SUMMA RY. The goal of Chapter 13 has been to use Newton s theory of gravity to understand the motion of satellites and planets. m b.

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1 Sua 371 SUA Y The goal of Chapte 13 has been to use Newton s theo of gavit to undestand the otion of satellites and planets. Geneal Pinciples Newton s Theo of Gavit 1. Two objects with asses and a distance apat eet attactive gavitational foces on each othe of agnitude F on = F on = G 2 F on whee the gavitational constant is G = 6.67 * N 2 /kg Gavitational ass and inetial ass ae equivalent. F on The foces ae an action/eaction pai. 3. Newton s thee laws of otion appl to all objects in the univese. Ipotant Concepts Obital otion of a planet (o satellite) is descibed b Keple s laws: 1. Obits ae ellipses with the sun (o planet) at one focus. 2. A line between the sun and the planet sweeps out equal aeas duing equal intevals of tie. 3. The squae of the planet s peiod T is popotional to the cube of the obit s seiajo ais. Cicula obits ae a special case of an ellipse. Fo a cicula obit aound a ass, v = B G Swept-out aea and T 2 = 1 4p2 G 2 3 b Seiajo ais v Consevation of angula oentu The angula oentu L = v sin b eains constant thoughout the obit. Keple s second law is a consequence of this law. Obital enegetics A satellite s echanical eneg E ech = K + U g is conseved, whee the gavitational potential eneg is U g = - G Fo cicula obits, K = U g and E ech = 1 2 U g. Negative total eneg is chaacteistic of a bound sste. Applications Fo a planet of ass and adius, The fee-fall acceleation on the suface is g suface = G 2 The escape speed is v escape = B 2G The adius of a geosnchonous obit is geo = 1 G 4p 2 T 2 2 1/3 v geo Tes and Notation cosolog Keple s laws gavitational foce Newton s law of gavit gavitational constant, G gavitational ass pinciple of equivalence Newton s theo of gavit escape speed satellite geosnchonous obit bound sste

2 372 chapte 13. Newton s Theo of Gavit C ONCEPTU AL QUESTI ONS 1. Is the eath s gavitational foce on the sun lage than, salle than, o equal to the sun s gavitational foce on the eath? Eplain. 2. The gavitational foce of a sta on obiting planet 1 is F 1. Planet 2, which is twice as assive as planet 1 and obits at twice the distance fo the sta, epeiences gavitational foce F 2. What is the atio F 1 /F 2? 3. A 1000 kg satellite and a 2000 kg satellite follow eactl the sae obit aound the eath. a. What is the atio F 1 /F 2 of the foce on the fist satellite to that on the second satellite? b. What is the atio a 1 /a 2 of the acceleation of the fist satellite to that of the second satellite? 4. How fa awa fo the eath ust an obiting spacecaft be fo the astonauts inside to be weightless? Eplain. 5. A space shuttle astonaut is woking outside the shuttle as it obits the eath. If he dops a hae, will it fall to eath? Eplain wh o wh not. 6. The fee-fall acceleation at the suface of planet 1 is 20 /s 2. The adius and the ass of planet 2 ae twice those of planet 1. What is g on planet 2? 7. Wh is the gavitational potential eneg of two asses negative? Note that saing because that s what the equation gives is not an eplanation. 8. The escape speed fo Planet X is 10,000 /s. Planet Y has the sae adius as Planet X but is twice as dense. What is the escape speed fo Planet Y? 9. The ass of Jupite is 300 ties the ass of the eath. Jupite obits the sun with T Jupite = 11.9 in an obit with Jupite = 5.2 eath. Suppose the eath could be oved to the distance of Jupite and placed in a cicula obit aound the sun. Which of the following descibes the eath s new peiod? Eplain. a. 1 b. Between 1 and 11.9 c d. oe than 11.9 e. It would depend on the eath s speed. f. It s ipossible fo a planet of eath s ass to obit at the distance of Jupite. 10. Satellites in nea-eath obit epeience a ve slight dag due to the eteel thin uppe atosphee. These satellites slowl but suel spial inwad, whee the finall bun up as the each the thicke lowe levels of the atosphee. The adius deceases so slowl that ou can conside the satellite to have a cicula obit at all ties. As a satellite spials inwad, does it speed up, slow down, o aintain the sae speed? Eplain. EXE C I SES AND POBLE S Pobles labeled Eecises Section 13.3 Newton s Law of Gavit integate ateial fo ealie chaptes. 1. What is the atio of the sun s gavitational foce on ou to the eath s gavitational foce on ou? 2. The centes of a 10 kg lead ball and a 100 g lead ball ae sepaated b 10 c. a. What gavitational foce does each eet on the othe? b. What is the atio of this gavitational foce to the gavitational foce of the eath on the 100 g ball? 3. What is the atio of the sun s gavitational foce on the oon to the eath s gavitational foce on the oon? 4. A 1.0--diaete lead sphee has a ass of 5900 kg. A dust paticle ests on the suface. What is the atio of the gavitational foce of the sphee on the dust paticle to the gavitational foce of the eath on the dust paticle? 5. Estiate the foce of attaction between a 50 kg woan and a 70 kg an sitting 1.0 apat. 6. The space shuttle obits 300 k above the suface of the eath. What is the gavitational foce on a 1.0 kg sphee inside the space shuttle? Section 13.4 Little g and Big G 7. a. What is the fee-fall acceleation at the suface of the sun? b. What is the sun s fee-fall acceleation at the distance of the eath? 8. What is the fee-fall acceleation at the suface of (a) the oon and (b) Jupite? 9. A sensitive gaviete at a ountain obsevato finds that the fee-fall acceleation is /s 2 less than that at sea level. What is the obsevato s altitude? 10. Suppose we could shink the eath without changing its ass. At what faction of its cuent adius would the fee-fall acceleation at the suface be thee ties its pesent value? 11. Planet Z is 10,000 k in diaete. The fee-fall acceleation on Planet Z is 8.0 /s 2. a. What is the ass of Planet Z? b. What is the fee-fall acceleation 10,000 k above Planet Z s noth pole? Section 13.5 Gavitational Potential Eneg 12. An astonaut on eath can thow a ball staight up to a height of 15. How high can he thow the ball on as? 13. What is the escape speed fo Jupite?

3 Eecises and Pobles A ocket is launched staight up fo the eath s suface at a speed of 15,000 /s. What is its speed when it is ve fa awa fo the eath? 15. A space station obits the sun at the sae distance as the eath but on the opposite side of the sun. A sall pobe is fied awa fo the station. What iniu speed does the pobe need to escape the sola sste? 16. You have been visiting a distant planet. You easueents have deteined that the planet s ass is twice that of eath but the fee-fall acceleation at the suface is onl one-fouth as lage. a. What is the planet s adius? b. To get back to eath, ou need to escape the planet. What iniu speed does ou ocket need? Section 13.6 Satellite Obits and Enegies 17. The asteoid belt cicles the sun between the obits of as and Jupite. One asteoid has a peiod of 5.0 eath eas. What ae the asteoid s obital adius and speed? 18. Use infoation about the eath and its obit to deteine the ass of the sun. 19. Planet X obits the sta Oega with a ea that is 200 eath das long. Planet Y cicles Oega at fou ties the distance of Planet X. How long is a ea on Planet Y? 20. You ae the science office on a visit to a distant sola sste. Pio to landing on a planet ou easue its diaete to be 1.8 * 10 7 and its otation peiod to be 22.3 hous. You have peviousl deteined that the planet obits 2.2 * fo its sta with a peiod of 402 eath das. Once on the suface ou find that the fee-fall acceleation is 12.2 /s 2. What is the ass of (a) the planet and (b) the sta? 21. Thee satellites obit a planet of adius, as shown in FIGUE EX Satellites S 1 and S 3 have ass. Satellite S 2 has ass 2. Satellite S 1 obits in 250 inutes and the foce on S 1 is 10,000 N. a. What ae the peiods of S 2 and S 3? b. What ae the foces on S 2 and S 3? c. What is the kinetic-eneg atio K 1 /K 3 fo S 1 and S 3? FIGUE EX A satellite obits the sun with a peiod of 1.0 da. What is the adius of its obit? 23. An eath satellite oves in a cicula obit at a speed of 5500 /s. What is its obital peiod? 24. What ae the speed and altitude of a geosnchonous satellite obiting as? as otates on its ais once eve 24.8 hous. Pobles S Two spheical objects have a cobined ass of 150 kg. The gavitational attaction between the is 8.00 * 10-6 N when thei centes ae 20 c apat. What is the ass of each? S 1 2 S FIGUE P13.26 shows thee asses. What ae the agnitude and the diection of the net gavitational foce on (a) the 20.0 kg ass and (b) the 5.0 kg ass? Give the diection as an angle cw o ccw fo the -ais. 20 c 20.0 kg 5.0 kg 10 c FIGUE P kg 10.0 kg FIGUE P What ae the agnitude and diection of the net gavitational foce on the 20.0 kg ass in FIGUE P13.27? 28. What is the total gavitational potential eneg of the thee asses in FIGUE P13.26? 29. What is the total gavitational potential eneg of the thee asses in FIGUE P13.27? 30. Two 100 kg lead sphees ae suspended fo 100--long assless cables. The tops of the cables have been caefull anchoed eactl 1 apat. What is the distance between the centes of the sphees? 31. A 20 kg sphee is at the oigin and a 10 kg sphee is at = 20 c. At what position on the -ais could ou place a sall ass such that the net gavitational foce on it due to the sphees is zeo? 32. a. At what height above the eath is the acceleation due to gavit 10% of its value at the suface? b. What is the speed of a satellite obiting at that height? 33. A 1.0 kg object is eleased fo est 500 k ( 300 iles) above the eath. a. What is its ipact speed as it hits the gound? Ignoe ai esistance. b. What would the ipact speed be if the eath wee flat? c. B what pecentage is the flat-eath calculation in eo? 34. An object of ass is dopped fo height h above a planet of ass and adius. Find an epession fo the object s speed as it hits the gound. 35. A pojectile is shot staight up fo the eath s suface at a speed of 10,000 k/h. How high does it go? 36. Two eteooids ae heading fo eath. Thei speeds as the coss the oon s obit ae 2.0 k/s. a. The fist eteooid is heading staight fo eath. What is its speed of ipact? b. The second isses the eath b 5000 k. What is its speed at its closest point? 37. A bina sta sste has two stas, each with the sae ass as ou sun, sepaated b 1.0 * A coet is ve fa awa and essentiall at est. Slowl but suel, gavit pulls the coet towad the stas. Suppose the coet tavels along a staight line that passes though the idpoint between the two stas. What is the coet s speed at the idpoint? 38. Suppose that on eath ou can jup staight up a distance of 50 c. Can ou escape fo a 4.0-k-diaete asteoid with a ass of 1.0 * kg? 10.0 kg 5.0 c 5.0 c 20.0 c 20.0 kg

4 374 chapte 13. Newton s Theo of Gavit 39. A pojectile is fied staight awa fo the oon fo a base on the fa side of the oon, awa fo the eath. What is the pojectile s escape speed fo the eath-oon sste? 40. Two spheical asteoids have the sae adius. Asteoid 1 has ass and asteoid 2 has ass 2. The two asteoids ae eleased fo est with distance 10 between thei centes. What is the speed of each asteoid just befoe the collide? Hint: You will need to use two consevation laws. 41. Two Jupite-size planets ae eleased fo est 1.0 * apat. What ae thei speeds as the cash togethe? 42. A staship is cicling a distant planet of adius. The astonauts find that the fee-fall acceleation at thei altitude is half the value at the planet s suface. How fa above the suface ae the obiting? You answe will be a ultiple of. 43. Thee stas, each with the ass and adius of ou sun, fo an equilateal tiangle 5.0 * 10 9 on a side. If all thee ae siultaneousl eleased fo est, what ae thei speeds as the cash togethe in the cente? 44. The two stas in a bina sta sste have asses 2.0 * kg and 6.0 * kg. The ae sepaated b 2.0 * What ae a. The sste s otation peiod, in eas? b. The speed of each sta? 45. A 4000 kg luna lande is in obit 50 k above the suface of the oon. It needs to ove out to a 300-k-high obit in ode to link up with the othe ship that will take the astonauts hoe. How uch wok ust the thustes do? 46. The space shuttle is in a 250-k-high cicula obit. It needs to each a 610-k-high cicula obit to catch the Hubble Space Telescope fo epais. The shuttle s ass is 75,000 kg. How uch eneg is equied to boost it to the new obit? 47. In 2000, NASA placed a satellite in obit aound an asteoid. Conside a spheical asteoid with a ass of 1.0 * kg and a adius of 8.8 k. a. What is the speed of a satellite obiting 5.0 k above the suface? b. What is the escape speed fo the asteoid? 48. NASA would like to place a satellite in obit aound the oon such that the satellite alwas eains in the sae position ove the luna suface. What is the satellite s altitude? 49. A satellite obiting the eath is diectl ove a point on the equato at 12:00 idnight eve two das. It is not ove that point at an tie in between. What is the adius of the satellite s obit? 50. FIGUE P13.50 shows two planets of ass obiting a sta of ass. The planets ae in the sae obit, with adius, but ae alwas at opposite ends of a diaete. Find an eact epession fo the obital peiod T. Hint: Each planet feels two foces. FIGUE P Figue showed a gaph of log T vesus log fo the planeta data given in Table Such a gaph is called a loglog gaph. The scales in Figue ae logaithic, not linea, eaning that each division along the ais coesponds to a facto of 10 incease in the value. Stictl speaking, the coect labels on the -ais should be 7, 8, 9, and 10 because these ae the logaiths of 10 7, p, a. Conside two quantities u and v that ae elated b the epession v p = Cu q, whee C is a constant. The eponents p and q ae not necessail integes. Define = log u and = log v. Find an epession fo in tes of. b. What shape will a gaph of vesus have? Eplain. c. What slope will a gaph of vesus have? Eplain. d. Use the epeientall deteined best-fit line in Figue to find the ass of the sun. 52. Lage stas can eplode as the finish buning thei nuclea fuel, causing a supenova. The eplosion blows awa the oute laes of the sta. Accoding to Newton s thid law, the foces that push the oute laes awa have eaction foces that ae inwadl diected on the coe of the sta. These foces copess the coe and can cause the coe to undego a gavitational collapse. The gavitational foces keep pulling all the atte togethe tighte and tighte, cushing atos out of eistence. Unde these etee conditions, a poton and an electon can be squeezed togethe to fo a neuton. If the collapse is halted when the neutons all coe into contact with each othe, the esult is an object called a neuton sta, an entie sta consisting of solid nuclea atte. an neuton stas otate about thei ais with a peiod of 1 s and, as the do so, send out a pulse of electoagnetic waves once a second. These stas wee discoveed in the 1960s and ae called pulsas. a. Conside a neuton sta with a ass equal to the sun, a adius of 10 k, and a otation peiod of 1.0 s. What is the speed of a point on the equato of the sta? b. What is g at the suface of this neuton sta? c. A stationa 1.0 kg ass has a weight on eath of 9.8 N. What would be its weight on the sta? d. How an evolutions pe inute ae ade b a satellite obiting 1.0 k above the suface? e. What is the adius of a geosnchonous obit about the neuton sta? 53. The sola sste is 25,000 light eas fo the cente of ou ilk Wa gala. One light ea is the distance light tavels in one ea at a speed of 3.0 * 10 8 /s. Astonoes have deteined that the sola sste is obiting the cente of the gala at a speed of 230 k/s. a. Assuing the obit is cicula, what is the peiod of the sola sste s obit? Give ou answe in eas. b. Ou sola sste was foed oughl 5 billion eas ago. How an obits has it copleted? c. The gavitational foce on the sola sste is the net foce due to all the atte inside ou obit. ost of that atte is concentated nea the cente of the gala. Assue that the atte has a spheical distibution, like a giant sta. What is the appoiate ass of the galactic cente? d. Assue that the sun is a tpical sta with a tpical ass. If galactic atte is ade up of stas, appoiatel how an stas ae in the cente of the gala? Astonoes have spent an eas ting to deteine how an stas thee ae in the ilk Wa. The nube of stas sees to be onl about 10% of what ou found in pat d. In othe wods, about 90% of the ass of the gala appeas to be in soe fo othe than stas. This is called the dak atte of the univese. No one knows what the dak atte is. This is one of the outstanding scientific questions of ou da. 54. Thee stas, each with the ass of ou sun, fo an equilateal tiangle with sides 1.0 * long. (This tiangle would just about fit within the obit of Jupite.) The tiangle has to otate, because othewise the stas would cash togethe in the cente. What is the peiod of otation? Give ou answe in eas.

5 Eecises and Pobles Pluto oves in a fail elliptical obit aound the sun. Pluto s speed at its closest appoach of 4.43 * 10 9 k is 6.12 k/s. What is Pluto s speed at the ost distant point in its obit, whee it is 7.30 * 10 9 k fo the sun? 56. ecu oves in a fail elliptical obit aound the sun. ecu s speed is 38.8 k/s when it is at its ost distant point, 6.99 * fo the sun. How fa is ecu fo the sun at its closest point, whee its speed is 59.0 k/s? 57. Coets ove aound the sun in ve elliptical obits. At its closet appoach, in 1986, Coet Halle was 8.79 * 10 7 k fo the sun and oving with a speed of 54.6 k/s. What was the coet s speed when it cossed Neptune s obit in 2006? 58. A spaceship is in a cicula obit of adius 0 about a planet of ass. A bief but intense fiing of its engine in the fowad diection deceases the spaceship s speed b 50%. This causes the spaceship to ove into an elliptical obit. a. What is the spaceship s new speed, just afte the ocket bun is coplete, in tes of, G, and 0? b. In tes of 0, what ae the spaceship s aiu and iniu distances fo the planet in its new obit? In Pobles 59 though 61 ou ae given the equation(s) used to solve a poble. Fo each of these, ou ae to a. Wite a ealistic poble fo which this is the coect equation(s). b. Daw a pictoial epesentation. c. Finish the solution of the poble. 59. (6.67 * N 2 /kg 2 )(5.68 * kg) 2 = (6.67 * N 2 /kg 2 )(5.98 * kg) (6.37 * 10 6 ) (6.67 * N 2 /kg 2 )(5.98 * kg)(1000 kg) 2 (1000 kg)(1997 /s)2 = (100 kg)v (6.67 * N 2 /kg 2 )(7.36 * kg)(100 kg) 1.74 * 10 6 = 0 - (6.67 * N 2 /kg 2 )(7.36 * kg)(100 kg) 3.48 * 10 6 Challenge Pobles 62. A satellite in a cicula obit of adius has peiod T. A satellite in a neab obit with adius +, whee V, has the ve slightl diffeent peiod T + T. a. Show that T T = 3 2 b. Two eath satellites ae in paallel obits with adii 6700 k and 6701 k. One da the pass each othe, 1 k apat, along a line adiall outwad fo the eath. How long will it be until the ae again 1 k apat? 63. In 1996, the Sola and Heliospheic Obsevato (SOHO) was paked in an obit slightl inside the eath s obit, as shown in FIGUE CP The satellite s peiod in this obit is eactl one ea, so it eains fied elative to the eath. At this point, called a Lagange point, the light fo the sun is neve blocked b the eath, et the satellite eains neab so that data ae easil tansitted to eath. What is SOHO s distance fo the eath? FIGUE CP13.63 Hint: Use the binoial appoiation. SOHO s distance fo the eath is uch less than the eath s distance fo the sun. 64. A pojectile is fied fo the eath in the diection of the eath s otion aound the sun. What iniu speed ust the pojectile have elative to the eath to escape the sola sste? Ignoe the eath s otation. Hint: This is a thee-pat poble. Fist find the speed a pojectile at the eath s distance needs to escape the sun. Tansfo that speed into the eath s efeence fae, then deteine how fast the pojectile ust be launched to have this speed when fa fo the eath. 65. You job with NASA is to onito satellite obits. One da, duing a outine suve, ou find that a 400 kg satellite in a 1000-khigh cicula obit is going to collide with a salle 100 kg satellite taveling in the sae obit but in the opposite diection. Knowing the constuction of the two satellites, ou epect the will becoe eneshed into a single piece of space debis. When ou notif ou boss of this ipending collision, he asks ou to quickl deteine whethe the space debis will continue to obit o cash into the eath. What will the outcoe be? 66. While visiting Planet Phsics, ou toss a ock staight up at 11 /s and catch it 2.5 s late. While ou visit the suface, ou cuise ship obits at an altitude equal to the planet s adius eve 230 in. What ae the (a) ass and (b) adius of Planet Phsics? 67. A oon lande is obiting the oon at an altitude of 1000 k. B what pecentage ust it decease its speed so as to just gaze the oon s suface one-half peiod late? 68. Let s look in oe detail at how a satellite is oved fo one cicula obit to anothe. FIGUE CP13.68 shows two cicula obits, of adii 1 and 2, and an elliptical obit that connects the. Points 1 and 2 ae at the ends of the seiajo ais of the ellipse. FIGUE CP13.68 Eath Satellite s otion is snchonized with the eath s SOHO Inne obit Oute obit a. A satellite oving along the elliptical obit has to satisf two consevation laws. Use these two laws to pove that the velocities at points 1 and 2 ae v = 1 = B 2G( 2 / 1 ) and v = 2 = B 2G( 1 / 2 ) Tansfe ellipse

6 376 chapte 13. Newton s Theo of Gavit The pie indicates that these ae the velocities on the elliptical obit. Both educe to Equation if 1 = 2 =. b. Conside a 1000 kg counications satellite that needs to be boosted fo an obit 300 k above the eath to a geosnchonous obit 35,900 k above the eath. Find the velocit v 1 on the inne cicula obit and the velocit v = 1 at the low point on the elliptical obit that spans the two cicula obits. c. How uch wok ust the ocket oto do to tansfe the satellite fo the cicula obit to the elliptical obit? d. Now find the velocit v = 2 at the high point of the elliptical obit and the velocit v 2 of the oute cicula obit. e. How uch wok ust the ocket oto do to tansfe the satellite fo the elliptical obit to the oute cicula obit? f. Copute the total wok done and copae ou answe to the esult of Eaple FIGUE CP13.69 shows a paticle of ass at distance fo the cente of a ve thin clinde of ass and length L. The paticle is outside the clinde, so 7 L/2. a. Calculate the gavitational potential eneg of these two asses. b. Use what ou know about the elationship between foce and potential eneg to find the agnitude of the gavitational foce on when it is at position. 70. FIGUE CP13.70 shows a paticle of ass at distance along the ais of a ve thin ing of ass and adius. a. Calculate the gavitational potential eneg of these two asses. b. Use what ou know about the elationship between foce and potential eneg to find the agnitude of the gavitational foce on when it is at position. z FIGUE CP13.69 L FIGUE CP13.70 STOP TO THINK ANSWES Stop to Think 13.1: e. The acceleation deceases invesel with the squae of the distance. At height e, the distance fo the cente of the eath is 2 e. Stop to Think 13.2: c. Newton s thid law equies F 1 on 2 = F 2 on 1. Stop to Think 13.3: b. g suface = G/ 2. Because of the squae, a adius twice as lage balances a ass fou ties as lage. Stop to Think 13.4: In absolute value, U e + U a U b U d + U c. 0 U g 0 is popotional to 1 2 /. Stop to Think 13.5: a. T 2 is popotional to 3, o T is popotional to 3/2. 4 3/2 = 8.