asp.net mvc 4 and the web api pdf free download : Add pdf files together reader application software tool html winforms web page online mechanics22-part1293

6|Orbits
217
Thisisabout4%largerthantheresultusuallyquotedforthiseect(531
:
6
00
). Oneerrorinthepresent
approximation is that the plane of Mercury’s s orbit t is s 7
o from the plane e of most other r planets.
Anotheristhatitsorbitismoreeccentricthantheotherplanets,
=0
:
21,sothatassumingasmall
deviationfromacirclemaynotbegoodenough.Also,theorbitsoftheotherplanetsarenotcircles,and
thelimitationsofreplacingthemovingplanetbyauniformringofmassmustbereexaminedbecause
planetary speeds vary overan orbit, , making theeective mass s density y biggerwhere e it moves more
slowly.Allofthesecontributetotheeectandaccountforthechangefrom553to531seconds. You
willhavetondthosecalculationselsewhere,astheygowellbeyondwhatyouseehere.
Computingellipticintegrals
Thisusesthe\arithmetic-geometricmean"method,andifyouthinkthatit’snotobviouswhyitworks,
you’reright.Itinvolvesmakingsomeincrediblycleverchangesofvariableintheoriginalintegrals.*
n
=0
0.
a
0
=1
b
0
=
k
2
1
=
2
c
0
=
k
d
=
c
2
0
1.
a
n
+1
=
1
2
(
a
n
+
b
n
)
b
n
+1
=
a
n
b
n
1
=
2
c
n
+1
=
1
2
(
a
n
b
n
)
d
=
d
+
c
2
n
+1
.
2
n
+1
2. if
c
n
+1
issmallenoughthenstop,elseincrement
n
andrepeatstep#1
K
(
k
)=
=
(2
a
n
+1
)
;
E
(
k
)=
K
(
k
)
.
(1 
d=
2)
Thisprocessconvergesveryfast.Forexample,if
k
=1
=
2,
b
0
=
p
3
=
4and
0.
a
0
=1
b
0
=0
:
86602540
c
0
=0
:
5
d
=0
:
25
1.
a
1
=0
:
933012702
b
1
=0
:
930604859
c
1
=0
:
066987298
d
=0
:
258974600
2.
a
2
=0
:
931808780
b
2
=0
:
931808003
c
2
=0
:
001203921
d
=0
:
258980394
3.
a
3
=0
:
931808392
b
3
=0
:
931808392
c
3
=0
:
00000039
d
=0
:
25898039
K
(0
:
5)=
2
a
3
=1
:
685750354812596
E
(0
:
5)=1
:
467462209339427
Thesenumberswerereallycomputedwithalmosttwicethenumberofdigitsshown,andittook
nomoresteps. Ifallthatyouwantis
K
,thenyoudon’tneedeither
c
n
or
d
. Here,however,computing
K
0
and
K
00
involves
E
.
Exercises
SketchaLissajousgureforthecase
!
2
=10
!
1
.
Forthetwo-dimensionalharmonicoscillatorofEqs.(6.1){(6.3),whatistheangularmomentum?Is
itconstant? I.e.take
z
=0here.
From the informationgiven atandjust afterEq.(6.17),whatis theratioof thesemi-minoraxis
tothesemi-majoraxisforEarthandforMercury? Whatistheratiooftheperiheliondistancetothe
apheliondistanceinthesametwocases?
Whatguredoyougetifthesumofthesquaresofthedistancesfromtwopointsaddtoaconstant?
ForthethinnestellipseatEq.(6.25),whatare
b
and
c
?
* SeetheWikipediaentryforarithmetic-geometricmean.
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6|Orbits
218
CommunicationssatellitesareplacedinorbitsabovetheEarth’sequatorsothatrelativetotheEarth,
theystayinaxedposition.HowfararetheyfromtheEarth’scenterandfromtheEarth’ssurface?
Amass is inorbitaboutaxedcenter,and the attractive force lawissuchthatitcanorbitina
circleatradius
r
0
fromthatcenter.Theangularspeedis
_
0
=
!
0
.Theorbitisslightlyperturbed,so
thatthemassorbitsinanearlycircularorbitandtheradiusoscillatesaboutthecirclewithfrequency
3
!
0
=
2. Drawtheresultingorbitandexplainthereasonsthatyourdrawingappearsasitdoes.
WhatistheratioofthesolarpowerreachingEarthatitsperihelion(January)tothatatitsaphelion
(July)?NowwhataboutMercuryandMars?
Checkthedimensionsoftheequations(6.27)and(6.28).
10 InthesetofellipsesatEq.(6.25),ifthecirclerepresentsEarth’sorbit,radius15010
6
km,what
istheperiheliondistanceforthemosteccentricoftheseorbits? Andwhatisitssemi-minoraxis? In
thesamepicture,whereistheorbitofMars?
11 Inlookingforplanetarytransitsofanotherstarasonpage 203,whatdothepictureslooklikein
theextremecasesthatsin
1andthecasethatsin
isalmostequaltoone.
12 Theexpression
L
z
=
mr
2
sin
2
_
followsEq.(6.37).Showhowtoderiveitinoneortwolines.
13 Jupiter’s orbital period is about 12 years. . Orbiting g atthe same distance, what would Jupiter’s
periodbeiftheJupiterhadthesamemassastheSun? OriftheSunhadthesamemassasJupiter?
14 Userthersttermsoftheseriesexpansionfor
K
tocheckthatEq.(6.75)producesEq.(6.70).
not
AllAboutEllipses
15 Ifyourememberthatthepolarequationforanellipseissomethinglike
r
=
K=
(1+
cos
),but
youdon’trememberwhat
K
is,evaluatethisexpressionatthetwoendsofthemajoraxis,at
=0
;
,
andaddtheresultstosolvefor
K
. Fortheanswer,seetherstequationinEq.(6.17).
16 WheredoesthesecondoftheequationsinEq.(6.17)comefrom?Imaginetwoequalpointmasses
atthetwoendsofthemajoraxis,andcomputetheircenterofmass,measuredfromtheorigin. Use
theequationfromtheimmediatelyprecedingexercise.
17 ThethirdequationinEq.(6.17) tells you
b
. Howso? ? Thelength h ofthesemi-minoraxisisthe
maximum
y
-coordinateontheellipse,somaximize
y
=
r
sin
bytaking
dy=d
=0.
18 ThefourthequationinEq.(6.17)followsfromthetwoprecedingexercises.
~r
~r
0
~
F
19 The fthequation in Eq.(6.17) is just alittle more involved. . Use
the same e gure e as s with the e four r preceding exercises, , but t now express
everythingintermsofvectors.
~r
0
=
~r
+
~
F
,and
~
F
=2
~
f
. Torelatethe
lengths,use
~r
0
.
~r
0
=(
~
F
~r
)
.
(
~
F
~r
)andeliminate
r
cos
byusingthe
precedingexercises.
20 Thesixthequationisproblem0.48.Ifyouhaven’tdoneit,nowwouldbeagoodtime.
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6|Orbits
219
Problems
6.1 InEq.(6.10),iftheforce
f
iszero,themotionshouldsomehowrepresentastraightline. Takethe
caseasrepresentedinrectangularcoordinatesby
x
=
x
0
and
y
=
v
0
t
;translateitintoanexpression
for
r
(
t
),andshowthatitworks.
6.2 TheLissajouscurveinEq.(6.5)canbeexpressedasapolynomialequationin
x
and
y
. Doso.
6.3_Referringtotheequations(6.3)andtothepicture,Figure6.2,showthattheanglebywhichthe
thirdellipseisrotatedis
,wheretan2
=
A
2
2
sin2
=
A
2
1
A
2
2
cos2
. Ifyouranswercomesoutina
dierentform,thencheckafewnumericalvaluesoftheparameterstoseeifthetwoexpressionsagree.
6.4 Transform Eq.(6.12) intotheform Eq.(6.13)toshowthatthis isthepolarformof anellipse.
Notreally. Findtheconditionson
A
,
B
,and
C
sothatit is anellipse. . Whatshapesdoyougetif
theseconditionsareviolated?Catalogandsketchthepossibilities,including
B
=
C
.
6.5 (a)Foraplanetincircularorbitaroundthesun,whatistheplanet’sspeedasafunctionofdistance
fromthesun?(b)AyearonMarsisveryclosetotwoEarthyears.Oneastronomicalunitis(veryclose
to)Earth’smeandistancefromthesun;HowmanyAUisMarsfromthesun?
6.6 InEq.(6.14),lettheforcebezero.Solve,andshowwhattheshapeoftheorbitis.
6.7 If
r
1
and
r
2
arethedistancesofperihelionandaphelionforaplanet,whataretheirarithmeticand
theirgeometricmeans? Ans:
a
and
b
Ifyoudidn’tdoproblems4.15and4.16,nowwouldbeagoodtime.Alsoproblem4.13.
6.8 TheEarth-Sundistanceis150Gm. Startfromelementaryprinciples,commonlyknowndata,and
thevalueof
G
todeducethemassoftheSun.
6.9 Aparticleinacentralforcemovesinaspiralorbitgivenby
r
=
a
. Findtheforceandndthe
timedependenceof
,i.e.,
(
t
).
6.10 ImproveontheresultcalculatedinEq.(6.11)byusingthefactthattheorbitofHalley’scomet
haseccentricity
=0
:
96714.
6.11 Startfrom Eq.(6.34)andexaminethe caseof theKeplerproblem. . Plot t the polarcoordinate
orbit
r
(
)andcomparetheresulttotheexactsolution.
6.12 Theforceonanorbitingplanetofmass
m
is givenasthesumoftwoterms: : Oneisacentral
force;theotherisavelocitydependentfrictionterm.
~
F
1
F
0
sech
3
(
r
)
K
3
(
r
)^
r;
~
F
2
~v
wheresechisthe hyperbolicsecant and
K
3
isthe Besselfunction of thethirdkindwith imaginary
argument. Theangularmomentumabouttheoriginisgivenattime
t
=0tobe
~
L
0
.Findtheangular
momentumatlatertimes.
6.13 Theforceonamass
m
isgiventobe
~
F
=
k^r=r
3
. Themassisgivenaninitialspeed
v
0
andis
aimedsothatintheabsenceofforces,itwillmissthecenteroftherepulsiveforcebyadistance
b
(the
impactparameter). Findthedistanceofclosestapproachtotheoriginandwhatisitsspeedwhenit
reachesthatpoint?(Don’tsolvefortheorbitunlessyouneedtheextrawork.) Ans:
p
b
2+
k=mv
2
0
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6|Orbits
220
6.14 Anobjectstartsatanegligiblysmallspeedfromalongdistanceawayfromthesun,fallingstraight
intothesun. (a)AsitpassesdistancetothesunequaltotheradiusoftheEarth’sorbit,howmuch
timedoesithavebeforeithitsthesun? ExpresstheanswerasaratiototheperiodofEarth’sorbit,
and then expressitin weeks ordays. . (b) And since you’vedone the labor r already,gobacktothe
exercise#4onpage176tondhowmuchtimeitwouldtakefromtheonsetofspaghetticationto
hitthecenterofattractioninthatcase,andwiththesameassumptionthatyou’refallingfromagreat
distance. Ans:(a)almostamonthand(b)about0
:
3second.
6.15_IftheEarthstoppedstillwithrespecttothesun,howmuchtimewouldittaketofallintothesun?
Setuptheequationofmotionforstraightlineradialmotion,gettheconservationofenergyequation
and rearrangetoget
dt
interms of
dr
. Set t up awell-deneddenite integral forthenalanswer,
thenevaluateit. Thisisanintegralyoumayhavetolookup. Expresstheanswerasamultipleofthe
orbitalperiodoftheplanet.HowmanyweeksordaysisthatfortheEarth? Ans:1
=
4
p
2years|about
2months
6.16_Acentralattractiveforceisgivenby the equation
~
F
r
^r. Here >0,butcanhave
eithersign.(a)Assumethattheorbitisalmostcircularandndthevaluesof
forwhichtheorbitis
stable.(b)Findthevaluesof
forwhichtheorbitisclosed|atleastintheperturbationapproximation
usedinthischapter. Ans:a:
>
3
6.17 Aparticleinacentralforcemovesinaspiralorbitgivenby
r
=
a=
. Findtheforcethatdoes
this.Andsketchthisorbit.
6.18 DeriveEq.(6.52).
6.19 Amass
m
ismovinginthecentralforce
~
F
k~r
. Foracircularorbit,whatisthefrequency
(orperiod)oftheorbitasafunctionoftheorbit’sradius?
6.20 Forthesameforceastheprecedingproblem,ndtheshapeofanalmostcircularorbitandexplain
whyitcomesoutasitdoes.
6.21_Twomasses
m
1
and
m
2
are attracted toeach otherby aforce directly proportional to their
distance apart. . Transform m to the centerof mass coordinate system and solve the problem oftheir
motionasafunctionoftime.
6.22Forthesameforceasinproblems6.19and6.20,transformtoasystemrotatingwiththefrequency
thatyoufoundintherstoftheseproblems. Onesolutionisofcoursethatthemassisatrestatits
initialradiusandangle,say
r
=
r
0
and
=0.Expandtheequationsaboutthistorstorderandget
lineardierentialequationsin
r
r
0
=
andin
. Forsmall
and
plottheorbit. Doyoubelieveit?
6.23_AssumeforthemomentthattheEarthandMoonareheldstationary. Asmallmassisplacedata
pointbetweentheEarthandMoonwherethetotalgravitationalforceonitiszero. Findthispointand
then(a)ThemassisdisplacedasmallamountalongthelinejoiningtheEarthandMoonandreleased
fromrest. Findtheforceon n itforsmalldistancesfromthatequilibriumpositionandthen solvethe
equationsofmotionforitsposition. (b)Like(a)butitisdisplacedalongalineperpendiculartothe
linejoiningtheEarthandMoon.
6.24 InthegureatEq.(6.17),iftheorbitrepresentsanasteroidthatcomesinjustcloseenoughto
thesuntograzeEarth’sorbit,to1AU,thenhowfaroutwillitgo?Whatplanethasanorbitofabout
thatradius? Ans:4
:
88AU
6.25 Twoparticlesofmass
m
1
and
m
2
arerepelled fromeach otherbyaforce ofmagnitude
k=r
2
.
Theystartatrest;ndthespeedofeachasafunctionofthedistancetravelled.
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6|Orbits
221
6.26_Imagineaspherical,non-rotatingplanetofmass
M
andradius
R
. Asatellite e is sentupfrom
thesurfaceoftheplanetwithaspeed
v
0
atangle30
fromthelocalvertical. Initssubsequentpathit
reachesamaximumdistance5
R=
2fromthecenteroftheplanet. (a)Find
v
0
.Ignoretheatmosphere.
(b)Nowassumethatthe planetis rotatingwithangularspeed
!
andthe satelliteisred fromthe
equator. Whatis
v
0
now? Twocases: redeast,redwest. WhatisthiseectfortheEarth,andin
particularwhatistheratiooftheenergiesrequiredtolaunchthesatellite? Alsocomparethesmaller
ofthesetotheenergyrequiredtolaunchfromtheNorthPole. Ans:a:
q
6
5
gR=
cos30
6.27 AnasteroidisheadedinthegeneraldirectionofEarth,anditsspeedwhenfarawayis
v
0
relative
totheEarth. ItisaimedsothatintheabsenceofgravityitwouldpasstheEarthatadistance
b
from
itscenter. (a)Inthepresenceofgravityhowfarfromthecenterwillitpass? Youcandothissimply
usingconservationlaws,orthehardwayusingalltheapparatusofthischapter. (b) Ifitjustbarely
scrapes the planet,whatwould
b
havetobe? Expressthislastresultinterms s of
v
0
andtheescape
speedfromtheplanet. Ans:(b)
b>r
1+
v
2
esc
=v
2
0
1
=
2
andofcourse,isthisanswerplausible?
6.28_Forthe sphericalpendulumwithan almostcircularorbit,whatis theshapeofthe orbitifyou
havesetitmovingsofastthatthecordholdingthemassisalmost,butnotquite,horizontal?
6.29_In anotedexperimentbyDicke, he concludedthat the sun is notexactly sphericalbuthas a
slightbulgetowardtheequator.Thiswouldcauseanalterationinthegravitationalpotentialenergyto
U
(
r
)= 
GMm
r
GMmR
2
r
3
where
1. (Thisformforthecorrectionappliesonlyforthecasethattheplanetisintheplaneof
thesun’sequator.) Foraplanetinanalmostcircularorbit,thisaddedtermwillcauseasmallprecession
oftheplanet’sorbit.Findthisprecessionbydoingaperturbationanalysisaboutacircularorbit.
6.30_Ifthegravitationaleldhasasmallinversecubepartinadditiontotheusualone,youcansolve
fortheorbitwithoutapproximation.
F
r
(
r
)= 
GMm
r
2
GMmR
r
3
Gobacktotheorbitaldierentialequationforageneralforceandapplyittothis.Justgettherelation
for
r
(
)insteadofthecoordinatesasfunctionsoftime. Plottheorbitforbothpositiveandnegative
small
.
6.31 \As the Earth and Moon move around the Sun together, , the Moon’s s orbit is always concave
towardtheSun."Isthistrue? Ifso,doesitmeanthattheMoonisorbitingtheSunandnottheEarth?
6.32 DeriveEq.(6.25). Also,whatistheradiusfortheorbitwithenergy
E
1
inthesamepicture?
6.33 Whatistheminimumenergy
E
1
fromEq.(6.24)andinterpretthisresultintermsofEq.(6.25).
6.34 Findtheforcelawforacentralforcethatallowsamasstomoveinaspiralorbit
r
=
k
2
where
k
isaconstant.
6.35 Whenyoutossan object straightupatthe surfaceof theEarthyouusuallyassumethat
g
is
constant,andofcourseitisn’t. Foracoordinate
y
measuredfromtheEarth’ssurface,
g
=
GM
r
2
=
GM
(
R
+
y
)2
=
GM
R
2(1+
y=R
)2
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6|Orbits
222
Expand thistorstorder r in
y=R
and use theresultingforce to nd the motion of amass thrown
straightup atinitial speed
v
0
,neglectingair resistance e and therotation ofthe Earth. . Whatis s the
maximumheightinthiscase? Comparetheresulttothesimplecase e thatassumes
g
remains atthe
samevalueithadatthesurface. Howbigistheeectifthesimplerversiongavearesultof10km?
100km?
r
2
r
1
6.36_Space debrisisaproblem in near-Earth orbit. . Ifyou u are in ashipinsuch
anorbit, thefactthatyou’removingatseveral kilometers s persecond is notthe
problem,becausemostdebriswillbemovingataboutthesamespeedinaboutthe
samedirection. Thekey y wordis\about". . Assumethatyourorbitiscircularwith
radius
r
1
andthatsomeoneoncelostaball-peenhammerthatwoundupinaorbit
withperigee
r
1
andapogee
r
2
=
r
1
+
. Ifithits s yourship,howfastwill itbe
movingrelativetoyou? Dothisforsmall
,andexpresstheresultintermsof
r
1
,
g
,and
. Here
g
is theEarth’sgravitationaleld atthis height. . Forloworbit,
if
=100km,whatisthisspeed? Thisspeedcorrespondstoafallfromwhatheightatthe e Earth’s
surface? Ans:
p
g=r
1
=
4,almost50meters
6.37 Amass
m
isorbitinginaeldwithacentralforce
F
r
Ar
4
. Ithasangularmomentum
L
.
A>
0,
L>
0. (a)Whatisthepotentialenergyforthisforce? Takeittobezeroat
r
=0. (b)Find
theenergyforwhichthisorbitwillbecircular,andwhatisitsradius? Ans:(b)
r
=
L
2
=mA
1
=
7
6.38_Amassismovinginacentralpotentialinacircularorbit. Thiscircularorbitpassesthroughthe
centeroftheforce. Thatis,thecenteroftheeldisonthecircumferenceofthecircle. Thistypeof
motionispossibleforexactlyoneformofcentralforce. Findthatform. (Itissomethinglike
r
n
for
some
n
.) Firstshowthattheorbitcanbedescribedbytheequation
r
=
R
max
cos
6.39 TheHohmanntransferorbitbetweenplanets(assumecircularplanetaryorbits)putsthespacecraft
inanellipticalorbitthatistangenttotheorbitofthestartingplanetandtangenttotheorbitofthe
destinationplanet. Howmuchtimewouldthisorbittakegettingfrom m EarthtoMars? ? This s neglects
theaccelerationtimeatthestartandtheaccelerationtimeattheend. Whatdirectionsarethesetwo
accelerations? Ans:8.5months
6.40Asmallmoonofmass
m
andradius
a
orbitsaplanetofmass
M
whilekeepingthesameface
toward the planet. . Showthatifthis s moonis tooclosetotheplanet, , thenlooserocks s lyingonthe
surfaceofthemoonwillbeliftedo.Whatisthisdistance,andwhatisitfortheEarth-moonsystem?
Themoon’smassis1/81oftheEarth’s;itsradiusis1740km. Ans:
r
(center-to-center)11000km
6.41_A planet t is s orbiting a star, but all that t you u know w about the e force of f attraction n is that t its
potential energyis
U
(
r
) and thatthe planetdoes havean orbit. . Whatisthe e angle throughwhich
the planetwill orbit as it goes from minimum
r
to maximum
r
? The e result will be an integral in
which
U
(
r
)appears. Testyourresultforthespecialcasethat
U
(
r
)= 
GMm=r
toseeifitgivesthe
correctanswerthere. Useconservationofenergyandangularmomentum. Eliminate
dt
betweenthem.
Ans:
R
r
2
r
1
‘=r
2
dr
q
2
E
U
(
r
m‘
2
=
2
r
2
=m
6.42 Insection 6.9there areplotsofmany orbits. . Figure e outwhichonesgowith which equations.
Thereisonepairthatistooclosetocall,butinthatcaseyoucanstilldeterminewhichpairofequations
are the candidates. . Theparameters
and
canmakeabigchangeintheshapes,andinturnthe
angularmomentumpermass,
,controlsthese|how?
6.43_Nowtakethephysicalsystemoftheprecedingproblemandndtheshapeoftheorbit,assuming
thatitisonlyalmosthorizontal. Usecylindricalcoordinates,
r
-
-
z
,andwritedownthetotalenergy
6|Orbits
223
of themass. . Writethe
z
-component of angularmomentum, which is conserved. . Because
r
and
z
are not t independent variables in this s situation, , eliminate
z
. Eliminate
and solve e for r the e almost
circularradialmotionasafunctionoftime. Combinethiswith
_
todeterminetheshapeoftheorbit.
Ans:
!
radial
=
_
0
=
p
3sin
6.44 Returntotheanalysisofsection6.9,butnowmaketheforcerepulsiveinsteadofattractiveand
analyzethesolutions.Theseorbitsarenotnearlyaswild.
6.45 Amassisslidingaroundtheinsideofaconethathasvertexangle2
.Assume
nofrictionandthatthemassisslidinginahorizontalcircle.Findtherelationbetween
itsangularspeedandtheheightofthecircleabovethevertexofthecone. Ans:
_
0
=
p
g=z
0
cot
6.46FortheellipticalorbitsofEq.(6.3),whatisthesemi-majoraxisforarbitrary
?
6.47 Aplanetisorbitingastarinanalmostcircularorbitofmeanradius
r
0
. Ifthere
is,inadditiontotheusual
GMm=r
2
gravitationalforce,asmallharmonicoscillatorforce
~
F
1
k~r
,
ndtheeectontheorbitintheapproximationthattheorbitdierslittlefromacircle.
6.48 Theproblem6.15canbedonewithnointegration,usingthreeorfourlinesofalgebraandKepler’s
laws. Doso.
6.49 (a)Fortwomasses
m
1
and
m
2
,whatisthetotalmomentumintermsofcoordinates
~r
and
~r
cm
(andtheirderivatives)asinEq.(6.47).
(b)WhatistheirtotalkineticenergyintermsofCMvariables.
(c)WhatistheirtotalangularmomentumintermsofCMvariables.
Inallthree cases you shouldhaveno crosstermsbetweentherelativeandtheCM coordinates. . Do
theanswersdependonwhichoneismass#1andwhichis#2? Ans:
m
tot
~v
cm
,
1
2
m
tot
v
2
cm
+
1
2
v
2
rel
,
m
tot
~r
cm
~v
cm
+
~r
rel
~v
rel
6.50_Setupthetwo-dimensionalisotropicharmonic oscillatorinpolarcoordinatesas insections 6.2
and6.3. (a)Writethedierentialequationfor
r
(
t
).(b)Whatistheeectivepotentialenergyforthis
system?(c)Youhavethesolutiontothisprobleminrectangularcoordinatesfromsection6.1,sofrom
thatyoucanconstructthepolarsolutionfor
r
(
t
). (d)Whyisthesolutionexpressedinproblem3.67
correctanddotheresultsthereagreewithwhatyougetfromthecalculationhere?
6.51Twomassesaretiedtothetwoendsof alightstring,andoneof
them is suspended through ahole ina a table. . Assume e thatthere is no
friction anywhere and that the mass
m
1
is moving on acircle,held at
aconstantradiusbythetensioninthestring. (a)Find
v
,
r
,
!
forthis
motion. (b) Forthegeneralcaseinwhich
m
1
hasarbitrarymotionin
r
and
,writethetotalenergy andtheangularmomentumandusethem
tondtheorbitof
m
1
ifitisalmostcircular.
Ans:
_
2
0
=!
2
0
=(
m
1
+
m
2
)
=
3
m
1
.
6.52_Asatelliteisinacircularorbitofradius
r
0
abovetheEarth. (a)Whatisitstotal(kineticplus
potential) energy? ? Ifthe e orbitis lowenough(say 200-300km orso) therewill be some very small
frictionwiththeatmosphere,applyingforceofmagnitude
F
fr
tothesatellite. (b)Inoneorbit,what
isthechangeinenergyofthesatellite? (c) Inthatorbit,whatwillbethechangein
r
? Whatthen
isthechangein
r
pertimeforthesatellite,
dr=dt
,thedecayrate? (d)Ifyoumaketheimplausible
assumptionthatthefrictionalforceisaconstantalltheway downtotheEarth’ssurface,howmuch
timewillitbebeforethesatellitehitstheground? (e)Howdoesthespeedofthesatellitevaryover
timebecauseofthisfrictionalforce?
6|Orbits
224
6.53 Insection6.8theequationsin
=
R
star
=R
orbit
appears.Fillinthemissingstepsinitsderivation.
6.54 Thehyperbolicorbitsinsection6.10ledtoapictureinwhichonlyonehalfthehyperbolawas
theorbit. Theotherhalf(thedashedlines)didn’tseemtocorrespondtoanything. Showthatthey
representthesolutioniftheinversesquareforceis repulsive insteadofattractive. . Foranexampleof
this,considertwopositivelychargedparticles,e.g.aprotonandanatomicnucleus.
6.55Forthose who knowthe mathematicaltechnique, start from
dt
=
r
2
d=‘
and (6.16), using
contourintegrationtoderiveKepler’slawEq.(6.20).
6.56_Fortheorbitalequation(6.14),whatistheequationiftheforcehastwoterms,theusualNew-
toniangravityplusa1
=r
4
term.
f
(
r
)= 
GMm
r
2
3
GMm‘
2
c
2
r
4
Here,
c
isthe speed of light. . Doesthiseven n havecorrectdimensions? ? This s is theorbitalequation
derivedfromEinstein’s theory ofgravity,the GeneralTheory ofrelativity. . Usethe e methodsstarting
withequation(6.33)tondtheanglebywhichtheellipticalorbitdescribedbythersttermalonewill
precessineachorbit. LookuptheorbitalparametersfortheplanetMercurytogetthenumericalvalue
forthis resultand then convertitintotheconventionally reportedvalueof theprecession perEarth
century(43
00
percentury). Cf.section6.12,whichaddressesthe(larger)eectofthegravitationalpull
bytheotherplanets.
6.57AddauniformmagneticeldtothespringforceofEq.(6.1),assumingthatallthespringconstants
are equal,andthatthemasshasacharge
q
. Choosethe
z
-axis alongthedirectionofthemagnetic
eld. Solveforthemotionofthemassandsketchsomesolutions,especiallyforweak
B
-elds.
6.58 Thecarbonmonoxidemolecule,CO,canbemodeledastwomassesontheendsofaspringhaving
unstretchedlength
. Solvefortheoscillationsofthismolecule,thenormalmodes,assumingthatthe
motionisalongthesinglelongaxisbetweentheatoms.Compareyourresulttowhatappearsinsection
6.7,especiallyEq.(6.46).
6.59 Estimatetheperiodforthecloseorbitofapebblearoundarock. More
thanoneasteroidhasbeenobservedwithsuchasatellite.Assumethedensity
oftherockiscomparabletotheaveragedensityofEarth. LookupIdaand
Dactyl,picturedontheright.
6.60 Insection6.9youseeavarietyofcomplicatedorbits. Nowexaminethe
casethattheinversecubeforceisrepulsiveandsolveforthoseorbits.
6.61 Verifythattheequations(6.57)describethehyperbolaasstatedthere.
6.62 Takeallthecomplicatedresultsinsection6.11andapplythemtothecaseofacircularorbitto
seeiftheygivethecorrectresults. (ThisishowIfoundsomemissingfactors.)
6.63_Forthespecialcasethatanasteroidstartsfromaninitialdistance
R
fromthesunatzerovelocity,
whatdotheequationsfor
r
(
s
),
t
(
s
),
(
s
)insection6.11become?Andanalyzetheresults.Checkout
thevariousversionsofEq.(6.64)tocomparetheirutility.Howmuchtimedoesittaketohitthesun?
6.64_FortheKeplerorbit,usetheequationfor
r
intermsof
,therstofEqs.(6.17)andcompute
thevelocityofthe planet,
~v
_
r
^
r
+
r
_
^
. Use
toexpressthis interms of
alone. Showthatthe
vectorcoecientof
istheconstantunitvector^
y
=^
r
sin
+
^
cos
. Finallyusethistoshowthat
thegraphof
~v
(
t
)isacirclecenteredat
‘
^
y=a
(1 
2
).Thisiscalledahodograph.
6.65 EasierthantheprecedingproblemfortheKeplerorbit,whatisthelocusof
~v
(
t
)fortheanisotropic
two-dimensionalharmonicoscillator?
6|Orbits
225
6.66 Insection6.3yousawthechangeofvariablesfrom
r
(
t
)and
(
t
)to
u
=1
=r
withtheindependent
variable
. Try y anotherapproach: : Againuse
astheindependentvariable,butuse
v
,theangular
componentofthevelocityasthedependentvariable. Whatdierentialequationdoyougetforthis?
Solveit.Ifyoucan’t,thengobackandndyourmistake.
6.67 Showthattheratioofthesolartidaleectatperihelionis(1+6
)1
:
1timesthesolartideat
aphelion.
6.68Avariationontheproblem6.41isactuallyuseful. Againforanarbitrary
U
(
r
),anobjectstarts
fromveryfarawayandisheadedtowardthegeneraldirectionofthecenteroftheforce,thoughaimed
soastomiss. Farawayithasaspeed
v
0
anditslineofmotionisaimedtopassatadistance
b
fromthe
origin,theimpactparameter.Followtheproceduredescribedinproblem6.41toeliminate
dt
between
thetwoconservationlaws,andndtheangle(the\scatteringangle")atwhichitwillleavethesystem
afterhavingcometoitsdistanceofclosestapproach. Sincethersthalfoftheorbitisamirrorimage
ofthesecondhalf,ndthedirectionchangebetweenthedistanceofclosestapproachand whenit’s
innitelyfaraway. Thendoubletheresult. . Theequationfor
r
min
comesfromconservationofenergy
andofangularmomentum. Check: : whatif
U
0? Whatif
v
0
!1? Inthischeck,youmay y need
theequation
R
1
0
d
sech
=
=
2,dependingonwhatchangeofvariablesyoumakefortheintegral.
Ans: 
+2
R
1
r
min
‘=r
2
dr
q
2
E
U
(
r
m‘
2
=
2
r
2
=m
, where
=
v
0
b
6.69Usetheresultoftheprecedingproblem tondthescatteringangleforan
-particle aimedat
agoldnucleus. Ignorethemotion n ofthenucleus,asitisquitemassive. . Thisresultisimportantin
understandingRutherford’sexperimentaldiscoveryoftheexistenceofthenucleus. Herethepotential
energyis
kq
1
q
2
=r
,where
q
1
=2
e
,
q
2
=
Ze
,and
Z
=79forgold.
6.70 Derive
V
1
(
r
)inEq.(6.66)fromEq.(6.65).Factor
R
fromthedenominatorandusethebinomial
expansionfor(1+
x
)
1
=
2
outtosecondorderin
x
.
6.71 Anellipsecanbewrittenastwoparametricequations:
x
(
)=
a
cos
and
y
(
)=
b
sin
Usethesetondthecircumferenceofanellipse:
ds
=
p
dx
2+
dy
2. Expressthe
resultintermsofan ellipticintegralasinEq.(6.71). . Is s the resultcorrectinthe
twocasesforwhichyoucanwritedowntheanswereasily? Ifyoustudymoreorbit
theoryyouwillencountertheterm\eccentricanomaly". That’swhatthis
is.
6.72 (Irrelevanttothismaterial,butfun)Anellipseisaconicsection. Thatmeansthat
theintersectionofaplaneandaconeisan ellipse(orahyperbolaoraparabolaifthe
anglesaredierent). Youcaninscribetwo o spheres insidetheconeand tangenttothe
intersectingplaneattwopoints.Whatarethesetwopoints?(Whatelsecouldtheybe?)
r
R
6.73 DeriveEq.(6.65).
dM
=
Md=
2
.
6.74 Showthat
E
(
ik
)=
p
1+
k
2
E
k=
p
1+
k
2
.
Also,
K
(
ik
)=
1
p
1+
k2
K
k=
p
1+
k
2
.
Thisneedsonlythesimplesttrigonometricidentities.
6.75Theexpressionforpotentialenergyofaring,Eq.(6.65),isexactlythesameif
r>R
,andyou
can expressitin termsofanellipticintegralbyfactoringthe largerradius
r
outofthedenominator
6|Orbits
226
insteadof
R
. (a)Showthatyouget
U
1
(
r
)= (2
GMm=r
)
K
(
R=r
)thistime. Also,checkthatit
behavesproperlyforlarge
r
andfor
r
nearto
R
. (b)ThelargestperturbationontheorbitofSaturn
comesfromJupiter;justlookatthenumbersfollowingEq.(6.70)toseewhy.Computetheprecession
oftheperihelion ofSaturnbecauseofthe presenceofJupiter. . (c)AddtheeectsfromUranus s and
Neptune,as they usethesameformsoftheequations thatappearinthetext,applyingtoMercury
(
r<R
).
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