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

Transcription

1 Chapte Gaity Conceptual Pobles [SS] ue o false: (a) o Keple s law of equal aeas to be alid, the foce of aity ust ay inesely with the squae of the distance between a ien planet and the Sun. (b) he planet closest to the Sun has the shotest obital peiod. (c) Venus s obital speed is lae than the obital speed of ath. (d) he obital peiod of a planet allows accuate deteination of that planet s ass. (a) alse. Keple s law of equal aeas is a consequence of the fact that the aitational foce acts alon the line joinin two bodies but is independent of the anne in which the foce aies with distance. (b) ue. he peiods of the planets ay with the thee-hales powe of thei distances fo the sun. So the shote the distance fo the sun, the shote the peiod of the planet s otion. (c) ue. Settin up a popotion inolin the obital speeds of the two planets in tes of thei obital peiods and ean distances fo the Sun (see able -) shows that.. Venus 7 ath (d) alse. he obital peiod of a planet is independent of the planet s ass. If the ass of a sall ath-obitin satellite is doubled, the adius of its obit can eain constant if the speed of the satellite (a) inceases by a facto of 8, (b) inceases by a facto of, (c) does not chane, (d) is educed by a facto of 8, (e) is educed by a facto of. Deteine the Concept We can apply Newton s nd law and the law of aity to the satellite to obtain an expession fo its speed as a function of the adius of its obit. Apply Newton s nd law to the satellite to obtain: adial whee is the ass of the object the satellite is obitin and is the ass of the satellite. 9

2 9 Chapte Solin fo yields: hus the speed of the satellite is independent of its ass and (c) is coect. [SS] Duin what season in the nothen heisphee does ath attain its axiu obital speed about the Sun? What season is elated to its iniu obital speed? HIN: he ajo facto deteinin the seasons on ath is not the aiation in distance fo the Sun. Deteine the Concept ath is closest to the Sun duin winte in the nothen heisphee. his is the tie of fastest obital speed. Sue would be the tie fo iniu obital speed. Haley s coet is in a hihly elliptical obit about the Sun with a peiod of about 76 y. Its last closest appoach to the Sun occued in 987. In what yeas of the twentieth centuy was it taelin at its fastest o slowest obital speed about the Sun? Deteine the Concept Haley s coet was taelin at its fastest obital speed in 987, and at its slowest obital speed 8 yeas peiously in Venus has no natual satellites. Howee atificial satellites hae been placed in obit aound it. o use one of thei obits to deteine the ass of Venus, what obital paaetes would you hae to easue? How would you then use the to do the ass calculation? Deteine the Concept o obtain the ass of Venus you need to easue the peiod and sei-ajo axis a of the obit of one of the satellites, substitute the a π (Keple s d law), and sole fo. easued alues into ( ) 6 A ajoity of the asteoids ae in appoxiately cicula obits in a belt between as and Jupite. Do they all hae the sae obital peiod about the Sun? xplain. Deteine the Concept No. As descibed by Keple s d law, the asteoids close to the Sun hae a shote yea and ae obitin faste. 7 [SS] At the suface of the oon, the acceleation due to the aity of the oon is a. At a distance fo the cente of the oon equal to fou ties the adius of the oon, the acceleation due to the aity of the oon is (a) 6a, (b) a/, (c) a/, (d) a/6, (e) None of the aboe. Pictue the Poble he acceleation due to aity aies inesely with the squae of the distance fo the cente of the oon.

3 Gaity 9 xpess the dependence of the acceleation due to the aity of the oon on the distance fo its cente: a' xpess the dependence of the acceleation due to the aity of the oon at its suface on its adius: a Diide the fist of these expessions a' by the second to obtain: a Solin fo a and siplifyin yields: a' a a 6 ( ) and (d) is coect. a 8 At a depth equal to half the adius of ath, the acceleation due to aity is about (a) (b) (c) /, (d) /, (e) /8, (f) You cannot deteine the answe based on the data ien. Pictue the Poble We can use Newton s law of aity and the assuption of unifo density to expess the atio of the acceleation due to aity at a depth equal to half the adius of ath to the acceleation due to aity at the suface of ath. he acceleation due to aity at a depth equal to half the adius of ath is ien by: he acceleation due to aity at the suface of ath is ien by: Diidin the fist of these equations by the second and siplifyin yields: ' ' ( ) whee is the ass of ath between the location of inteest and the cente of ath. ' ' ()

4 9 Chapte xpess in tes of the aeae density of ath ρ and the olue V of ath between the location of inteest and the cente of ath: [ ( ) ] π πρ ' ρ V' ρ 6 xpess in tes of the aeae density of ath ρ and the olue V of ath: ( ) π πρ ρ V ρ Substitute fo and in equation () and siplify to obtain: and ( c) ( πρ ) πρ is coect. 6 9 wo stas obit thei coon cente of ass as a binay sta syste. If each of thei asses wee doubled, what would hae to happen to the distance between the in ode to aintain the sae aitational foce? he distance would hae to (a) eain the sae (b) double (c) quaduple (d) be educed by a facto of (e) You cannot deteine the answe based on the data ien. Pictue the Poble We can use Newton s law of aity to expess the atio of the foces and then sole this popotion fo the sepaation of the stas that would aintain the sae aitational foce. he aitational foce actin on the stas befoe thei asses ae doubled is ien by: he aitational foce actin on the stas afte thei asses ae doubled is ien by: Diidin the second of these equations by the fist yields: G ( )( ) G G ' ' ' G ' ' G ' Solin fo yields: ' and ( b) is coect. If you had been wokin fo NASA in the 96 s and plannin the tip to the oon, you would hae deteined that thee exists a unique location soewhee between ath and the oon, whee a spaceship is, fo an instant, tuly weihtless. [Conside only the oon, ath and the Apollo spaceship, and

5 Gaity 95 nelect othe aitational foces.] xplain this phenoenon and explain whethe this location is close to the oon, idway on the tip, o close to ath. Deteine the Concept Between ath and the oon, the aitational pulls on the spaceship ae oppositely diected. Because of the oon s elatiely sall ass copaed to the ass of ath, the location whee the aitational foces cancel (thus poducin no net aitational foce, a weihtless condition) is consideably close to the oon. [SS] Suppose the escape speed fo a planet was only slihtly lae than the escape speed fo ath, yet it was consideably lae than ath. How would the planet s (aeae) density copae to ath s (aeae) density? (a) It ust be oe dense. (b) It ust be less dense. (c) It ust be the sae density. (d) You cannot deteine the answe based on the data ien. Pictue the Poble he densities of the planets ae elated to the escape speeds fo thei sufaces thouh e. he escape speed fo the planet is ien by: planet planet planet he escape speed fo ath is ien by: ath ath ath xpessin the atio of the escape speed fo the planet to the escape speed fo ath and siplifyin yields: planet ath planet ath planet ath ath planet planet ath Because planet ath : ath planet planet ath Squain both sides of the equation yields: ath planet planet ath

6 96 Chapte xpess planet and ath in tes of thei densities and siplify to obtain: ath planet ρ ρ planet ath V V planet ath ath planet ρ ρ planet ath V V planet ath ath planet ρ ρ planet ath π π planet ath ρ ρ planet ath planet ath Solin fo the atio of the densities ρplanet yields: ρ ath ath planet Because the planet is consideably lae than ath: ρ ρ planet ath and ( b) << is coect. Suppose that, usin telescope in you backyad, you discoeed a distant object appoachin the Sun, and wee able to deteine both its distance fo the Sun and its speed. How would you be able to pedict whethe the object will eain bound to the Sola Syste, o if it is an intestella intelope and would coe in, tun aound and escape, nee to etun? Deteine the Concept You could take caeful easueents of its position as a function of tie in ode to deteine whethe its tajectoy is an ellipse, a hypebola, o a paabola. If the path is an ellipse, it will etun; if its path is hypebolic o paabolic, it will not etun. Altenatiely, by easuin its distance fo the Sun, you can estiate the aitational potential eney (pe k of its ass, and nelectin the planets) of the object, and by deteinin its position on seeal successie nihts, the speed of the object can be deteined. o this, its kinetic eney (pe k) can be deteined. he su of these two ies the coet s total eney (pe k) and if it is positie, it will likely swin once aound the Sun and then leae the Sola Syste foee. [SS] Nea the end of thei useful lies, seeal lae ath-obitin satellites hae been aneueed so as to bun up as they ente ath s atosphee. hese aneues hae to be done caefully so lae faents do not ipact populated land aeas. You ae in chae of such a poject. Assuin the satellite of inteest has on-boad populsion, in what diection would you fie the ockets fo a shot bun tie to stat this downwad spial? What would happen to the kinetic eney, aitational potential eney and total echanical eney followin the bun as the satellite cae close and close to ath? Deteine the Concept You should fie the ocket in a diection to oppose the obital otion of the satellite. As the satellite ets close to ath afte the bun, the potential eney will decease as the satellite ets close to ath. Howee, the total echanical eney will decease due to the fictional da foces tansfoin echanical eney into theal eney. he kinetic eney will

7 Gaity 97 incease until the satellite entes the atosphee whee the da foces slow its otion. Duin a tip back fo the oon, the Apollo spacecaft fies its ockets to leae its luna obit. hen it coasts back to ath whee it entes the atosphee at hih speed, suies a blazin e-enty and paachutes safely into the ocean. In what diection do you fie the ockets to initiate this etun tip? xplain the chanes in kinetic eney, aitational potential and total echanical eney that occu to the spacecaft fo the beinnin to the end of this jouney. Deteine the Concept Nea the oon you would fie the ockets to acceleate the spacecaft with the thust actin in the diection of you ship s elocity at the tie. When the ockets hae shut down, as you leae the luna obit, you kinetic eney will initially decease (the oon s aitational pull exceeds that of ath), and you potential eney will incease. When you each a cetain point, ath s aitational attaction will bein acceleatin the ship and its kinetic eney will incease at the expense of the aitational potential eney of the spacecaft- ath-oon syste. he spacecaft will ente ath s atosphee with its axiu kinetic eney. entually, landin in the ocean, the kinetic eney will be zeo, the aitational potential eney a iniu, and the total echanical eney of the ship will hae been daatically educed due to ai da foces poducin heat and liht duin e-enty. 5 xplain why the aitational field inside a solid sphee of unifo ass is diectly popotional to athe than inesely popotional to. Deteine the Concept At a point inside the sphee a distance fo its cente, the aitational field stenth is diectly popotional to the aount of ass within a distance fo the cente, and inesely popotional to the squae of the distance fo the cente. he ass within a distance fo the cente is popotional to the cube of. hus, the aitational field stenth is diectly popotional to. 6 In the oie A Space Odyssey, a spaceship containin two astonauts is on a lon-te ission to Jupite. A odel of thei ship could be a unifo pencil-like od (containin the populsion systes) with a unifo sphee (the cew habitat and fliht deck) attached to one end. he desin is such that the adius of the sphee is uch salle than the lenth of the od. At a location a few etes away fo the ship, on the pependicula bisecto of the od-like section, what would be the diection of the aitational field due to the ship alone (that is, assuin all othe aitational fields ae neliible)? xplain you answe. At a lae distance fo the ship, what would be the dependence of its aitational field on the distance fo the ship?

8 98 Chapte Deteine the Concept he pictoial epesentation shows the point of inteest P and the aitational fields od and sphee due to the od and the sphee as well as the esultant field net. Note that the net field (the su of od and sphee ) points slihtly towad the habitat end of the ship. At ey lae distances, the od-sphee ass distibution looks like a point ass and so the field s distance dependence is an inese squae dependence. P sphee od net stiation and Appoxiation 7 [SS] stiate the ass of ou alaxy (the ilky Way) if the Sun obits the cente of the alaxy with a peiod of 5 illion yeas at a ean distance of, c y. xpess the ass in tes of ultiples of the sola ass S. (Nelect the ass fathe fo the cente than the Sun, and assue that the ass close to the cente than the Sun exets the sae foce on the Sun as would a point paticle of the sae ass located at the cente of the alaxy.) Pictue the Poble o appoxiate the ass of the alaxy we ll assue the alactic cente to be a point ass with the sun in obit about it and apply Keple s d law. Usin Keple s d law, elate the peiod of the sun to its ean distance fo the cente of the alaxy: π alaxy π s G alaxy s Sole fo and siplify to obtain: G π alaxy s s alaxy π s s If we easue distances in AU and ties in yeas: π y s and ( AU) alaxy s ( AU) y

9 Gaity 99 Substitute nueical alues and ealuate alaxy / s : o alaxy s. 9.6 c y c y 5 6 ( 5 y) AU.5 alaxy.8 s y ( AU).8 8 Besides studyin saples of the luna suface, the Apollo astonauts had seeal ways of deteinin that the oon is not ade of een cheese. Aon these ways ae easueents of the aitational acceleation at the luna suface. stiate the aitational acceleation at the luna suface if the oon wee, in fact, a solid block of een cheese and copae it to the known alue of the aitational acceleation at the luna suface. Pictue the Poble he density of a planet o othe object deteines the stenth of the aitational foce it exets on othe objects. We can use Newton s law of aity to expess the acceleation due to aity at the suface of the oon as a function of the density of the oon. stiatin the density of cheese will then allow us to calculate what the acceleation due to aity at the suface of the oon would be if the oon wee ade of cheese. inally, we can copae this alue to the easued alue of.6 /s. Apply Newton s law of aity to an object of ass at the suface of the oon to obtain: Assuin the oon to be ade of cheese, substitute fo its ass to obtain: G a a a oon oon,cheese Gρ V cheese oon oon Substitutin fo the olue of the oon and siplifyin yields: a,cheese Gρ cheese oon π Gρ π cheese oon oon Substitute nueical alues and ealuate a : a,cheese π 6 k c 6 ( 6.67 N / k ).8 (.78 ).9 /s c

10 Chapte xpess the atio of a,cheese to the a,cheese.88 /s easued alue of a,oon :. a,oon.6 /s o, cheese. a a,oon 9 You ae in chae of the fist anned exploation of an asteoid. You ae concened that, due to the weak aitational field and esultin low escape speed, tethes iht be equied to bind the exploes to the suface of the asteoid. heefoe, if you do not wish to use tethes, you hae to be caeful about which asteoids to choose to exploe. stiate the laest adius the asteoid can hae that would still allow you to escape its suface by jupin. Assue spheical eoety and easonable ock density. Pictue the Poble he density of an asteoid deteines the stenth of the aitational foce it exets on othe objects. We can use the equation fo the escape speed fo an asteoid of ass asteoid and adius asteoid to deie an expession fo the adius of an asteoid as a function of its escape speed and density. We can appoxiate the escape speed fo the asteoid by deteinin one s push-off speed fo a jup at the suface of ath. he escape speed fo an asteoid is ien by: e,asteoid asteoid asteoid In tes of the density of the asteoid, e,asteoid becoes: e,asteoid Gρ asteoid asteoid π asteoid 8 πgρ asteoid asteoid Solin fo asteoid yields: Usin a constant-acceleation equation, elate the heiht h to which you can jup on the suface of ath to you push-off speed: e,asteoid asteoid () 8 πgρ asteoid h o, because, h h Lettin e,asteoid, substitute in equation () and siplify to obtain: h asteoid πgρ 8 asteoid h πgρ asteoid

11 Gaity Assuin that you can jup.75 and that the aeae density of an asteoid is. /c, substitute nueical alues and ealuate asteoid : asteoid π ( )(.75 ) 9.8/s ( 6.67 N / k ). c 6 k c. k One of the eat discoeies in astonoy in the past decade is the detection of planets outside the Sola Syste. Since 996, planets hae been detected obitin stas othe than the Sun. While the planets theseles cannot be seen diectly, telescopes can detect the sall peiodic otion of the sta as the sta and planet obit aound thei coon cente of ass. (his is easued usin the Dopple effect, which is discussed in Chapte 5.) Both the peiod of this otion and the aiation in the speed of the sta oe the couse of tie can be deteined obseationally. he ass of the sta is found fo its obseed luinance and fo the theoy of stella stuctue. Iota Daconis is the 8th bihtest sta in the constellation Daco. Obseations show that a planet, with an obital peiod of.5 y, is obitin this sta. he ass of Iota Daconis is.5 Sun. (a) stiate the size (in AU) of the seiajo axis of this planet s obit. (b) he adial speed of the sta is obseed to ay by 59 /s. Use conseation of oentu to find the ass of the planet. Assue the obit is cicula, we ae obsein the obit ede-on, and no othe planets obit Iota Daconis. xpess the ass as a ultiple of the ass of Jupite. Pictue the Poble We can use Keple s d law to find the size of the seiajo axis of the planet s obit and the conseation of oentu to find its ass. (a) Usin Keple s d law, elate the peiod of this planet to the lenth of its sei-ajo axis and siplify to obtain: π Iota Daconis π s G Iota Daconis s π Iota Daconis s s If we easue tie in yeas, distances in AU, and asses in tes of the ass of the sun: π G s and Iota Daconis s Solin fo yields: Iota Daconis s

12 Chapte Substitute nueical alues and.5 s s ealuate : (.5 y).au (b) Apply conseation of oentu to the planet (ass and speed ) and the sta (ass Iota Daconis and speed V) to obtain: Sole fo to obtain: Iota Daconis V V Iota Daconis () he speed of the obitin planet is ien by: Δd Δt π Substitute nueical alues and ealuate :.5 π.au AU 65.d h 6s.5 y y d h.68 /s Substitute nueical alues in equation () and ealuate : (.5sun ) (.5)(.99 k)(.).6 96 /s.68 /s xpess as a faction of the ass 8.6 k J of Jupite:. 7 J.9 k o. eaks: A oe sophisticated analysis, usin the eccenticity of the obit, leads to a lowe bound of 8.7 Joian asses. (Only a lowe bound can be established, as the plane of the obit is not known.) One of the biest unesoled pobles in the theoy of the foation of the sola syste is that, while the ass of the Sun is 99.9 pecent of the total ass of the Sola Syste, it caies only about pecent of the total anula oentu. he ost widely accepted theoy of sola syste foation has as its cental hypothesis the collapse of a cloud of dust and as unde the foce of aity, with ost of the ass foin the Sun. Howee, because the net anula oentu of this cloud is conseed, a siple theoy would indicate that the Sun J 8 k

13 Gaity should be otatin uch oe apidly than it cuently is. In this poble, you will show why it is ipotant that ost of the anula oentu was soehow tansfeed to the planets. (a) he Sun is a cloud of as held toethe by the foce of aity. If the Sun wee otatin too apidly, aity couldn t hold it toethe. Usin the known ass of the Sun (.99 k) and its adius ( ), estiate the axiu anula speed that the Sun can hae if it is to stay intact. What is the peiod of otation coespondin to this otation ate? (b) Calculate the obital anula oentu of Jupite and of Satun fo thei asses (8 and 95. ath asses, espectiely), ean distances fo the Sun (778 and illion k, espectiely), and obital peiods (.9 and 9.5 y, espectiely). Copae the to the expeientally easued alue of the Sun s anula oentu of.9 k /s. (c) If we wee to soehow tansfe all of Jupite s and Satun s anula oentu to the Sun, what would be the Sun s new otational peiod? he Sun is not a unifo sphee of as, and its oent of inetia is ien by the foula I.59. Copae this to the axiu otational peiod of Pat (a). Pictue the Poble We can apply Newton s law of aity to estiate the axiu anula speed which the sun can hae if it is to stay toethe and use the definition of anula oentu to find the obital anula oenta of Jupite and Satun. In Pat (c) we can elate the final anula speed of the sun to its initial anula speed, its oent of inetia, and the obital anula oenta of Jupite and Satun. (a) Gaity ust supply the centipetal foce which keeps an eleent of the sun s ass otatin aound it. Lettin the adius of the sun be, apply Newton s law of aity to an object of ass on the suface of the Sun to obtain: ω < o ω < ω < whee we e used the inequality because we e estiatin the axiu anula speed which the sun can hae if it is to stay toethe. Substitute nueical alues and ealuate ω: ω < (.67 N /k )(.99 k) 8 ( 6.96 ) 6 Calculate the axiu peiod of this otion fo its anula speed:. 6.8 s h 6s ad/s π π ω 6.8 ad/s ax.78h

14 Chapte (b) xpess the obital anula oenta of Jupite and Satun: L J JJ J and L S SS S xpess the obital speeds of Jupite and Satun in tes of thei peiods and distances fo the sun: J π J and J S π S S Substitute to obtain: L J π JJ and J L S π S S S Substitute nueical alues and ealuate L J and L S : and L L S J π π 9 ( 8 ) π ( 8)( 5.98 k)( 778 ) J.9 J k /s 65.d h 6s.9 y y d h 9 ( 95. ) π ( 95.)( 5.98 k)( ) S 7.85 S k /s 65.d h 6s 9.5y y d h xpess the anula oentu of the sun as a faction of the su of the anula oenta of Jupite and Satun: Lsun L + L J S ( ).7%.9 k /s k /s (c) he Sun s otational peiod depends on its otational speed: elate the final anula oentu of the sun to its initial anula oentu and the anula oenta of Jupite and Satun: Sole fo ω f to obtain: Substitute fo ω i and I Sun : π Sun () ω f L f Li + LJ + LS o I ω I ω + L + L Sun f Sun L J + L ω f ωi + I Sun i sun LJ + LS ω f π +.59 S J sun S sun

15 Gaity 5 Substitute nueical alues and ealuate ω f : π ωf h 6s d d h.798 ad/s ( ) k /s 8 ( k)( 6.96 ) Substitute nueical alues in equation () and ealuate Sun : Sun π ad 6 s.798 s h.6 h Copae this to the axiu otational peiod of Pat (a). o ax.6 h.78 h Sun Sun. ax. Keple s Laws he new coet Alex-Casey has a ey elliptical obit with a peiod of 7. y. If the closest appoach of Alex-Casey to the Sun is. AU, what is its eatest distance fo the Sun? Pictue the Poble We can use the elationship between the sei-ajo axis and the distances of closest appoach and eatest sepaation, toethe with Keple s d law, to find the eatest sepaation of Alex-Casey fo the sun. Lettin x epesent the eatest distance fo the sun, expess the elationship between x, the distance of closest appoach, and its seiajo axis : Apply Keple s d law, with the peiod easued in yeas and in AU to obtain: Substitutin fo in equation () yields: Substitute nueical alues and ealuate x: x +.AU x.au () x.au x ( 7. y).au 5.5AU

16 6 Chapte he adius of ath s obit is.96 and that of Uanus is.87. What is the obital peiod of Uanus? Pictue the Poble We can use Keple s d law to elate the obital peiod of Uanus to the obital peiod of ath. Usin Keple s d law, elate the obital peiod of Uanus to its ean distance fo the sun: Usin Keple s d law, elate the obital peiod of ath to its ean distance fo the sun: C. Uanus C ath Uanus ath Diidin the fist of these equations Uanus CUanus by the second and siplifyin yields: C ath ath Uanus ath Sole fo Uanus to obtain: Uanus ath Uanus ath Substitute nueical alues and ealuate Uanus : (. y) Uanus 8. y he asteoid Hekto, discoeed in 97, is in a nealy cicula obit of adius 5.6 AU about the Sun. Deteine the peiod of this asteoid. Pictue the Poble We can use Keple s d law to elate the obital peiod of Hekto to its ean distance fo the sun. Usin Keple s d law, elate the obital peiod of Hekto to its ean distance fo the sun: Hecto CHecto π whee C.97 Hecto C Hecto 9 s s /.

17 Gaity 7 Substitute nueical alues and ealuate Hecto : Hecto 9 (.97 s / ) 8.7 s h 6s AU AU d h y 65.d.8y 5 [SS] One of the so-called Kikwood aps in the asteoid belt occus at an obital adius at which the peiod of the obit is half that of Jupite s. he eason thee is a ap fo obits of this adius is because of the peiodic pullin (by Jupite) that an asteoid expeiences at the sae place in its obit eey othe obit aound the sun. epeated tus fo Jupite of this kind would eentually chane the obit of such an asteoid theefoe all asteoids that would othewise hae obited at this adius hae pesuably been cleaed away fo the aea due to this esonance phenoenon. How fa fo the sun is this paticula : Kikwood ap? Pictue the Poble he peiod of an obit is elated to its sei-ajo axis (fo cicula obits this distance is the obital adius). Because we know the obital peiods of Jupite and a hypothetical asteoid in the Kikwood ap, we can use Keple s d law to set up a popotion elatin the obital peiods and aeae distances of Jupite and the asteoid fo the Sun fo which we can obtain an expession fo the obital adius of an asteoid in the Kikwood ap. Use Keple s d law to elate Jupite s obital peiod to its ean distance fo the Sun: Use Keple s d law to elate the obital peiod of an asteoid in the Kikwood ap to its ean distance fo the Sun: C Jupite Kikwood Jupite C Kikwood Diidin the second of these Kikwood CKikwood equations by the fist yields: C Jupite Jupite Kikwood Jupite Solin fo Kikwood yields: Kikwood Kikwood Jupite Jupite

18 8 Chapte Because the peiod of the obit of an asteoid in the Kikwood ap is half that of Jupite s: Kikwood Jupite Jupite.9 ( 77.8 ) 6 he tiny Satunian oon, Atlas, is locked into what is known as an obital esonance with anothe oon, ias, whose obit lies outside of Atlas s. he atio between peiods of these obits is : that eans, fo eey obits of Atlas, ias copletes obits. hus, Atlas, ias and Satun ae alined at inteals equal to two obital peiods of Atlas. If ias obits Satun at a adius of 86, k, what is the adius of Atlas s obit? Pictue the Poble he peiod of an obit is elated to its sei-ajo axis (fo cicula obits this distance is the obital adius). Because we know the obital peiods of Atlas and ias, we can use Keple s d law to set up a popotion elatin the obital peiods and aeae distances fo Satun of Atlas and ias fo which we can obtain an expession fo the adius of Atlas s obit. Use Keple s d law to elate Atlas s obital peiod to its ean distance fo Satun: Use Keple s d law to elate the obital peiod of ias to its ean distance fo Satun: C Atlas ias Atlas C ias Diidin the second of these ias Cias equations by the fist yields: C Atlas Atlas ias Atlas Solin fo Atlas yields: Atlas Atlas ias ias Because fo eey obits of Atlas, 5 ias has copleted : Atlas (.86 k). 5 k 7 he asteoid Icaus, discoeed in 99, was so naed because its hihly eccentic elliptical obit bins it close to the Sun at peihelion. he eccenticity e of an ellipse is defined by the elation p a( e), whee p is the peihelion distance and a is the seiajo axis. Icaus has an eccenticity of.8 and a peiod of. y. (a) Deteine the seiajo axis of the obit of Icaus.

19 Gaity 9 (b) ind the peihelion and aphelion distances of the obit of Icaus. Pictue the Poble Keple s d law elates the peiod of Icaus to the lenth of its seiajo axis. he aphelion distance a is elated to the peihelion distance p and the seiajo axis by + a. a p (a) Usin Keple s d law, elate the peiod of Icaus to the lenth a of its seiajo axis: Substitute nueical alues and ealuate a: C Ca a π whee C a.6 9 s.97 s / 65.d h 6s.y y d h 9.97 s /. (b) Use the definition of the eccenticity of an ellipse to deteine the peihelion distance of Icaus: p a( ) (.59 )(.8) e.7.7 xpess the elationship between p and a fo an ellipse: Substitute nueical alues and ealuate a : a + p a a a p a ( ) A anned ission to as and its attendant pobles due to the exteely lon tie the astonauts would spend weihtless and without supplies hae been extensiely discussed. o exaine this issue in a siple way, conside one possible tajectoy fo the spacecaft: the Hohann tansfe obit. his obit consists of an elliptical obit tanent to the obit of ath at its peihelion and tanent to the obit of as at its aphelion. Gien that as has a ean distance fo the Sun of.5 ties the ean Sun ath distance, calculate the tie spent by the astonauts duin the out-bound pat of the tip to as. any adese bioloical effects (such as uscle atophy, deceased bone density, etc.) hae been obseed in astonauts etunin fo nea-ath obit afte only a few onths in space. As the fliht docto, ae thee any health issues you should be awae of?

20 Chapte Pictue the Poble he Hohann tansfe obit is shown in the diaa. We can apply Keple s d law to elate the tie-in-obit to the peiod of the spacecaft in its Hohann ath-to-as obit. he peiod of this obit is, in tun, a function of its sei-ajo axis which we can find fo the aeae of the lenths of the sei-ajo axes of ath and as obits. Usin Keple s d law, elate the peiod of the spacecaft to the sei-ajo axis of its obit: whee is in yeas and is in AU. elate the out-bound tansit tie to the peiod of this obit: t out-bound xpess the sei-ajo axis of the Hohann tansfe obit in tes of the ean sun-as and sun-ath distances:.5 AU +. AU.6 AU Substitute nueical alues and ealuate t out-bound : t out-bound (.6 AU) 65. d.77 y y 58 d In ode fo bones and uscles to aintain thei health, they need to be unde copession as they ae on ath. Due to the lon duation (well oe a yea) of the ound tip, you would want to desin an execise poa that would aintain the stenth of thei bones and uscles. 9 [SS] Keple deteined distances in the Sola Syste fo his data. o exaple, he found the elatie distance fo the Sun to Venus (as copaed to the distance fo the Sun to ath) as follows. Because Venus s obit is close to the Sun than is ath s, Venus is a onin o eenin sta its position in the sky is nee ey fa fo the Sun (see iue -). If we

21 Gaity conside the obit of Venus as a pefect cicle, then conside the elatie oientation of Venus, ath, and the Sun at axiu extension when Venus is fathest fo the Sun in the sky. (a) Unde this condition, show that anle b in iue - is 9º. (b) If the axiu elonation anle a between Venus and the Sun is 7º, what is the distance between Venus and the Sun in AU? (c) Use this esult to estiate the lenth of a Venusian yea. Pictue the Poble We can use a popety of lines tanent to a cicle and adii dawn to the point of contact to show that b 9. Once we e established that b is a iht anle we can use the definition of the sine function to elate the distance fo the Sun to Venus to the distance fo the Sun to ath. (a) he line fo ath to Venus' obit is tanent to the obit of Venus at the point of axiu extension. Venus will appea close to the sun in eath s sky when it passes the line dawn fo ath and tanent to its obit. Hence b 9 (b) Usin tionoety, elate the distance fo the sun to Venus d SV to the anle a: d sin a d d SV SV S d S sin a Substitute nueical alues and ealuate d SV : d SV (. AU).7AU sin 7.7AU (c) Use Keple s d law to elate Venus s obital peiod to its ean distance fo the Sun: Use Keple s d law to elate ath s obital peiod to its ean distance fo the Sun: C Venus C ath Venus ath Diidin the fist of these equations Venus CVenus by the second yields: C ath ath Venus ath Solin fo Venus yields: Venus Venus ath ath Usin the esult fo pat (b) yields: Venus.7 AU. AU (. y).6 y

22 Chapte eaks: he coect distance fo the sun to Venus is close to.7 AU. At apoee the oon is 6,95 k fo ath and at peiee it is 57,6 k. What is the obital speed of the oon at peiee and at apoee? Its obital peiod is 7. d. Pictue the Poble Because the aitational foce ath exets on the oon is alon the line joinin thei centes, the net toque actin on the oon is zeo and its anula oentu is conseed in its obit about ath. Because eney is also conseed, we can cobine these two expessions to sole fo eithe p o a initially and then use conseation of anula oentu to find the othe. Lettin be the ass of the oon, apply conseation of anula oentu to the oon at apoee and peiee to obtain: Apply conseation of eney to the oon-ath syste to obtain: Substitute fo a to obtain: pp aa a o p p p p p p a a a p p a a p p p a p a a a Solin fo p yields: p p + p a Substitute nueical alues and ealuate p : p ( 6.67 N /k )( 5.98 k).9 k/s

23 Substitute nueical alues in equation () and ealuate a : a (.9 k/s) 8 Newton s Law of Gaity 959 /s Gaity [SS] Jupite s satellite uopa obits Jupite with a peiod of.55 d at an aeae obital adius of (a) Assuin that the obit is cicula, deteine the ass of Jupite fo the data ien. (b) Anothe satellite of Jupite, Callisto, obits at an aeae adius of with an obital peiod of 6.7 d. Show that this data is consistent with an inese squae foce law fo aity (Note: DO NO use the alue of G anywhee in Pat (b)). Pictue the Poble While we could apply Newton s Law of Gaitation and nd Law of otion to sole this poble fo fist pinciples, we ll use Keple s d law (deied fo these laws) to find the ass of Jupite in Pat (a). In Pat (b) we can copae the atio of the centipetal acceleations of uopa and Callisto to show that they ae consistent with an inese squae law fo aity. (a) Assuin a cicula obit, apply Keple s d law to the otion of uopa to obtain: π J J π G Substitute nueical alues and ealuate J : 8 π ( 6.7 ) ( 6.67 N /k ).55d 7 J.9 h 6s d h Note that this esult is in excellent aeeent with the accepted alue of.9 7 k. (b) xpess the centipetal π acceleation of both of the oons to π obtain: acentipetal whee and ae the adii and peiods of thei otion. k Usin this esult, expess the centipetal acceleations of uopa and Callisto: a π π and ac C C

24 Chapte Diide the fist of these equations by the second and siplify to obtain: a a C π π C C C C Substitute fo the peiods of Callisto and uopa usin Keple s d law to obtain: a a C C C C C C his esult, toethe with the fact that the aitational foce is diectly popotional to the acceleation of the oons, deonstates that the aitational foce aies inesely with the squae of the distance. Soe people think that shuttle astonauts ae weihtless because they ae beyond the pull of ath s aity. In fact, this is copletely untue. (a) What is the anitude of the aitational field in the icinity of a shuttle obit? A shuttle obit is about k aboe the ound. (b) Gien the answe in Pat (a), explain why shuttle astonauts do suffe fo adese bioloical affects such as uscle atophy een thouh they ae actually not weihtless? Deteine the Concept he weiht of anythin, includin astonauts, is the eadin of a scale fo which the object is suspended o on which it ests. hat is, it is the anitude of the noal foce actin on the object. If the scale eads zeo, then we say the object is weihtless. he pull of ath s aity, on the othe hand, depends on the local alue of the acceleation of aity and we can use Newton s law of aity to find this acceleation at the eleation of the shuttle. (a) Apply Newton s law of aitation to an astonaut of ass in a shuttle at a distance h aboe the suface of ath: shuttle G ( h + ) Solin fo shuttle yields: shuttle ( h + ) Substitute nueical alues and ealuate shuttle : shuttle ( 6.67 N /k )( 5.98 k) ( k + 67 k) 8.7/s

25 Gaity 5 (b) In obit, the astonauts expeience only one (the aitational foce) of the two foces (the second bein the noal foce a copessie foce exeted by ath) that noally acts on the. Lackin this copessie foce, thei bones and uscles, the absence of an execise poa, will weaken. In obit the astonauts ae not weihtless, they ae noal-foceless. [SS] he ass of Satun is k. (a) ind the peiod of its oon ias, whose ean obital adius is (b) ind the ean obital adius of its oon itan, whose peiod is.8 6 s. Pictue the Poble While we could apply Newton s Law of Gaitation and nd Law of otion to sole this poble fo fist pinciples, we ll use Keple s d law (deied fo these laws) to find the peiod of ias and to elate the peiods of the oons of Satun to thei ean distances fo its cente. (a) Usin Keple s d law, elate the peiod of ias to its ean distance fo the cente of Satun: π S π S Substitute nueical alues and ealuate : 8 π (.86 ) 6 ( N /k )( 5.69 k) 8.8 s.7 h (b) Usin Keple s d law, elate the peiod of itan to its ean distance fo the cente of Satun: π S S π Substitute nueical alues and ealuate : 6 6 (.8 s) ( N /k )( 5.69 k) 9. π Calculate the ass of ath fo the peiod of the oon, 7. d; its ean obital adius,.8 8 ; and the known alue of G. Pictue the Poble While we could apply Newton s law of aitation and nd law of otion to sole this poble fo fist pinciples, we ll use Keple s d law (deied fo these laws) to elate the peiod of the oon to the ass of ath and the ean ath-oon distance.

26 6 Chapte Usin Keple s d law, elate the peiod of the oon to its ean obital adius: π π G Substitute nueical alues and ealuate : 8 π (.8 ) ( N /k ) 7.d 6. h 6 s d h eaks: his analysis nelects the ass of the oon; consequently the ass calculated hee is slihtly too lae. 5 Suppose you leae the Sola Syste and aie at a planet that has the sae ass-to-olue atio as ath but has ties ath s adius. What would you weih on this planet copaed with what you weih on ath? Pictue the Poble You weiht is the local aitational foce exeted on you. We can use the definition of density to elate the ass of the planet to the ass of ath and the law of aity to elate you weiht on the planet to you weiht on ath. k Usin the definition of density, elate the ass of ath to its adius: ρ V ρ π elate the ass of the planet to its adius: ( ) P ρ V P ρ π ρ π P Diide the second of these equations by the fist to expess P in tes of : Lettin w epesent you weiht on the planet, use the law of aity to elate w to you weiht on ath: P ρ π ( ) ρ P ρ π G w' P P G G ( ) ( ) w You weiht would be ten ties you weiht on ath.

27 Gaity 7 6 Suppose that ath etained its pesent ass but was soehow copessed to half its pesent adius. What would be the alue of at the suface of this new, copact planet? Pictue the Poble We can elate the acceleation due to aity of a test object at the suface of the new planet to the acceleation due to aity at the suface of ath thouh use of the law of aity and Newton s nd law of otion. Lettin a epesent the acceleation due to aity at the suface of this new planet and the ass of a test object, apply Newton s nd law and the law of aity to obtain: G a adial ( ) ( ) a Siplify this expession to obtain: a 9./s 7 A planet obits a assie sta. When the planet is at peihelion, it has a speed of 5. /s and is. 5 fo the sta. he obital adius inceases to. 5 at aphelion. What is the planet s speed at aphelion? Pictue the Poble We can use conseation of anula oentu to elate the planet s speeds at aphelion and peihelion. Usin conseation of anula oentu, elate the anula oenta of the planet at aphelion and peihelion: Substitute nueical alues and ealuate a : L a L p o p pp aa a a 5 ( 5. /s)(. ).. /s 5 a p 8 What is the anitude of the aitational field at the suface of a neuton sta whose ass is.6 ties the ass of the Sun and whose adius is.5 k? Pictue the Poble We can use Newton s law of aity to expess the aitational foce actin on an object at the suface of the neuton sta in tes of the weiht of the object. We can then siplify this expession be diidin out the ass of the object leain an expession fo the anitude of the aitational field at the suface of the neuton sta.

28 8 Chapte Apply Newton s law of aity to an object of ass at the suface of the neuton sta to obtain: Sole fo and substitute fo the ass of the neuton sta: Neuton Sta Neuton Sta whee epesents the anitude of the aitational field at the suface of the neuton sta. (. ) Neuton Sta G 6 sun Neuton Sta Neuton Sta Substitute nueical alues and ealuate : ( N /k )(.99 k) (.5 k).9 /s 9 he speed of an asteoid is k/s at peihelion and k/s at aphelion. (a) Deteine the atio of the aphelion to peihelion distances. (b) Is this asteoid fathe fo the Sun o close to the Sun than ath, on aeae? xplain. Pictue the Poble We can use conseation of anula oentu to elate the asteoid s aphelion and peihelion distances. (a) Usin conseation of anula oentu, elate the anula oenta of the asteoid at aphelion and peihelion: L o a Lp a a pp a p p a Substitute nueical alues and ealuate the atio of the asteoid s aphelion and peihelion distances: p k/s k/s a. (b) It is fathe fo the Sun than ath. Keple s thid law ( Ca ) tells us that lone obital peiods toethe with lae obital adii eans slowe obital speeds, so the speed of objects obitin the Sun deceases with distance fo the Sun. he aeae obital speed of ath, ien by πs S, is appoxiately k/s. Because the ien axiu speed of the asteoid is only k/s, the asteoid is futhe fo the Sun. A satellite with a ass of k oes in a cicula obit 5. 7 aboe ath s suface. (a) What is the aitational foce on the satellite? (b) What is the speed of the satellite? (c) What is the peiod of the satellite?

29 Gaity 9 Pictue the Poble We ll use the law of aity to find the aitational foce actin on the satellite. he application of Newton s nd law will lead us to the speed of the satellite and its peiod can be found fo its definition. (a) Lettin epesent the ass of the satellite and h its eleation, use the law of aity to expess the aitational foce actin on it: Because : Diide the nueato and denoinato of this expession by to obtain: Substitute nueical alues and ealuate : G ( + h) ( + h) h + ( k)( 9.8N/k) N N (b) Usin Newton s nd law, elate the aitational foce actin on the satellite to its centipetal acceleation: Substitute nueical alues and ealuate : 6 7 ( 7.58 N)( ).657 k/s.66 k/s k (c) he peiod of the satellite is ien by: Substitute nueical alues and ealuate : π π 6 7 ( ) s h 6s /s 7.h

30 Chapte [SS] A supeconductin aity ete can easue chanes in aity of the ode Δ/.. (a) You ae hidin behind a tee holdin the ete, and you 8-k fiend appoaches the tee fo the othe side. How close to you can you fiend et befoe the ete detects a chane in due to his pesence? (b) You ae in a hot ai balloon and ae usin the ete to deteine the ate of ascent (assue the balloon has constant acceleation). What is the sallest chane in altitude that esults in a detectable chane in the aitational field of ath? Pictue the Poble We can deteine the axiu ane at which an object with a ien ass can be detected by substitutin the equation fo the aitational field in the expession fo the esolution of the ete and solin fo the distance. Diffeentiatin () with espect to, sepaatin aiables to obtain d/, and appoxiatin Δ with d will allow us to deteine the etical chane in the position of the aity ete in ath s aitational field is detectable. (a) xpess the aitational field of ath: xpess the aitational field due to the ass (assued to be a point ass) of you fiend and elate it to the esolution of the ete: () G.. Solin fo yields:. Substitute nueical alues and 6. ealuate : ( ) ( 8 k) k (b) Diffeentiate () and siplify to obtain: d G G d Sepaate aiables to obtain: d d Appoxiatin d with Δ, ealuate Δ with : Δ 6 (. )( 6.7 ).9 μ

31 Gaity Suppose that the attactie inteaction between a sta of ass and a planet of ass << is of the fo K/, whee K is the aitational constant. What would be the elation between the adius of the planet s cicula obit and its peiod? Pictue the Poble We can use the law of aity and Newton s nd law to elate the foce exeted on the planet by the sta to its obital speed and the definition of the peiod to elate it to the adius of the obit. he peiod of the planet is elated to its obital speed: π () Usin the law of aity and Newton s nd law, elate the foce exeted on the planet by the sta to its centipetal acceleation: K K net Substitute fo in equation () to obtain: π K [SS] ath s adius is 67 k and the oon s adius is 78 k. he acceleation of aity at the suface of the oon is.6 /s. What is the atio of the aeae density of the oon to that of ath? Pictue the Poble We can use the definitions of the aitational fields at the sufaces of ath and the oon to expess the acceleations due to aity at these locations in tes of the aeae densities of ath and the oon. xpessin the atio of these acceleations will lead us to the atio of the densities. xpess the acceleation due to aity at the suface of ath in tes of ath s aeae density: Gρ π Gρ V Gρ π he acceleation due to aity at the suface of the oon in tes of the oon s aeae density is: Gρ π Diide the second of these equations by the fist to obtain: ρ ρ ρ ρ

32 Chapte Substitute nueical alues and ealuate 6 ρ (.6 /s )( 6.7 ) ρ 6 : ρ ( 9.8/s )(.78 ρ.65 Gaitational and Inetial ass he weiht of a standad object defined as hain a ass of exactly. k is easued to be 9.8 N. In the sae laboatoy, a second object weihs 56.6 N. (a) What is the ass of the second object? (b) Is the ass you deteined in Pat (a) aitational o inetial ass? Pictue the Poble Newton s nd law of otion elates the weihts of these two objects to thei asses and the acceleation due to aity. (a) Apply Newton s nd law to the standad object: Apply Newton s nd law to the object of unknown ass: liinate between these two equations and sole fo : Substitute nueical alues and ealuate : net w net w w w 56.6 N 9.8N (. k) 5.77 k (b) Because this esult is deteined by the effect on of ath s aitational field, it is the aitational ass of the second object. 5 he Pinciple of quialence states that the fee-fall acceleation of any object in a aitational field is independent of the ass of the object. his can be deduced fo the law of uniesal aitation, but how well does it hold up expeientally? he oll-kotko-dicke expeient pefoed in the 96s indicates that the fee-fall acceleation is independent of ass to at least pat in. Suppose two objects ae siultaneously eleased fo est in a unifo aitational field. Also, suppose one of the objects falls with a constant acceleation of exactly 9.8 /s while the othe falls with a constant acceleation that is eate than 9.8 /s by one pat in. How fa will the fist object hae fallen when the second object has fallen. fathe than it has? Note that this estiate poides only an uppe bound on the diffeence in the acceleations; ost physicists beliee that thee is no diffeence in the acceleations.

33 Gaity Pictue the Poble Notin that ~ ~, let the acceleation of aity on the fist object be, and on the second be. We can use a constant-acceleation equation to expess the diffeence in the distances fallen by each object and then elate the aeae distance fallen by the two objects to obtain an expession fo which we can appoxiate the distance they would hae to fall befoe we iht easue a diffeence in thei fall distances eate than. xpess the diffeence Δd in the distances fallen by the two objects in tie t: Δ d d d xpess the distances fallen by each of the objects in tie t: d t and d t Substitute fo d and d to obtain: ( ) Δ d t t t elate the aeae distance d fallen by the two objects to thei tie of fall: d t t d Substitute fo t to obtain: Δd d Δ Δ d d Δd Δ Substitute nueical alues and ealuate d: Gaitational Potential ney d 9 ( )( ) 6 (a) If we take the potential eney of a -k and ath to be zeo when the two ae sepaated by an infinite distance, what is the potential eney when the object is at the suface of ath? (b) ind the potential eney of the sae object at a heiht aboe ath s suface equal to ath s adius. (c) ind the escape speed fo a body pojected fo this heiht. Pictue the Poble Choosin the zeo of aitational potential eney to be at infinite sepaation yields, as the potential eney of a two-body syste in which the objects ae sepaated by a distance, U ( ), whee and ae the asses of the two bodies. In ode fo an object to just escape a aitational field fo a paticula location, it ust hae enouh kinetic eney so that its total eney is zeo. (a) Lettin U( ), expess the aitational potential eney of ath-object syste: U () ()

34 Chapte Substitute fo and siplify to obtain: U ( ) Substitute nueical alues and ealuate U( ): U 6 9 ( ) ( k)( 9.8N / k)( 6.7 ) 6.5 J (b) aluate equation () with : U ( ) Substitute nueical alues and ealuate U( ): U ( ) ( k)( 9.8N / k)( 6.7 ). J. J (c) xpess the condition that an object ust satisfy in ode to escape fo ath s aitational field fo a heiht aboe its suface: K ( ) + U ( ) e o ( ) e + U Solin fo e yields: U ( ) e Substitute nueical alues and ealuate e : 9 (. J) 7.9k/s e k 7 [SS] ind the escape speed fo a pojectile leain the suface of the oon. he acceleation of aity on the oon is.66 ties that on ath and the oon s adius is.7. Pictue the Poble he escape speed fo the oon o ath is ien by e, whee and epesent the asses and adii of the oon o ath. xpess the escape speed fo the oon: e. ()

35 Gaity 5 xpess the escape speed fo ath: e. () Diide equation () by equation () to obtain: e. e. Solin fo e, yields: e. e. Substitute nueical alues and ealuate e, : e. (.66)(.7)(. k/s).8 k/s 8 What initial speed would a paticle hae to be ien at the suface of ath if it is to hae a final speed that is equal to its escape speed when it is ey fa fo ath? Nelect any effects due to ai esistance. Pictue the Poble Let the zeo of aitational potential eney be at infinity, epesent the ass of the paticle, and the subscipt efe to ath. When the paticle is ey fa fo ath, the aitational potential eney of the ath-paticle syste is zeo. We ll use conseation of eney to elate the initial potential and kinetic eneies of the paticle-ath syste to the final kinetic eney of the paticle. Use conseation of eney to elate the initial eney of the syste to its eney when the paticle is ey fa away: Substitute in equation () to obtain: Solin fo i yields: K f Ki + U f U i o, because U f, K K U () ( ) ( ) ( ) i +, i + o, because i +

36 6 Chapte Substitute nueical alues and ealuate i : 6 (. /s) + ( 9.8/s )( 6.7 ) 5.8k/s i 9 While tyin to wok out its budet fo the next fiscal yeas, NASA would like to epot to the nation a ouh estiate of the cost (pe kiloa) to launch a oden satellite into nea-ath obit. You ae chosen fo this task, because you know physics and accountin. (a) Deteine the eney, in kw h, necessay to place.-k object in low-ath obit. In low-ath obit, the heiht of the object aboe the suface of ath is uch salle than ath s adius. ake the obital heiht to be k. (b) If this eney can be obtained at a typical electical eney ate of $.5/kW h, what is the iniu cost of launchin a -k satellite into low-ath obit? Nelect any effects due to ai esistance. Pictue the Poble We can use the expession fo the total eney of a satellite to find the eney equied to place in a low-ath obit. (a) he total eney of a satellite in a low-ath obit is ien by: Substitutin fo U yields: o a nea-ath obit, ath and the aount of eney equied to place the satellite in obit becoes: K + U U athsatellite whee is the obital adius and the inus sin indicates the satellite is bound to ath. ath ath satellite Substitute nueical alues and ealuate : ( 6.67 N / k )( 5.98 k)(. k) 6 ( 6.7 ) 8.7 kw h kw h. J.6J (b) xpess the cost of this poject in tes of the ass of the satellite: equied eney Cost ate k satellite