Essential University Physics Volume 1 Richard Wolfson Second Edition

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1 Essential Univesity Physics Volume 1 Richad Wolfson Second Edition

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3 CHAPTER SUMMARY Big Pictue The big idea hee is univesal gavitation an attactive foce that acts between all matte with a stength that depends diectly on the poduct of two inteacting masses and invesely on the squae of the distance between them. Gavitation is esponsible fo the familia behavio of falling objects and also fo the obits of planets and satellites. Depending on enegy, obits may be closed (elliptical/cicula) o open (hypebolic/paabolic). Gavity govens both the falling apple and the obiting Moon. Open (hypebola) Closed (ellipse/ cicle) Bodeline (paabola) Key Concepts and Equations Mathematically, Newton s law of univesal gavitation descibes the attactive foce F between two masses and located a distance apat: m 1 m 2 F 5 Gm 1 m 2 1univesal gavitation2 F F m 1 m 2 This equation applies to point masses of negligible size and to spheically symmetic masses of any size. It s an excellent appoximation fo any objects whose size is much smalle than thei sepaation. In all cases, is measued fom the centes of the gavitating objects. Because the stength of gavity vaies with distance, potential-enegy changes ove lage distances aen t just a poduct of foce and distance. Integation shows that the potential enegy change DU involved in moving a mass m oiginally a distance fom the cente of a mass M to a distance is With gavity, it s convenient to choose the zeo of potential enegy at infinity; then fo the potential enegy of a mass m located a distance fom the cente of a mass M. Applications DU 5 GMm a b 1change in potential enegy2 GMm U 52 1potential enegy, U 5 0 at infinity2 U 5 0 Potential enegy, U Eath Distance, Satellite The satellite s potential enegy is negative. It would take this much enegy fo the satellite to escape infinitely fa fom Eath. A total enegy kinetic plus potential of zeo maks the dividing line between closed and open obits. An object located a distance fom a gavitating mass M must have at least the escape speed to achieve an open obit and escape M s vicinity foeve: 2GM v esc 5 A The gavitational field concept povides a way to descibe gavity that avoids the toublesome actionat-a-distance. A gavitating mass ceates a field in the space aound it, and a second mass esponds to the field in its immediate vicinity. Gavitational field F Foce aises fom field at Moon s location. Cicula obits ae eadily analyzed using Newton s laws and concepts fom cicula motion. A cicula obit of adius about a mass M has a peiod given by T 2 5 4p2 3 GM Its kinetic and potential enegies ae elated by U 522K. Total enegy is negative, as appopiate fo a closed obit, and the object actually moves faste the lowe its total enegy. A special obit is the geosynchonous obit, paallel to Eath s equato at an altitude of about 36,000 km. Hee the obital peiod is 24 h, so a satellite in geosynchonous obit appeas fom Eath s suface to be fixed in the sky. TV, communications, and weathe satellites use geosynchonous obits. 157

4 Gavity Answes to Chapte Questions Answe to Chapte Opening Question The satellite obits Eath in 24 hous, so fom Eath s suface it appeas at a fixed position in the sky. Answes to GOT IT? Questions 1. (d) Quadupled. If the oiginal distance wee, the oiginal foce would be popotional to 1/. At half that distance, the foce is popotional to 1/1/ /. 2. Hypebolic. paabolic. elliptical. cicula. 3. (a), (c), and (d). Since B has highe total enegy, it must have lowe kinetic enegy and is theefoe moving slowe. B is fathe fom the gavitating body, so its potential enegy is highe still negative, but less so than A s. Fo cicula obits, the atio of potential enegy to total enegy is always the same namely, U 5 2E. 158

5 Gavity Poblem Set Fo homewok assigned on MasteingPhysics, go to Biology and/o medical-elated poblems C Compute poblems BIO Fo Thought and Discussion 1. What do Newton s apple and the Moon have in common? 2. Explain the diffeence between G and g. 3. When you stand on Eath, the distance between you and Eath is zeo. So why isn t the gavitational foce infinite? 4. The foce of gavity on an object is popotional to the object s mass, yet all objects fall with the same gavitational acceleation. Why? 5. A fiend who knows nothing about physics asks what keeps an obiting satellite fom falling to Eath. Give an answe that will satisfy you fiend. 6. Could you put a satellite in an obit that keeps it stationay ove the south pole? Explain. 7. Why ae satellites geneally launched eastwad and fom low latitudes? (Hint: Think about Eath s otation.) 8. Given Eath s mass, the Moon s distance and obital peiod, and the value of G, could you calculate the Moon s mass? If yes, how? If no, why not? 9. How should a satellite be launched so that its obit takes it ove evey point on the (otating) Eath? 10. Does the gavitational foce of the Sun do wok on a planet in a cicula obit? In an elliptical obit? Explain. Execises and Poblems Execises Section 2 Univesal Gavitation 11. Space exploes land on a planet that has the same mass as Eath, but find they weigh twice as much as they would on Eath. What s the planet s adius? 12. Use data fo the Moon s obit fom Appendix: Astophysical Data to compute the Moon s acceleation in its cicula obit, and veify that the esult is consistent with Newton s law of gavitation. 13. To what faction of its cuent adius would Eath have to shink (with no change in mass) fo the gavitational acceleation at its suface to tiple? 14. Calculate the gavitational acceleation at the suface of (a) Mecuy and (b) Satun s moon Titan. 15. Two identical lead sphees with thei centes 14 cm apat attact each othe with a 0.25-mN foce. Find thei mass. 16. What s the appoximate value of the gavitational foce between a 67-kg astonaut and a 73,000-kg spacecaft when they e 84 m apat? 17. A sensitive gavimete is caied to the top of Chicago s Willis (fomely Seas) Towe, whee its eading fo the acceleation of gavity is 1.36 mm/s 2 lowe than at steet level. Find the building s height. Section 3 Obital Motion 18. At what altitude will a satellite complete a cicula obit of Eath in 2.0 h? 19. Find the speed of a satellite in geosynchonous obit. 20. Mas s obit has a diamete.52 times that of Eath s obit. How long does it take Mas to obit the Sun? 21. Calculate the obital peiod fo Jupite s moon Io, which obits km fom the planet s cente. 22. An astonaut hits a golf ball hoizontally fom the top of a luna mountain so fast that it goes into cicula obit. What s its obital peiod? 23. The Mas Reconnaissance Obite cicles the ed planet with a 112-min peiod. What s the spacecaft s altitude? Section 4 Gavitational Enegy 24. Eath s distance fom the Sun vaies fom 147 Gm at peihelion to 152 Gm at aphelion because its obit isn t quite cicula. Find the change in potential enegy as Eath goes fom peihelion to aphelion. 25. So-called subobital missions take scientific instuments into space fo bief peiods without the expense of getting into obit; thei tajectoies ae often simple up and down vetical paths. How much enegy does it take to launch a 230-kg instument on a vetical tajectoy that peaks at 1800 km altitude? 26. A ocket is launched vetically upwad fom Eath s suface at 5.1 km/s. What s its maximum altitude? 27. What vetical launch speed is necessay to get a ocket to an altitude of 1100 km? 28. Find the enegy necessay to put 1 kg, initially at est on Eath s suface, into geosynchonous obit. 29. What s the total mechanical enegy associated with Eath s obital motion? 30. The escape speed fom a planet of mass kg is 7.1 km/s. Find the planet s adius. 31. Detemine escape speeds fom (a) Jupite s moon Callisto and (b) a neuton sta, with the Sun s mass cammed into a sphee of adius 6.0 km. 32. To what adius would Eath have to shink, with no change in mass, fo escape speed at its suface to be 30 km/s? Poblems 33. The gavitational acceleation at a planet s suface is 22.5 m/s 2. Find the acceleation at an altitude equal to half the planet s adius. 34. One of the longest-standing athletic ecods is Cuban Javie BIO Sotomayo s 2.45-m high jump. How high could Sotomayo jump on (a) Mas and (b) Eath s Moon? 35. You e the navigato on a spaceship studying an unexploed planet. You ship has just gone into a cicula obit aound the planet, and you detemine that the gavitational acceleation at Fom Chapte 8 of Essential Univesity Physics, Second Edition, Richad Wolfson. Copyight 2012 by Peason Education, Inc. Published by Peason Addison-Wesley. All ights eseved. 159

6 Gavity: Poblem Set you obital altitude is half what it would be at the suface. What do you epot fo you altitude, in tems of the planet s adius? 36. If you e standing on the gound 15 m diectly below the cente of a spheical wate tank containing kg of wate, by what faction is you weight educed due to the wate s gavitational attaction? 37. Given the Moon s obital adius of 384,400 km and peiod of 27.3 days, calculate its acceleation in its cicula obit, and compae with the acceleation of gavity at Eath s suface. Show that the Moon s acceleation is lowe by the atio of the squae of Eath s adius to the squae of the Moon s obital adius, thus confiming the invese-squae law fo the gavitational foce. 38. The equation F x 52 du elates foce to the deivative of dx potential enegy. Use this fact to diffeentiate Equation 6, GMm U12 52, fo gavitational potential enegy, and show that you ecove Newton s law of gavitation. 39. Duing the Apollo Moon landings, one astonaut emained with the command module in luna obit, about 130 km above the suface. Fo half of each obit, this astonaut was completely cut off fom the est of humanity as the spacecaft ounded the fa side of the Moon. How long did this peiod last? 40. A white dwaf is a collapsed sta with oughly the Sun s mass compessed into the size of Eath. What would be (a) the obital speed and (b) the obital peiod fo a spaceship in obit just above the suface of a white dwaf? 41. Given that ou Sun obits the galaxy with a peiod of 200 My at m fom the galactic cente, estimate the galaxy s mass. Assume (incoectly) that the galaxy is essentially spheical and that most of its mass lies inteio to the Sun s obit. 42. You e pepaing an exhibit fo the Golf Hall of Fame, and you ealize that the longest golf shot in histoy was Astonaut Alan Shepad s luna dive. Shepad, swinging single-handed with a golf club attached to a luna sample scoop, claimed his ball went miles and miles. The ecod fo a single-handed golf shot on Eath is 257 m. Could Shepad s ball eally have gone miles and miles? Assume the ball s initial speed is independent of gavitational acceleation. 43. Exact solutions fo gavitational poblems involving moe than two bodies ae notoiously difficult. One solvable poblem involves a configuation of thee equal-mass objects spaced in an equilateal tiangle. Foces due to thei mutual gavitation cause the configuation to otate. Suppose thee identical stas, each of mass M, fom a tiangle of side L. Find an expession fo the peiod of thei obital motion. 44. Satellites A and B ae in cicula obits, with A twice as fa fom Eath s cente as B. How do thei obital peiods compae? 45. The asteoid Pasachoff obits the Sun with peiod 1417 days. Find the semimajo axis of its obit fom Keple s thid law. Use Eath s obital adius and peiod, espectively, as you units of distance and time. 46. We still don t have a pemanent solution fo the disposal of adioactive waste. As a nuclea waste specialist with the Depatment of Enegy, you e asked to evaluate a poposal to shoot waste canistes into the Sun. You need to epot the speed at which a caniste, dopped fom est in the vicinity of Eath s obit, would hit the Sun. What s you answe? 47. At peihelion in Febuay 1986, Comet Halley was km fom the Sun and was moving at 54.6 km/s. What was Halley s speed when it cossed Neptune s obit in 2006? 48. Neglecting ai esistance, to what height would you have to fie a ocket fo the constant-acceleation equations to give a height in eo by 1%? Would those equations oveestimate o undeestimate the height? 49. Show that an object eleased fom est vey fa fom Eath eaches Eath s suface at essentially escape speed. 50. By what facto must an object s speed in cicula obit be inceased to each escape speed fom its obital altitude? 51. You e in chage of tacking celestial objects that might pose a dange to Eath. Astonomes have discoveed a new comet that s moving at 45 km/s as it cosses Eath s obit. Detemine whethe the comet will again etun to Eath s vicinity. 52. Two meteooids ae 250,000 km fom Eath s cente and moving at 2.1 km/s. One is headed staight fo Eath, while the othe is on a path that will come within 8500 km of Eath s cente (Fig. 15). Find the speed (a) of the fist meteooid when it stikes Eath and (b) of the second meteooid at its closest appoach. (c) Will the second meteooid eve etun to Eath s vicinity? 8500 km FIGURE 15 Poblem Neglecting Eath s otation, show that the enegy needed to launch a satellite of mass m into cicula obit at altitude h is a GM Em ba R E 1 2h R E 21R E 1 h2 b. 54. A pojectile is launched vetically upwad fom a planet of mass M and adius R; its initial speed is twice the escape speed. Deive an expession fo its speed as a function of the distance fom the planet s cente. 55. A spacecaft is in cicula obit 5500 km above Eath s suface. How much will its altitude decease if it moves to a new cicula obit whee (a) its obital speed is 10% highe o (b) its obital peiod is 10% shote? 56. Two meteooids ae 160,000 km fom Eath s cente and heading staight towad Eath, one at 10 km/s, the othe at 20 km/s. At what speeds will they stike Eath? 57. Two ockets ae launched fom Eath s suface, one at 12 km/s and the othe at 18 km/s. How fast is each moving when it cosses the Moon s obit? 58. A satellite is in an elliptical obit at altitudes anging fom 230 to 890 km. At its highest point, it s moving at 7.23 km/s. How fast is it moving at its lowest point? 59. A missile s tajectoy takes it to a maximum altitude of 1200 km. If its launch speed is 6.1 km/s, how fast is it moving at the peak of its tajectoy? 60. A 720-kg spacecaft has total enegy TJ and is in cicula obit aound the Sun. Find (a) its obital adius, (b) its kinetic enegy, and (c) its speed. 160

7 Gavity: Poblem Set 61. Mecuy s obital speed vaies fom 38.8 km/s at aphelion to 59.0 km/s at peihelion. If the planet is m fom the Sun s cente at aphelion, how fa is it at peihelion? 62. Show that the fom DU 5 mg D follows fom Equation 5 GMm DU d 5 GMm d 5 GMm 21 ` 5 GMma b when.. [Hint: Wite 5 1Dand apply the binomial appoximation.] 63. Two satellites ae in geosynchonous obit but in diametically opposite positions (Fig. 16). In ode to catch up with the othe, one satellite descends into a lowe cicula obit (see Conceptual Example 1 fom the chapte, Gavity fo a desciption of this maneuve). How fa should it descend if it s to catch up in 10 obits? Neglect ocket fiing times and time spent moving between the two cicula obits. 64. We deived Equation 4 T 2 5 4p2 3 on the assumption that the GM massive gavitating cente emains fixed. Now conside two objects with equal mass M obiting each othe, as shown in Fig. 17. Show that the obital peiod is given by T 2 5 2p 2 d 3 /GM, whee d is the distance between the objects. FIGURE 16 Poblem 63 simplifying assumption that asteoids ae spheical, with aveage density 2500 kg/m 3. Fo safety, make sue even a jumpe capable of 3 m on Eath will etun to the suface. What do you epot fo the minimum asteoid diamete? 67. The Olympic Committee is keeping you busy! You e now asked to conside a poposal fo luna hockey. The ecod speed fo a hockey puck is 168 km/h. Is thee any dange that hockey pucks will go into luna obit? 68. Tidal foces ae popotional to the vaiation in gavity with position. By diffeentiating Equation 1, F 5 Gm 1 m 2, estimate the atio of the tidal foces due to the Sun and the Moon. Compae you answe with the atio of the gavitational foces that the Sun and Moon exet on Eath. 69. Spacecaft that study the Sun ae often placed at the so-called L1 C Lagange point, located sunwad of Eath on the Sun-Eath line. L1 is the point whee Eath s and Sun s gavity togethe poduce an obital peiod of one yea, so that a spacecaft at L1 stays fixed elative to Eath as both planet and spacecaft obit the Sun. This placement ensues an uninteupted view of the Sun, without being peiodically eclipsed by Eath as would occu in Eath obit. Find L1 s location elative to Eath. (Hint: This poblem calls fo numeical methods o solving a highe-ode polynomial equation.) Passage Poblems The Global Positioning System (GPS) uses a constellation of some 30 satellites to povide accuate positioning fo any point on Eath (Fig. 18). GPS eceives time adio signals taveling at the speed of light fom thee of the satellites to find the eceive s position. Signals fom one o moe additional satellites povide coections, eliminating the need fo high-accuacy clocks in individual 20,200 km FIGURE 18 GPS satellites (Passage Poblems 70 73) FIGURE 17 Poblem Tidal effects in the Eath-Moon system cause the Moon s obital peiod to incease at a cuent ate of about 35 ms pe centuy. Assuming the Moon s obit is cicula, to what ate of change in the Eath-Moon distance does this coespond? (Hint: Diffeentiate Keple s thid law, Equation 4.) 66. As a membe of the 2040 Olympic committee, you e consideing a new spot: asteoid jumping. On Eath, wold-class high BIO jumpes outinely clea m. You job is to make sue athletes jumping fom asteoids will etun to the asteoid. Make the GPS eceives. GPS satellites ae in cicula obits at 20,200 km altitude. 70. What s the appoximate obital peiod of GPS satellites? a. 90 min b. 8 h c. 12 h d. 24 h e. 1 week 71. What s the appoximate speed of GPS satellites? a. 9.8 m/s b. 500 m/s c. 1.7 km/s d. 4 km/s e. 12 km/s 161