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CH 13 

Gravitation

I. Newton’sLawofGravitation

A. Everyparticleattractsanyotherparticlewithagravitationalforceofmagnitude:

 

 

 

1. Herem1andm

2arethemassesoftheparticles,risthedistance

betweenthem,andGisthegravitationalconstant.

G =6.67 x10-11

Nm2/kg

2

2. Drawingtheforcedirection

 

a) Thegravitationalforceonparticle1duetoparticle2isanattractiveforcebecauseparticle1isattractedtoparticle2.

b) Forceisdirectedalongaradialcoordinateaxisrextendingfromparticle1throughparticle2.

c) Forceisinthedirectionofaunitvector alongtheraxis. 

r̂

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B. Auniformsphericalshellofmatterattractsaparticlethatisoutsidetheshellasifalltheshell’smasswereconcentratedatitscenter.

 

C. GravitationandthePrincipleofSuperposition

1. Givenagroupofparticles,wefindtheNet(orresultant)gravitationalforceonanyoneofthemfromtheothersbyusingthePrincipleofSuperposition.Thisisageneralprinciplethatsaystheneteffectisthesumoftheindividualeffects.

2. Here,thatprinciplemeansthatwefirstcomputetheindividualgravitationalforcesthatactonourselectedparticleduetoeachoftheotherparticles.Wethenfindthenetforcebyaddingtheseforcestogethervectorially.

a) Forninteractingparticles,wecanwritetheprincipleofsuperpositionforthegravitationalforcesonparticle1as

 

 

b) HereF1,net

isthenetforceonparticle1duetotheotherparticles

and,forexample,F13istheforceonparticle1fromparticle3,etc.

Therefore,

c) Thegravitationalforceonaparticlefromareal(extended)objectcanbeexpressedas:

 

Heretheintegralistakenovertheentireextendedobject.

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D. SampleProblems: 

1. Thefigurebelowshowsanarrangementofthreeparticles,particle1ofmassm

1=6.0kgandparticles2and3ofmassm

2=m

3=4.0kg,anddistancea

=2.0cm.Whatisthenetgravitationalforce1,netonparticle1duetotheotherparticles?

 

a) Solution: 

 

 

 

 

 

 

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E. InFig.below,asquareofedgelength20cmisformedbyfourspheresofmassesm1=5.00g,m2=3.00g,m3=1.00gandm4=5.00g.Whatis(a)thex‐componentand(b)they‐componentofthenetgravitationalforcefromthemonacentralspherewithmassm5=2.50g?

1. Solution: 

 

 

 

 

 

 

 

 

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II. GravitationnearEarth’sSurface

A. Iftheparticleisreleased,itwillfalltowardthecenterofEarth,asaresultofthegravitationalforce,withanaccelerationweshallcallthegravitationalaccelerationa

g.Newton’ssecondlawtellsusthat

magnitudesFandagarerelatedby

B. IftheEarthisauniformsphereofmassM,themagnitudeofthegravitationalforcefromEarthonaparticleofmassm,locatedoutsideEarthadistancerfromEarth’scenter,is

 

 

 

C. Therefore, 

 

D. Thus,

E. Annapolisag=9.80171m/s2

F. TableofVariationofag

 

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G. Anygvaluemeasuredatagivenlocationwilldifferfromtheag

valuegivenbeforeforthatlocationforthreereasons:

1. Earth’smassisnotdistributeduniformly,

2. Earthisnotaperfectsphere,and

3. Earthrotates.

a) =

b) g= 

H. Forthesamethreereasons,themeasuredweightmgofaparticlealsodiffersfromthemagnitudeofthegravitationalforceontheparticle.

 

I. SampleProblems:

1. Atwhataltitude(inkm)aboveEarth'ssurfacewouldthegravitationalaccelerationbe4.9m/s2?(TaketheEarth'sradiusas6370km.)

 

a) Solution: 

 

 

 

 

 

 

 

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2. PlanetAhasmassM=3.00×1024kgandradiusR=2.00×107m,anditcompletesafullrotationintimeT=35.0h.Whatisthefree‐fallaccelerationgonitsequator?

a) Solution: 

  

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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III. GravitationalPotentialEnergy

A. Thegravitationalpotentialenergyofthetwo‐particlesystemis: 

 

 

1. U(r)approacheszeroasrapproachesinfinityandthatforanyfinitevalueofr,thevalueofU(r)isnegative.

 

a) ProofofEquation:(nottestable)

(1) LetusshootabaseballdirectlyawayfromEarthalongthepathinthefigure.WewanttofindthegravitationalpotentialenergyUoftheballatpointPalongitspath,atradialdistanceRfromEarth’scenter.

(2) TheworkWdoneontheballbythegravitationalforceastheballtravelsfrompointPtoagreat(infinite)distancefromEarthis:

 

 

(3) whereWistheworkrequiredtomovetheballfrompointP(atdistanceR)toinfinity.Workcanalsobeexpressedintermsofpotentialenergiesas

 

 

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B. HowdowesolveforPotentialEnergyiftherearemorethantwoparticles?

1. Ifthesystemcontainsmorethantwoparticles,considereachpairofparticlesinturn,calculatethegravitationalpotentialenergyofthatpairwiththeaboverelation,asiftheotherparticleswerenotthere,andthenalgebraicallysumtheresults.Thatis,

 

 

 

2. Drawingforthreeparticlesasexample

 

C. GravitationalPotentialEnergyPathIndependence

1. Theworkdonealongeachcirculararciszero,becausethedirectionofFisperpendiculartothearcateverypoint.Thus,WisthesumofonlytheworksdonebyFalongthethreeradiallengths.

 

2. Thegravitationalforceisaconservativeforce.Thus,theworkdonebythegravitationalforceonaparticlemovingfromaninitialpointitoafinalpointfisindependentofthepathtakenbetweenthepoints.ThechangeUinthegravitationalpotentialenergyfrompointitopointfisgivenby

 

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3. SincetheworkWdonebyaconservativeforceisindependentoftheactualpathtaken,thechangeUingravitationalpotentialenergyisalsoindependentofthepathtaken.

 

D. GravitationalPotentialEnergy:PotentialEnergyandForce 

1. RememberfromChapter8: 

 

2. Thus

Theminussignindicatesthattheforceonmassmpointsradiallyinward,towardmassM.

 

E. EscapeSpeed

1. Ifyoufireaprojectileupward,thereisacertainminimuminitialspeedthatwillcauseittomoveupwardforever,theoreticallycomingtorestonlyatinfinity.

a) Thisminimuminitialspeediscalledthe(Earth)escapespeed.

2. Consideraprojectileofmassm,leavingthesurfaceofaplanet(massM,radiusR)withescapespeedv.TheprojectilehasakineticenergyKgiven

by½mv2,andapotentialenergyUgivenby:

 

 

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3. Whentheprojectilereachesinfinity,itstopsandthushasnokineticenergy.Italsohasnopotentialenergybecauseaninfiniteseparationbetweentwobodiesisourzero‐potential‐energyconfiguration.Itstotalenergyatinfinityisthereforezero.Fromtheprincipleofconservationofenergy,itstotalenergyattheplanet’ssurfacemustalsohavebeenzero,andso

 

 

 

4. Thisgivestheescapespeed:

 

 

5. SomeKnownEscapeSpeeds

F. SampleProblems:

1. ThemeandiametersofplanetsAandBare9.4×103kmand2.2×104km,respectively.TheratioofthemassofplanetAtothatofplanetBis0.10.WhatistheratioofescapespeedonAtothatonB?

a) Solution: 

 

 

 

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2. Whatisthegravitationalpotentialenergyofatwo‐particlesystemwithmasses5.2kgand2.4kg,iftheyareseparatedby19m?Ifyoutripletheseparationbetweentheparticles,howmuchworkisdonebythegravitationalforcebetweentheparticles?

 

a) Solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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IV. PlanetsandSatellites:Kepler’sLaws

A. Terminology

1. Ellipse

a) Anellipse.Theshapeoftheellipseisdeterminedbytheratioofthedistancebetweenthetwofoci(F)tothelengthofthemajoraxis(theeccentricity).Howstretchedoutthatellipseisfromaperfectcircleisknownasitseccentricity.

b) Ifthefociareclosertogether,theellipsewillhaveasmallereccentricityandwillmorecloselyresembleacircle.

c) Ifthefociarefartherapart,theywillhaveagreatereccentricityandwillmorecloselyresembleastraightline.

2. Suffix

a) Peri

b) ApoorAp

3. Usesinourcontext

a) PerihelionPointinorbitclosest

b) AphelionPointinorbitfurthest

c) PerigeePointinorbitclosestto

d) ApogeePointinorbitfurthestfrom

4. TableofusesinAstro‐Physics(nottestable)

a) http://en.wikipedia.org/wiki/Apsis 

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B. Kepler’sFirstLaw

1. THELAWOFORBITS:______________________________________________________________________________________________________________________________________________________________________________________________.

 

 

 

 

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C. Kepler’sSecondLaw

1. THELAWOFAREAS:AlinethatconnectsaplanettotheSunsweepsoutequalareasintheplaneoftheplanet’sorbitinequaltimeintervals;thatis,theratedA/dtatwhichitsweepsoutareaAisconstant.

 

a) Asaplanetmovesinitsorbit,thelinefromthecenteroftheSuntothecenteroftheplanetsweepsoutequalareasinequaltimes,soiftheareaSAB(withcurvedsideAB)equalstheareaSCD,theplanettakesthesametimetomovefromAtoBasitdoesfromCtoD.

2. ProvewithCalculus

 

 

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a)  

 

b) Angularmomentum,L: 

 

 

 

c) Thus

CONSTANT 

D. Kepler’sThirdLaw

1. THELAWOFPERIODS:Thesquareoftheperiodofanyplanetisproportionaltothecubeofthesemimajoraxisofitsorbit.

2. Consideracircularorbitwithradiusr(theradiusofacircleisequivalenttothesemimajoraxisofanellipse).ApplyingNewton’ssecondlawtotheorbitingplanetyields

 

 

 

 

3. Usingtherelationoftheangularvelocity,,andtheperiod,T; 

 

 

 

4. rmust(forellipses)bereplacedbya,buttheformulastillholds. 

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5. Examplesforoursolarsystem

V. Sampleproblems:

A. The ellipticalorbitofaplanetaroundtheSunisshownonthediagram.Whichletterlabelsthepositionofthesun?

 

1. A 

2. B 

3. C 

4. D 

5. E 

 

 

 

 

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B. AplanettravelsinanellipticalorbitaboutastarXasshown.Themagnitudeoftheangularvelocityoftheplanetis:

 

1. greatest at point Q 

2. greatest at point S 

3. greatest at point U 

4. greatest at point W 

5. the same at all points 

 

 

 

 

 

C. Atperihelionaplanetinanothersolarsystemis175x106kmfromitssunandistravelingat40km/s.Ataphelionitis250x106kmdistantandistravelingat:

 

1) 20 km/s 

2) 28 km/s 

3) 34 km/s 

4) 40 km/s 

5) 57 km/s 

 

 

 

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D. Twoplanetsareorbitingastarinadistantgalaxy.Thefirsthasasemimajoraxisof150x106km,aneccentricityof0.20,andaperiodof1.0Earthyears.Thesecondhasasemimajoraxisof250x106km,aneccentricityof0.30,andaperiodof:

1)0.46Earthyr

2)0.57Earthyr

3)1.4Earthyr

4)1.8Earthyr

5)2.2Earthyr 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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E. Supposethatacometthatwasseenin672A.D.byChineseastronomerswasspottedagaininyear1949.Assumethetimebetweenobservationsistheperiodofthecometandtakeitseccentricityas0.055.Whatare(a)thesemimajoraxisofthecomet'sorbitand(b)itsgreatestdistancefromtheSun?

 

a) Solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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VI. Satellites:OrbitsandEnergy

A. AsasatelliteorbitsEarthinanellipticalpath,themechanicalenergyEofthesatelliteremainsconstant.Assumethatthesatellite’smassissomuchsmallerthanEarth’smass.

B. Thepotentialenergyofthesystemisgivenby 

 

C. Forasatelliteinacircularorbit, 

 

1. Thus,onegets: 

 

 

 

2. Foranellipticalorbit(semimajoraxisa),