(d) False. The orbital period of a planet is independent of the planet s mass.

Transcription

1 hpte Gvity onceptul Pobles ue o flse: () (b) (c) (d) o Keple s lw of equl es to be vlid, the foce of vity ust vy invesely with the sque of the distnce between iven nd the. he closest to the hs the shotest obitl peiod. Venus s obitl speed is le thn the obitl speed of th. he obitl peiod of llows ccute deteintion of tht s ss. () lse. Keple s lw of equl es is consequence of the fct tht the vittionl foce cts lon the line joinin two bodies but is independent of the nne in which the foce vies with distnce. (b) ue. he peiods of the s vy with the thee-hlves powe of thei distnces fo the. So the shote the distnce fo the, the shote the peiod of the s otion. (c) ue. Settin up popotion involvin the obitl speeds of the two s in tes of thei obitl peiods nd en distnces fo the (see ble -) shows tht v. v. Venus 7 th (d) lse. he obitl peiod of is independent of the s ss. Duin wht seson in the nothen heisphee does th ttin its iu obitl speed bout the? Wht seson is elted to its iniu obitl speed? Hint: th is t peihelion in ely Jnuy. Deteine the oncept th is closest to the duin winte in the nothen heisphee. his is the tie of fstest obitl speed. Sue would be the tie fo iniu obitl speed. 7 At the sufce of the oon, the cceletion due to the vity of the oon is. At distnce fo the cente of the oon equl to fou ties the dius of the oon, the cceletion due to the vity of the oon is () 6, (b) /, (c) /, (d) /6, (e) None of the bove. Pictue the Poble he cceletion due to vity vies invesely with the sque of the distnce fo the cente of the oon.

2 hpte pess the dependence of the cceletion due to the vity of the oon on the distnce fo its cente: pess the dependence of the cceletion due to the vity of the oon t its sufce on its dius: ' Divide the fist of these epessions ' by the second to obtin: Solvin fo nd siplifyin yields: ' 6 ( ) nd (d) is coect. Suppose the escpe speed fo ws only slihtly le thn the escpe speed fo th, yet the is considebly le thn th. How would the s (vee) density cope to th s (vee) density? () It ust be dense. (b) It ust be less dense. (c) It ust be the se density. (d) You cnnot deteine the nswe bsed on the dt iven. Pictue the Poble he densities of the s e elted to the escpe speeds fo thei sufces thouh v e G. he escpe speed fo the is iven by: v G he escpe speed fo th is iven by: v th G th th pessin the tio of the escpe speed fo the to the escpe speed fo th nd siplifyin yields: v v th G G th th th th Becuse v v th : th th

3 Gvity Squin both sides of the eqution yields: th th pess nd th in tes of thei densities nd siplify to obtin: th th V V th th th V V th th th π π th th th Solvin fo the tio of the densities yields: th th Becuse the is considebly le thn th: th nd ( b) << is coect. Ne the end of thei useful lives, sevel le th-obitin stellites hve been neuveed so they bun up s they ente th s tosphee. hese neuves hve to be done cefully so le fents do not ipct populted lnd es. You e in che of such poject. Assuin the stellite of inteest hs on-bod populsion, in wht diection would you fie the ockets fo shot bun tie to stt this downwd spil? Wht would hppen to the kinetic eney, vittionl potentil eney nd totl echnicl eney followin the bun s the stellite ce close nd close to th? Deteine the oncept You should fie the ocket in diection to oppose the obitl otion of the stellite. As the stellite ets close to th fte the bun, the potentil eney will decese. Howeve, the totl echnicl eney will decese due to the fictionl d foces tnsfoin echnicl eney into thel eney. he kinetic eney will incese until the stellite entes the tosphee whee the d foces slow its otion. stition nd Appoition 7 stite the ss of ou ly (the ilky Wy) if the obits the cente of the ly with peiod of 5 illion yes t en distnce of, c y. pess the ss in tes of ultiples of the sol ss S. (Nelect the ss fthe fo the cente thn the, nd ssue tht the ss close to the cente thn the eets the se foce on the s would point pticle of the se ss locted t the cente of the ly.) Pictue the Poble o ppoite the ss of the ly we ll ssue the lctic cente to be point ss with the in obit bout it nd pply Keple s thid lw. et G epesent the ss of the ly.

4 hpte sin Keple s thid lw, elte the peiod of the to its en distnce fo the cente of the ly: π π G G G S G S Substitute nueicl vlues nd evlute G / s : G s π. N k. ly c y c y y.56 (.99 k) s s y Keple s ws 5 [SS] One of the so-clled Kikwood ps in the steoid belt occus t n obitl dius t which the peiod of the obit is hlf tht of Jupite s. he eson thee is p fo obits of this dius is becuse of the peiodic pullin (by Jupite) tht n steoid epeiences t the se plce in its obit evey othe obit ound the sun. epeted tus fo Jupite of this kind would eventully chne the obit of such n steoid. heefoe, ll steoids tht would othewise hve obited t this dius hve pesubly been cleed wy fo the e due to this esonnce phenoenon. How f fo the is this pticul : esonnce Kikwood p? Pictue the Poble he peiod of n obit is elted to its sei-jo is (fo cicul obits this distnce is the obitl dius). Becuse we know the obitl peiods of Jupite nd hypotheticl steoid in the Kikwood p, we cn use Keple s thid lw to set up popotion eltin the obitl peiods nd vee distnces of Jupite nd the steoid fo the fo which we cn obtin n epession fo the obitl dius of n steoid in the Kikwood p. se Keple s thid lw to elte Jupite s obitl peiod to its en distnce fo the : Jupite π G Jupite

5 Gvity 5 se Keple s thid lw to elte the obitl peiod of n steoid in the Kikwood p to its en distnce fo the : Kikwood π G Kikwood Dividin the second of these π Kikwood equtions by the fist nd siplifyin Kikwood G yields: Jupite π Jupite G Solvin fo Kikwood yields: Kikwood Kikwood Jupite Jupite Kikwood Jupite Becuse the peiod of the obit of n steoid in the Kikwood p is hlf tht of Jupite s: Kikwood ( 77.8 ) A Jupite Jupite A.5 eks: hee e lso sinificnt Kikwood ps t :, 5:, nd 7: nd esonnces t.5 A,.8 A, nd.95 A. 9 Keple deteined distnces in the Sol Syste fo his dt. o eple, he found the eltive distnce fo the to Venus (s coped to the distnce fo the to th) s follows. Becuse Venus s obit is close to the thn is th s obit, Venus is onin o evenin st its position in the sky is neve vey f fo the (iue -). If we suppose the obit of Venus is pefect cicle, then conside the eltive oienttion of Venus, th, nd the t iu etension, tht is when Venus is fthest fo the in the sky. () nde this condition, show tht nle b in iue - is 9º. (b) If the iu elontion nle between Venus nd the is 7º, wht is the distnce between Venus nd the in A? (c) se this esult to estite the lenth of Venusin ye. Pictue the Poble We cn use popety of lines tnent to cicle nd dii dwn to the point of contct to show tht b 9. Once we ve estblished tht b is iht nle we cn use the definition of the sine function to elte the distnce fo the to Venus to the distnce fo the to th.

6 6 hpte () he line fo th to Venus' obit is tnent to the obit of Venus t the point of iu etension. Venus will ppe close to the sun in th s sky when it psses the line dwn fo th nd tnent to its obit. Hence b 9 Deleted: eth (b) sin tionoety, elte the distnce fo the sun to Venus d SV to the nle : Substitute nueicl vlues nd evlute d SV : dsv sin dsv d d d SV S (. A).7A S sin sin 7.7A (c) se Keple s thid lw to elte Venus s obitl peiod to its en distnce fo the : se Keple s thid lw to elte th s obitl peiod to its en distnce fo the : Venus th π G π G Venus th Dividin the fist of these equtions by the second nd siplifyin yields: Venus th π G π G Venus th Venus th Solvin fo Venus yields: Venus Venus th th sin the esult fo Pt (b) yields: Venus (. y).6 y.7 A. A eks: he coect distnce fo the sun to Venus is close to.7 A. Newton s w of Gvity Jupite s stellite uop obits Jupite with peiod of.55 d t n vee obitl dius of () Assuin tht the obit is cicul, deteine the ss of Jupite fo the dt iven. (b) Anothe stellite of Jupite, llisto, obits t n vee dius of with n obitl peiod of

7 Gvity d. Show tht these dt e consistent with n invese sque foce lw fo vity (Note: DO NO use the vlue of G nywhee in Pt (b)). Pictue the Poble While we could pply Newton s lw of vittion nd second lw of otion to solve this poble fo fist pinciples, we ll use Keple s thid lw (deived fo these lws) to find the ss of Jupite in Pt (). In Pt (b) we cn cope the tio of the centipetl cceletions of uop nd llisto to show tht these dt e consistent with n invese sque lw fo vity. () Assuin cicul obit, pply Keple s thid lw to the otion of uop to obtin: π G J J π G Substitute nueicl vlues nd evlute J : 8 π ( 6.7 ) ( 6.67 N /k ).55d 7 J.9 h 6s d h Note tht this esult is in ecellent eeent with the ccepted vlue of.9 7 k. (b) pess the centipetl π cceletion of both of the oons to v π obtin: centipetl whee nd e the dii nd peiods of thei otion. k sin this esult, epess the centipetl cceletions of uop nd llisto: π π nd Divide the fist of these equtions by the second nd siplify to obtin: π π Substitute fo the peiods of llisto nd uop usin Keple s thid lw to obtin:

8 8 hpte his esult, toethe with the fct tht the vittionl foce is diectly popotionl to the cceletion of the oons, deonsttes tht the vittionl foce vies invesely with the sque of the distnce. he ss of Stun is k. () ind the peiod of its oon is, whose en obitl dius is (b) ind the en obitl dius of its oon itn, whose peiod is.8 6 s. Pictue the Poble While we could pply Newton s lw of vittion nd second lw of otion to solve this poble fo fist pinciples, we ll use Keple s thid lw (deived fo these lws) to find the peiod of is nd to elte the peiods of the oons of Stun to thei en distnces fo its cente. () sin Keple s thid lw, elte the peiod of is to its en distnce fo the cente of Stun: π G S π G S Substitute nueicl vlues nd evlute : 8 π (.86 ) 6 ( N /k )( 5.69 k) 8.8 s.7 h (b) sin Keple s thid lw, elte the peiod of itn to its en distnce fo the cente of Stun: π G G S S π Substitute nueicl vlues nd evlute : 6 6 (.8 s) ( N /k )( 5.69 k) 9. π A supeconductin vity ete cn esue chnes in vity of the ode Δ/.. () You e hidin behind tee holdin the ete, nd you 8-k fiend ppoches the tee fo the othe side. How close to you cn you fiend et befoe the ete detects chne in due to his pesence? (b) You e in hot i blloon nd e usin the ete to deteine the te of scent (ssue the blloon hs constnt cceletion). Wht is the sllest chne in ltitude tht esults in detectble chne in the vittionl field of th?

9 Gvity 9 Pictue the Poble We cn deteine the iu ne t which n object with iven ss cn be detected by substitutin the eqution fo the vittionl field in the epession fo the esolution of the ete nd solvin fo the distnce. Diffeentitin () with espect to, septin vibles to obtin d/, nd ppoitin Δ with d will llow us to deteine the veticl chne in the position of the vity ete in th s vittionl field is detectble. () th s vittionl field is G iven by: pess the vittionl field due to the ss (ssued to be point ss) of you fiend nd elte it to the esolution of the ete: () G. G. Solvin fo yields:. Substitute nueicl vlues nd 6. evlute : ( ) ( 8k) k (b) Diffeentite () nd siplify to obtin: d G G d Septe vibles to obtin: d d Appoitin d with Δ, evlute Δ with : Δ 6 (. )( 6.7 ).9μ th s dius is 67 k nd the oon s dius is 78 k. he cceletion of vity t the sufce of the oon is.6 /s. Wht is the tio of the vee density of the oon to tht of th? Pictue the Poble We cn use the definitions of the vittionl fields t the sufces of th nd the oon to epess the cceletions due to vity t these loctions in tes of the vee densities of th nd the oon. pessin the tio of these cceletions will led us to the tio of the densities.

10 hpte pess the cceletion due to vity t the sufce of th in tes of th s vee density: G G π G V G π he cceletion due to vity t the sufce of the oon in tes of the oon s vee density is: G π Divide the second of these equtions by the fist to obtin: Substitute nueicl vlues nd evlute 6 (.6 /s )( 6.7 ) 6 : ( 9.8/s )( Gvittionl Potentil ney 7 Knowin tht the cceletion of vity on the oon is.66 ties tht on th nd tht the oon s dius is.7, find the escpe speed fo pojectile levin the sufce of the oon. Pictue the Poble he escpe speed fo the oon is iven by v e, G, whee nd epesent the ss nd dius of the oon, espectively. pess the escpe speed fo the oon: Becuse. 66 nd. 7 : G v e. v (.66 )(. ) e. 7 Substitute nueicl vlues nd evlute v e, : 6 (.66)( 9.8 /s )(.7)( 6.7 ).8 k/s v e. 5 An object is dopped fo est fo heiht of. 6 bove the sufce of th. If thee is no i esistnce, wht is its speed when it stikes th?

11 Gvity Pictue the Poble et the zeo of vittionl potentil eney be t infinity nd let epesent the ss of the object. We ll use consevtion of eney to elte the initil potentil eney of the object-th syste to the finl potentil nd kinetic eneies. se consevtion of eney to elte the initil potentil eney of the syste to its eney s the object is bout to stike th: pess the potentil eney of the object-th syste when the object is t distnce fo the sufce of th: Substitute in eqution () to obtin: Solvin fo v yields: K f Ki f i o, becuse K i, K ( ) ( ) ( h) () whee h is the initil heiht bove th s sufce. () G G G v h v G G h h h Substitute nueicl vlues nd evlute v: v 6 6 ( )( 6.7 )(. ) 6.9k/s 9.8/s Gvittionl Obits 59 ny stellites obit th with iu ltitudes of k o less. Geosynchonous stellites, howeve, obit t n ltitude of 5 79 k bove th s sufce. How uch oe eney is equied to lunch 5-k stellite into eosynchonous obit thn into n obit k bove the sufce of th? Pictue the Poble We cn epess the eney diffeence between these two obits in tes of the totl eney of stellite t ech elevtion. he ppliction of Newton s second lw to the foce ctin on stellite will llow us to epess

12 hpte the totl eney of ech stellite s function of its ss, the dius of th, nd its obitl dius. pess the eney diffeence: Δ eo () pess the totl eney of n obitin stellite: Apply Newton s second lw to stellite to elte the vittionl foce to the obitl speed: tot K G v whee is the obitl dius. G dil v () Solvin fo v yields: v Substitute in eqution () to obtin: tot Substitutin in eqution () nd siplifyin yields: Δ eo eo k 5 79 k Substitute nueicl vlues nd evlute Δ: Δ 6 ( 5k)( 9.8N / k)( 6.7 ).GJ he Gvittionl ield ( ) A point pticle of ss is on the is t nd n identicl point pticle is on the y is t y. () Wht is the diection of the vittionl field t the oiin? (b) Wht is the nitude of this field?

13 Gvity Pictue the Poble We cn use the definition of the vittionl field due to point ss to find the nd y coponents of the field t the oiin nd then dd these coponents to find the esultnt field. We cn find the nitude of the field fo its coponents usin the Pythoen theoe. () he vittionl field t the oiin is the su of its nd y coponents: pess the vittionl field due to the point ss t : pess the vittionl field due to the point ss t y : Substitute in eqution () to obtin: y () y G iˆ G ˆj G iˆ G ˆ y j the diection of the vittionl field is lon line t 5 bove the is. (b) he nitude of is iven by: y Substitute fo nd y nd siplify G G G to obtin: 67 A nonunifo thin od of lenth lies on the is. One end of the od is t the oiin, nd the othe end is t. he od s ss pe unit lenth λ vies s λ, whee is constnt. (hus, n eleent of the od hs ss d λ d.) () Wht is the totl ss of the od? (b) ind the vittionl field due to the od on the is t, whee >. Pictue the Poble We cn find the ss of the od by intetin d ove its lenth. he vittionl field t > cn be found by intetin d t ove the lenth of the od. () he totl ss of the stick is iven by: λ d Substitute fo λ nd evlute the intel to obtin: d

14 hpte (b) pess the vittionl field due to n eleent of the stick of ss d: d Gd ( ) ( ) Gd ( ) i iˆ ˆ Gλ d iˆ Intete this epession ove the G lenth of the stick to obtin: ( ) d iˆ G ln he Gvittionl ield ( ) due to Spheicl Objects iˆ 7 wo widely septed solid sphees, S nd S, ech hve dius nd ss. Sphee S is unifo, whees the density of sphee S is iven by () /, whee is the distnce fo its cente. If the vittionl field stenth t the sufce of S is, wht is the vittionl field stenth t the sufce of S? Pictue the Poble he vittionl field stenth t the sufce of sphee is iven by G, whee is the dius of the sphee nd is its ss. pess the vittionl field stenth on the sufce of S : pess the vittionl field stenth on the sufce of S : G G Divide the second of these equtions G by the fist nd siplify to obtin: G 75 Suppose you e stndin on spin scle in n elevto tht is descendin t constnt speed in ine shft locted on the equto. odel th s hooeneous sphee. () Show tht the foce on you due to th s vity lone is popotionl to you distnce fo the cente of the. (b) Assue tht the ine shft locted on the equto nd is veticl. Do not nelect th s ottionl otion. Show tht the edin on the spin scle is popotionl to you distnce fo the cente of the.

15 Gvity 5 Pictue the Poble hee e two foces ctin on you s you descend in the elevto nd e t distnce fo the cente of th; n upwd nol foce ( N ) eeted by the scle, nd downwd vittionl foce () eeted by th. Becuse you e in equilibiu (you e descendin t constnt speed) unde the influence of these foces, the nol foce eeted by the scle is equl in nitude to the vittionl foce ctin on you. We cn use Newton s lw of vity to epess this vittionl foce. () pess the foce of vity ctin on you when you e distnce fo the cente of th: sin the definition of density, epess the density of th between you nd the cente of th nd the density of th s whole: G ( ) () whee () is the ss of th enclosed within the dius : tht is, lose to the cente of th thn you position. ( ) () V π ( ) he density of th is lso iven by: V π qutin these two epessions fo nd solvin fo () yields: () Substitute fo () in eqution () nd siplify to obtin: G G () Apply Newton s lw of vity to youself t the sufce of th to obtin: Substitute fo in eqution () to obtin: G G whee is the nitude of the vittionl field t the sufce of th. ht is, the foce of vity on you is popotionl to you distnce fo the cente of th.

16 6 hpte (b) Apply Newton s second lw to you body to obtin: N ω whee the net foce ( ω ), diected towd the cente of th, is the centipetl foce ctin on you body. Solvin fo N yields: N ω Note tht this eqution tells us tht you effective weiht inceses linely with distnce fo the cente of th. Howeve, due just to the effect of ottion, s you ppoch the cente the centipetl foce deceses linely nd, doin so, inceses you effective weiht. 77 A solid sphee of dius hs its cente t the oiin. It hs unifo ss density, ecept tht thee is spheicl cvity in it of dius centeed t s in iue -7. ind the vittionl field t points on the is fo >. Hint: he cvity y be thouht of s sphee of ss (/)π plus sphee of netive ss. Pictue the Poble We cn use the hint to find the vittionl field lon the is. sin the hint, epess ( ) : ( ) solid sphee hollow sphee Substitute fo ( ) solid sphee nd hollow sphee G solid sphee π G nd siplify to obtin: G ( ) 8 hollow sphee G ( ) [ ] ( π ) G π ( ) ( ) 8 A sll diete hole is dilled into the sphee of Poble 8 towd the cente of the sphee to depth of. below the sphee s sufce. A sll ss is dopped fo the sufce into the hole. Deteine the speed of the sll ss s it stikes the botto of the hole. Pictue the Poble We cn use consevtion of eney to elte the wok done by the vittionl field to the speed of the sll object s it stikes the botto of the hole. Becuse we e iven the ss of the sphee, we cn find by

17 Gvity 7 epessin the ss of the sphee in tes of. We cn then use the definition of the vittionl field to find the vittionl field of the sphee inside its sufce. he wok done by the field equls the netive of the chne in the potentil eney of the syste s the sll object flls in the hole. se consevtion of eney to elte the wok done by the vittionl field to the speed of the sll object s it stikes the botto of the hole: pess the ss of diffeentil eleent of the sphee: K f Ki Δ o, becuse K i nd W Δ, W W v v () whee v is the speed with which the object stikes the botto of the hole nd W is the wok done by the vittionl field. d dv ( π d) Intete to epess the ss of π d ( 5 ) π the sphee in tes of : 5. Solvin fo yields: ( 5 )π Substitute nueicl vlues nd. evlute : ( 5 ) k 6.7 π 8 k/ se its definition to epess the vittionl field of the sphee t distnce fo its cente less thn its dius: π d G G G pess the wok done on the sll object by the vittionl foce ctin on it: π d π d G π G. W d ( ) 5. Substitute in eqution () nd siplify to obtin: v (.) ( π G) ( 8.) π G

18 8 hpte Substitute nueicl vlues nd evlute v: v 8 ( 8.) π ( 6.67 N /k )( 6.7 k/ )./s 8 wo identicl spheicl cvities e de in led sphee of dius. he cvities ech hve dius /. hey touch the outside sufce of the sphee nd its cente s in iue -8. he ss of solid unifo led sphee of dius is. ind the foce of ttction on point pticle of ss locted t distnce d fo the cente of the led sphee. Pictue the Poble he foce of ttction of the sll sphee of ss to the led sphee of ss is the su of the foces due to the solid sphee ( ) nd the cvities ( ) of netive ss. S pess the foce of ttction: S () se the lw of vity to epess the foce due to the solid sphee: pess the nitude of the foce ctin on the sll sphee due to one cvity: elte the netive ss of cvity to the ss of the sphee befoe hollowin: ettin θ be the nle between the is nd the line joinin the cente of the sll sphee to the cente of eithe cvity, use the lw of vity to epess the foce due to the two cvities: S G iˆ d G' d whee is the netive ss of cvity. ' V 8 π ( π ) G cos iˆ θ 8 d becuse, by syety, the y coponents dd to zeo. 8 se the fiue to epess cosθ : cosθ d d

19 Gvity 9 Substitute fo cosθ nd siplify to obtin: G d Gd d / d iˆ d iˆ Substitute in eqution () nd siplify: Genel Pobles G iˆ Gd / d d d G d d iˆ / iˆ 89 A neuton st is hihly condensed ennt of ssive st in the lst phse of its evolution. It is coposed of neutons (hence the ne) becuse the st s vittionl foce cuses electons nd potons to colesce into the neutons. Suppose t the end of its cuent phse, the collpsed into neuton st (it cn t in ctulity becuse it does not hve enouh ss) of dius. k, without losin ny ss in the pocess. () lculte the tio of the vittionl cceletion t the sufce of the followin its collpse coped to its vlue t the sufce of the tody. (b) lculte the tio of the escpe speed fo the sufce of the neuton- to its vlue tody. Pictue the Poble We cn pply Newton s second lw nd the lw of vity to n object of ss t the sufce of the nd the neuton- to find the tio of the vittionl cceletions t thei sufces. Siilly, we cn epess the tio of the coespondin epessions fo the escpe speeds fo the two suns to deteine thei tio. () pess the vittionl foce ctin on n object of ss t the sufce of the : Solvin fo yields: G G ()

20 hpte he vittionl foce ctin on n object of ss t the sufce of neuton- is: Solvin fo ' yields: G ' neuton- neuton- neuton- neuton- G ' () Divide eqution () by eqution () to obtin: ' G G neuton- neuton- neuton- neuton- Becuse : neuton- ' neuton- Substitute nueicl vlues nd evlute the tio ' : ' (b) he escpe speed fo the neuton- is iven by: ' e v G neuton- neuton- he escpe speed fo the is iven by: v e G Dividin the fist of these equtions by the second nd siplifyin yields: ' e v v e neuton- Substitute nueicl vlues nd 8 v' e 6.96 evlute v ' e ve : v. 95 nus, the seventh in the Sol Syste, ws fist obseved in 78 by Willi Heschel. Its obit ws then nlyzed in tes of Keple s ws. By the 8s, obsevtions of nus clely indicted tht its tue obit ws diffeent fo the Keplein clcultion by n ount tht could not be ccounted fo by obsevtionl uncetinty. he conclusion ws tht thee ust be nothe influence othe thn the nd the known s lyin inside nus s obit. his influence ws hypothesized to be due to n eihth, whose pedicted obit ws descibed independently in 85 by two stonoes: John Ads (no eltion to the foe pesident of the nited Sttes) nd bin eveie. In Septebe of 86, John Glle, sechin in the sky t the plce pedicted by e

21 Gvity Ads nd eveie, de the fist obsevtion of Neptune. nus nd Neptune e in obit bout the with peiods of 8. nd 6.8 yes, espectively. o see the effect tht Neptune hd on nus, deteine the tio of the vittionl foce between Neptune nd nus to tht between nus nd the, when Neptune nd nus e t thei closest ppoch to one nothe (i.e. when lined with the ). he sses of the, nus, nd Neptune e,,.5 nd 7. ties tht of th, espectively. Pictue the Poble We cn use the lw of vity nd Keple s thid lw to epess the tio of the vittionl foce between Neptune nd nus to tht between nus nd the, when Neptune nd nus e t thei closest ppoch to one nothe. he tio of the vittionl foce between Neptune nd nus to tht between nus nd the, when Neptune nd nus e t thei closest ppoch to one nothe is iven by:,n-,-s G N ( N ) N () G S ( ) S N Applyin Keple s thid lw to nus yields: Applyin Keple s thid lw to Neptune yields: () N () N Divide eqution () by eqution () N N N to obtin: N Substitute fo N in eqution () to,n- N obtin:,-s N S N Siplifyin this epession yields:,n-,-s N N S

22 hpte Becuse N 7. nd S, :,N-,-S N 7. 5 N. Substitute nueicl vlues nd evlute,n-,-s :,N-,-S y. 8. y Becuse this tio is so sll, duin the tie t which Neptune is closest to nus, the foce eeted on nus by Neptune is uch less thn the foce eeted on nus by the. 97 [SS] ou identicl s e ned in sque s shown in iue -9. If the ss of ech is nd the ede lenth of the sque is, wht ust be thei speed if they e to obit thei coon cente unde the influence of thei utul ttction? Pictue the Poble Note tht, due to the syeticl neent of the s, ech epeiences the se centipetl foce. We cn pply Newton s second lw to ny of the fou s to elte its obitl speed to this net (centipetl) foce ctin on it. Applyin the s ives: dil θ θ dil to one of c cosθ

23 Substitutin fo,, nd θ nd G siplifyin yields: c ( ) G G G cos5 G Gvity Becuse v v c v G Solve fo v to obtin: G v. 6 G In this poble you e to find the vittionl potentil eney of the thin od in ple -8 nd point pticle of ss tht is on the is t. () Show tht the potentil eney shed by n eleent of the od of ss d (shown in iue -) nd the point pticle of ss locted t is iven by Gd G d ds s ( s ) whee t. (b) Intete you esult fo Pt () ove the lenth of the od to find the totl potentil eney fo the syste. Genelize you function ( ) to ny plce on the is in the eion > / by eplcin by enel coodinte nd wite it s (). (c) opute the foce on t enel point usin d/d nd cope you esult with, whee is the field t clculted in ple -8. Pictue the Poble et t. he potentil eney of n eleent of the stick d nd the point ss is iven by the definition of vittionl potentil eney: d Gd whee is the seption of d nd. () pess the potentil eney of the sses nd d: he ss d is popotionl to the size of the eleent d : s Gd d d λ d s whee λ. s Substitute fo d nd λ to epess Gλ d d d in tes of s : ( ) s s G d s s

24 hpte (b) Intete d to find the totl potentil eney of the syste: ln ln ln / / s s G G d G (c) Becuse is enel point lon the is: ( ) G d d uthe siplifiction yields: ( ) G his nswe nd the nswe iven in ple -8 e the se.