaplanetisproportionaltothecubeofitsmean
distancetotheSun(itisproportionaltothecubeofthelengthoftheorbitâsmajoraxis).
ClassicalDynamicsP32 ORBITSININVERSESQUARELAWFORCE
â˘Inversesquarelawforce:F=âA
r2.
â˘Angularmomentum:J=mr2ĎEnergy:
12mr
2+
J2
2mr2âA
r=E.
â˘Slightlyeasiertoworkwithu=1/r:r=dr
dĎĎ=âĎr
2du
dĎ=â
J
m
du
dĎ.
â˘Substituteintotheenergyequation
(du
dĎ
)2
+u2â
2m
J2(E+Au)=0.
Completesquare:
(du
dĎ
)2
=e2
r20
â(
uâ1
r0
)2
,wheree2
r20
âĄ2mE
J2+1
r20
andwehavedefinedr0=J
2
mA,theradiusofthecircularorbitwiththesameJ.
Standardintegral:du
â
e2
r20
â(
uâ1
r0
)2=dĎâu=
1
r0
(1+ecos(ĎâĎ0))
â˘Equationofconicsection:r0=r(1+ecosĎ)
â˘Forarepulsivepotentialr0=r(ecosĎâ1)
ClassicalDynamicsP33 INVERSESQUARELAWâELLIPTICALORBITS(E<0)
O
P
fr
semi-latus rectum0r
rmin=r0
1+e
rmax=r0
1âe
â˘Ellipseofeccentricitye(0<e<1).
Centreofattractionatonefocus.
â˘Polarequation:r0=r(1+ecosĎ)
r0iscalledthesemi-latusrectum.
â˘Cartesianequation:r=r0âex
y2
+x2(1âe
2)+2er0x=r
20.
Setxâ˛=x+
r0e
1âe2ây2
+(xâ˛)2(1âe
2)=
r20
1âe2
Ellipse:(x
â˛)2
a2+y2
b2=1;a=r0
1âe2;b=r0
â1âe2
O
P
r
rĎ â˘AreaofanellipseisĎab=Ďr
20
(1âe2)3/2
â˘PeriodTisArea
Rateofsweepingoutarea.
â˘Rateofsweepingoutarea:12r
2Ď=
J
2m,
henceperiodT=2Ďr
20m
J(1âe2)3/2=2Ď
â
ma3
A(Keplerâs3rdLaw).
ClassicalDynamicsP34 INVERSESQUAREORBITSâALTERNATIVE
(SeeSection4.3.1)
â˘Shapeoforbit.SincethevectorsJ,vandËerhavemagnitudesmr2Ď,A/mr
2andĎ
respectivelyandaremutuallyperpendicular,wemaywrite
JĂĂĂĂĂv=âAËer(thesignisobtainedbyinspection).
â˘SinceJisconstant,theequationmaybeintegratedtogiveJĂĂĂĂĂv+A(er+e)=0,
whereeisavectorintegrationconstant.
â˘Takingthedot-productofthisequationwithrgives
JĂĂĂĂĂv¡r ︸︡︡︸+A(r+e¡r)=0
=J¡vĂĂĂĂĂr=âJ2/m
â˘Thereforer(1+e¡er)=r(1+ecosĎ)=J
2
mA=r0,
whichisthepolarequationofaconicwithfocusatr=0(Keplerâs1stLaw).
â˘Themajoraxisisinthedirectionofe;eistheeccentricity:acircle(e=0);ellipse(e<1),
parabola;(e=1)orhyperbola(e>1).
ClassicalDynamicsP35 ALTERNATIVEâENERGYOFTHEORBIT
â˘TogettheenergytakethescalarproductofAe=â(JĂĂĂĂĂv+Aer)withitself
(notethatJandvareperpendicular).
A2e2
=J2v2
+2JĂĂĂĂĂv¡er︸︡︡︸
A+A2
=J¡vĂĂĂĂĂer=âJ2/mr Therefore
A2(e
2â1)=J
2
(
v2â
2A
mr
)
=2EJ
2
m=2AEr0
whereEisthetotalenergy.Themajoraxisoftheorbitisgivenby
2a=r0
(1
1+e+
1
1âe
)
=2r0
1âe2=âA
E
i.e.E=âA/2a,independentofeccentricity.
â˘ThedistanceofclosestapproachoccursatĎ=0:rmin=r0
1+eandthedistanceof
furthestapproachoccursatĎ=Ď,atrmax=r0
1âe.
â˘Thesemi-majoraxisasatisfies2a=rmin+rmax,sothata=r0
1âe2.
â˘Substituting,wefindtheusefulrelationsrmax=a(1+e)andrmin=a(1âe).
ClassicalDynamicsP36 ELLIPTICALORBITSâIMPORTANTTHINGSTOREMEMBER
OC
P
semi-major axis
semi-latus rectum
fsemi-minor axisb
a
0r
r
â˘Theequationofanellipseinpolarcoordinates
r0=r(1+ecosĎ)
â˘Thedistancesofclosestandfurthestapproachfollow
fromthis:rmin=r0
1+eandrmax=
r0
1âe
â˘Thesemi-majoraxisasatisfies2a=rmin+rmax
âa=r0
1âe2sothatrmax,min=a(1Âąe).Alsob=r0
â1âe2
â˘Thesemi-majoraxisadeterminestheenergyandtheperiodoftheorbit
E=âA
2a;T=
2Ď
Ď;Ď
2=
A
ma3
â˘Thesemi-latusrectumr0determinestheangularmomentumoftheorbit:J2
=Amr0
â˘Ifyouneedtoderiveanyoftheseformulaeinahurryconsiderasillycase.
Usethesimplebalanceofforcesargumentforcircularmotion:A
r2=mĎ2r.
ClassicalDynamicsP37 TIMEINORBITâ(DETAILSNON-EXAMINABLE)
â˘Wehavenâtsofargotaformulaforthecoordinates(r,Ď)asafunctionoftime.Thereisan
easywayofdoingthis,whichiscomingabitlater,butwecaninfactgetaformulafort(Ď),
ratherthanĎ(t).Itâsjustnotverypretty...
â˘FromtheequationoftheellipseandhâĄJ
m=r
2Ďweget
dĎ
(1+ecosĎ)2=
hdt
r20
â˘Theintegralisfoundasformula2.551ofGradshteyn&Ryzhik:
t=r20
h(1âe2)
(
âesinĎ
1+ecosĎ+
2â
1âe2tan
â1
(tan(Ď+
14Ď)+e
â1âe2
))
O CA
P
â˘Youcanalsogetaformulaforitbysubtracting
theareaofthetriangleCPOfromthesectorCPA.
ClassicalDynamicsP38 ELLIPTICALORBITSâANOTHER(NON-EXAMINABLE)WAY
â˘Returntotheenergyequation:12mr
2+
J2
2mr2âA
r=E.
â˘Changetheindependentvariable,definingaradially-scaledâtimeârds=dtsothat
dr
dt=
1
r
dr
dsWethenget
(dr
ds
)2
â2E
mr2â
2A
mr+J
2=0
â˘DefiningΊ2
=â2E
m(rememberEisnegative)wegetthesameformofequationas
before,butforrinsteadofu.Thedistancesoffurthestandclosestapproacharea(1Âąe),
sothesolutionisjustr=a(1âecosΊs).
â˘Wecannowintegrater=dt
dstofindaniceparametricformforthetimet
t=asâa
ΊsinΊs.TheperiodisT=
2Ďa
Ί=2Ď
â
ma3
A.
â˘ThereisalsoasimpleclosedformforĎ(s):tan
(â
1âe2Ď
2
)
=
â
1+e
1âetan
Ίs
2.
â˘ThisisrelatedtotheâsquarerootâoftheKeplerproblem,whichtransformstheorbittoa
centralellipse(2-SHM).ManyessentialfeaturesofthissolutionwereknowntoNewton,but
theprocedureiscalledtheâKustaanheimo-Stiefelâtransformation.
ClassicalDynamicsP39 EXAMPLE:THEHOHMANNTRANSFERORBIT
Dv
2
2
Dv1
1
a
a
â˘TheHohmanntransferorbitisonehalfofanellipticorbit
thattouchesboththeinitialorbitandthedesiredorbit.
â˘ForgravitationalcaseputA=GMm.
â˘IncircularorbitsT=âE=â12U,forelliptical
orbitsăTă=âăEă=â12ăUă(VirialTheorem).
â˘Theinitialenergy(foraspacecraftofunitmass)is
E1=âGM
2a1
,andthevelocityhastobeincreased
untilthespacecrafthastheenergyofthetransferorbitEt=âGM
a1+a2
.
â˘Theimportantthingtoknowisâv1,sincethatdeterminestheamountoffuelused,butwe
canworkallthatoutfromtheenergies:
Et=âGM
a1+a2
=âGM
a1
+12v
2t1â
12v
2t1=
GMa2
a1(a2+a1)
â˘Therestofrelationsareeasyenough,butnotparticularlyinformative...
â˘TheHohmanntransferisthemostfuelefficientorbit,unlessthereareothermassivebodies
inthevicinity,inwhichcaseyoucanusethegravitationalslingshot.
ClassicalDynamicsP40 GRAVITATIONALSLINGSHOT
â˘Theescapevelocityofaspacecraft
fromtheSolarsystemattheradius
oftheEarthâsorbitis42kmsâ1
,
whichshouldbecomparedtoits
orbitalvelocityof30kmsâ1
.
â˘Wecanuseagravitationalâslingshotâ
aroundplanetstoincreasekinetic
energyand/orchangedirection
inordertovisitotherbodies
intheSolarsystem.
â˘Voyager2madeaâgrandtourâ.
ClassicalDynamicsP41 GRAVITATIONALSLINGSHOTâVOYAGER2
ClassicalDynamicsP42 THETWO-BODYPROBLEMANDREDUCEDMASS
â˘ThetwomassesM1andM2orbitthecentreofmass.
â˘Eachorbitisanellipseinacommonplanewiththecentreofmassatonefocus.
Theellipseshavethesameeccentricityandphase.
â˘Theimportantcaseiscircularmotion.
MassM1isdistanceM2r
M1+M2
fromtheCoM.
â˘BalanceofforcesforM1:GM1M2
r2=M1Ď2M2r
M1+M2
âĎ2
=G(M1+M2)
r3
â˘Thisisthereallyimportantresultandisusuallythebeststartingpoint(e.g.inproblemQ10).
â˘YougetthesameresultbyconsideringthebalanceofforcesforM2.
â˘WecanrearrangetheresultasÂľrĎ2
=GM1M2
r2,inwhichwehavethetrueseparationr
andtheactualforceGM1M2
r2,butamodifiedreducedmasstermÂľâĄM1M2
M1+M2
.
â˘Idonotadvocatetheuseofreducedmass,despiteitswidespreaduseinthetextbooks...
ClassicalDynamicsP43 THETWO-BODYPROBLEM
â˘TwoparticlesofmassesM1andM2orbitingeachotherâpositionsr1andr2.
M1
M2
2r
1r
CoM
â˘Theenergy,angularmomentumandequationsofmotioncanbeexpressed
intermsofthereducedmassÂľâĄM1M2
M1+M2
andr1âr2.
â˘ThecentreofmassisatR0=M1r1+M2r2
M1+M2
.
â˘DefineĎ1âĄr1âR0=M2
M1+M2
(r1âr2)
Ď2âĄr2âR0=M1
M1+M2
(r2âr1)
â˘Kineticenergyinthecentreofmassframe:
T=1
2M1Ď2
1+1
2M2Ď2
2=1
2
(M1M
2
2
(M1+M2)2+
M2
1M2
(M1+M2)2
)
(r1âr2)2
=Âľ
2(r1âr2)
2
â˘Angularmomentum:J=M1Ď1ĂĂĂĂĂĎ1+M2Ď2ĂĂĂĂĂĎ2=Âľ(r1âr2)ĂĂĂĂĂ(r1âr2).
â˘Equationsofmotion:M1r1=F12;M2r2=F21=âF12
r1âr2=
(1
M1+
1
M2
)
F12=1
ÂľF12;R0=0
â˘Reducedtotheone-bodyprobleminthecentreofmassframe.
ClassicalDynamicsP44HYPERBOLICORBITS
OC
P
impact parameter
semi-latus rectum
semi-major axis
ff
Path for repulsive force
b
a
0r
r
8
â˘Attractivepotential:allpreviousformulaestillvalid,
bute>1soa<0andenergy
E=âA
2a=
(e2â1)A
2r0
>0.
â˘Impactparameterbandvelocityatinfinityvâ
determineangularmomentumJ=mbvâ
andenergyE=12mv
2â.
â˘Mostproblems(e.g.RutherfordscatteringQ14)
requirethetotalangleofdeflection
Ď=2ĎââĎ(Ďpositive).
â˘AsymptotesareatÂąĎâ;fromtheequationoftheellipsewehave
cosĎâ=â1/eâsecĎâ=âeâtan2Ďâ=e
2â1
. â˘NotethatĎ/2<Ďâ<Ď.
ClassicalDynamicsP45 HYPERBOLICORBITSII
(SeeSection4.4ofHandout)
OC
P
impact parameter
semi-latus rectum
semi-major axis
ff
Path for repulsive force
b
a
0r
r
8
Attractivepotential
â˘Weneedtofindtheeccentricity
fromthephysicalparametersEandJ.
â˘Fromthedefinitionofr0wecanwrite
(e2â1)=
2r0E
A=
2J2E
mA2.
â˘Intermsofbandvâthismeans
tan2Ďâ=e
2â1=
m2v4âb
2
A2âtanĎâ=mv
2âb
A.
Repulsivepotential:
â˘ChangeA
r2ââB
r2,defineJ2
=Bmr0anduseotherbranchr0=r(ecosĎâ1).
â˘aispositiveagainandthetotalangleofdeflectionisnowĎ=Ďâ2Ďâ(Ďnegative).
â˘TheasymptotesarestillrelatedtothephysicalparametersbytanĎâ=mv
2âb
B.
â˘Thedistanceofclosestapproachisa(1+e).
ClassicalDynamicsP46 RUTHERFORDSCATTERING
ClassicalDynamicsP47 RUTHERFORDSCATTERINGII
ClassicalDynamicsP48
HYPERBOLICORBITSINREPULSIVEPOTENTIALâANOTHERWAY
OC
P
impact parameter
semi-latus rectum
semi-major axis
ff
Path for repulsive force
b
a
0r
r
8
â˘Wecansimplyusetheresultsfortheattractive
potentialcase(r0=r(1+ecosĎ))
andletĎexceedĎâ.
â˘Theradiusristhennegativeandtheparticle
tracesouttherepulsivebranch,gettingclosest
toOatĎ=Ď,sothatr(Ď)=a(1+e),
whereaisnownegative.
â˘ThisworksbecausethepotentialenergyâA
rispositivewhenr<0andsorepresents
arepulsivepotential.
â˘Thisapproachhastheconsiderableadvantagethatnosignchangesareneeded,butithas
somethingofaâAlicethroughthelookingglassâqualitytoit.
ClassicalDynamicsP49 THETHREE-BODYPROBLEM
â˘Somehierarchicalsystemscanbestableindefinitelye.g.Sun,EarthandtheMoon.
â˘Ageneral3-bodyencountercanbeverycomplicated,butageneralfeatureemerges.
â˘If3bodiesareallowedtoattracteachotherfromadistance(a),theywillspeedupand
interactstrongly(b).Eventuallytheinteractionislikelytoformaclosebinary(negative
gravitationalbindingenergy)releasingkineticenergy,whichmaybeenoughforthebodiesto
escapetoinfinity(c).
â˘Thismechanismisresponsibleforâevaporationâofstarsfromstarclusters.(maybealso
invalidatesthevirialtheorem?)
â˘TheplanetPlutohasaclosecompanionCharon,andhasaneccentricorbitwhichtakesit
insideNeptuneâsorbit.A3-bodycollisionamongstNeptuneâsmoonsisthemostlikelycause.
ClassicalDynamicsP50 GRAVITATIONALPOTENTIAL,FIELDANDTIDALFORCES
Therethreeimportantaspectstogravitation(NewtonianorGR).
â˘GravitationalpotentialĎ(r)Thisdeterminesenergiesandredshifts;velocitiesof
objectsandtemperatureofgases.Alwaysrelativeâcanâthaveanabsolutevalue.
ForpointmassĎ=âGM
R.[NewtonianpotentialisonepartofthemetricofGR.]
â˘Gravitationalfieldg(r)=ââĎThisdeterminesaccelerationsandorbits.The
fieldisalsorelative(perhapssurprisingly).e.g.we(andtheLocalGroupofgalaxies)could
allbeacceleratingtoaâGreatAttractorâandnothingwouldchangehere.Forpointmass
|g|=GM
R2.[GravitationalfieldisonepartoftheâconnectionâinGR].
â˘GravitationaltidalfieldR(a)=a¡âgThisisallonecanfeelandmeasure
locallyâitdescribeshowthegravitationalfieldvariesinspace.Incomponents
[R(a)]i=RijajwhereRijâĄâgi
âxj.Thegravitationaltidalfieldvariesas1/R
3.
ForapointmassitisR(ur)=2GM
R3ur;R(uθ)=âGM
R3uθ;R(uĎ)=âGM
R3uĎ;
Thereisaradialstretching,andasquashingbyhalfasmuchinthetransversedirections.
ThetidalaccelerationtensorRijgivesthecoefficientsofthequadratictermintheTaylor
expansionofthegravitationalpotential.[TidalfieldispartoftheRiemanntensorofGR].
ClassicalDynamicsP51 MOREABOUTTIDALFORCES
â˘Thegravitationalfieldnearapointmassisdirectedradiallyandisproportionalto1/r2.The
tidalforcesconsistofaradialstretching2GM/r3
andasidewayscompressionâGM/r3.
â˘Forthetwo-bodypotentialwemustalsoaddthecontributionfromcentrifugalforceâthisis
astretchingintwodirectionsintheplaneofrotation,andnocontributioninthedirectionof
therotationaxis.Thisassumesourstickmaniscorotatingwiththeorbit(i.e.keepingthe
samerelativeorientationwithrespecttothemassM).
â˘Thesumisastretchof3GM/R3
alongtheradialdirection,nocontributionintheorbital
planeandâGM/R3
perpendiculartotheplane.
â˘TidalforcesareweakonEarthandnotverystronginthesolarsystem(exception:Jupiter
andIo),buttidalforcescanbecolossalnearcompactobjectssuchasneutronstars.
ClassicalDynamicsP52 ORIGINOFTHETIDES
â˘Lookingdownontheorbitalplane,wesee
theEarthrotatingunderatidalbulgeofwater,
makingtwotidesaday(â1hrlaterperday).
â˘ThetidalaccelerationintheEarth-Moondirection
is3GM2z
r3,wherezisthedistancefromthecentreoftheEarth.
â˘IntegratingtogetthetidalpotentialwefindĎtide=â3GM2a
2
2r3atthesurfaceatpointA.
â˘ThereisnotidalpotentialatpointB,sotheheighthofthetideisgivenbyĎtide=âgh,
wherethegravityg=GM1
a2.
â˘Eliminatingg,theheightoftidesish=3M2a
4
2M1r3=0.5m.
â˘Canbeless(Mediterranean)ormuchmore,duetolimitedflow(e.g.St.MaloandBristol
Channel)oramplificationbyresonances(e.g.Solent).
â˘TidefromtheSunabouthalfthatfromtheMoonâexplainsâspringâandâneapâtides.
â˘TheMoonnowkeepsthesamefacetowardsusâitsinitialadditionalrotationwas
dissipatedagainstEarthtidesof16m.Itwasoncemuchclosertous,anditisstillreceding.
(TherotationoftheEarthisalsoslowingdown.)