University Physics Volume I Unit 1: Mechanics Chapter 13: Gravitation Conceptual Questions

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1 OpenStax Univesity Physics Volume I Univesity Physics Volume I Conceptual Questions 1. Action at a distance, such as is the case fo gavity, was once thought to be illogical and theefoe untue. What is the ultimate deteminant of the tuth in science, and why was this action at a distance ultimately accepted? The ultimate tuth is expeimental veification. Field theoy was developed to help explain how foce is exeted without objects being in contact fo both gavity and electomagnetic foces that act at the speed of light. It has only been since the twentieth centuy that we have been able to measue that the foce is not conveyed immediately.. Must enginees take Eath s otation into account when constucting vey tall buildings at any location othe than the equato o vey nea the poles? The centipetal acceleation is not diected along the gavitational foce and theefoe the coect line of the building (i.e., the plumb bob line) is not diected towads the cente of Eath. But enginees use eithe a plumb bob o a tansit, both of which espond to both the diection of gavity and acceleation. No special consideation fo thei location on Eath need be made. 5. It was shown that the enegy equied to lift a satellite into a low Eath obit (the change in potential enegy) is only a small faction of the kinetic enegy needed to keep it in obit. Is this tue fo lage obits? Is thee a tend to the atio of kinetic enegy to change in potential enegy as the size of the obit inceases? As we move to lage obits, the change in potential enegy inceases, wheeas the obital velocity deceases. Hence, the atio is highest nea Eath s suface (technically infinite if we obit at Eath s suface with no elevation change), moving to zeo as we each infinitely fa away. 7. Many satellites ae placed in geosynchonous obits. What is special about these obits? Fo a global communication netwok, how many of these satellites would be needed? The peiod of the obit must be 4 hous. But in addition, the satellite must be located in an equatoial obit and obiting in the same diection as Eath s otation. All thee citeia must be met fo the satellite to emain in one position elative to Eath s suface. At least thee satellites ae needed, as two on opposite sides of Eath cannot communicate with each othe. (This is not technically tue, as a wavelength could be chosen that povides sufficient diffaction. But it would be totally impactical.) 9. In the diagam below fo a satellite in an elliptical obit about a much lage mass, indicate whee its speed is the geatest and whee it is the least. What consevation law dictates this behavio? Indicate the diections of the foce, acceleation, and velocity at these points. Daw vectos fo these same thee quantities at the two points whee the y-axis intesects (along the semi-mino axis) and fom this detemine whethe the speed is inceasing deceasing, o at a max/min. Page 1 of 8

2 OpenStax Univesity Physics Volume I The speed is geatest whee the satellite is closest to the lage mass and least whee fathe away at the peiapsis and apoapsis, espectively. It is consevation of angula momentum that govens this elationship. But it can also be gleaned fom consevation of enegy, the kinetic enegy must be geatest whee the gavitational potential enegy is the least (most negative). The foce, and hence acceleation, is always diected towads M in the diagam, and the velocity is always tangent to the path at all points. The acceleation vecto has a tangential component along the diection of the velocity at the uppe location on the y-axis; hence, the satellite is speeding up. Just the opposite is tue at the lowe position. 11. The pinciple of equivalence states that all expeiments done in a lab in a unifom gavitational field cannot be distinguished fom those done in a lab that is not in a gavitational field but is unifomly acceleating. Fo the latte case, conside what happens to a lase beam at some height shot pefectly hoizontally to the floo, acoss the acceleating lab. (View this fom a nonacceleating fame outside the lab.) Relative to the height of the lase, whee will the lase beam hit the fa wall? What does this say about the effect of a gavitational field on light? Does the fact that light has no mass make any diffeence to the agument? The lase beam will hit the fa wall at a lowe elevation than it left, as the floo is acceleating upwad. Relative to the lab, the lase beam falls. So we would expect this to happen in a gavitational field. The mass of light, o even an object with mass, is not elevant. Poblems 1. Evaluate the magnitude of gavitational foce between two 5-kg spheical steel balls sepaated by a cente-to-cente distance of 15 cm N 15. Astology makes much of the position of the planets at the moment of one s bith. The only known foce a planet exets on Eath is gavitational. (a) Calculate the gavitational foce exeted on a 4.0-kg baby by a 0-kg fathe 0.00 m away at bith (he is assisting, so he is close to the child). (b) Calculate the foce on the baby due to Jupite if it is at its closest distance to Eath, some m away. How does the foce of Jupite on the baby compae to the foce of the fathe on the baby? Othe objects in the oom and the hospital building also exet simila Page of 8

3 OpenStax Univesity Physics Volume I gavitational foces. (Of couse, thee could be an unknown foce acting, but scientists fist need to be convinced that thee is even an effect, much less that an unknown foce causes it.) 7 a N ; b. The mass of Jupite is m F J = kg 11 7 ( 6.67 N m kg ) ( 1.90 kg ) ( 4.0 kg) 11 ( 6.9 m) = = J N -7 Ff 7.01 N = = FJ 1.5 N 17. The Intenational Space Station has a mass of appoximately 70,000 kg. (a) What is the foce on a 150-kg suited astonaut if she is 0 m fom the cente of mass of the station? (b) How accuate do you think you answe would be? 6 a. 9.5 N ; b. Not vey, as the ISS is not even symmetical, much less spheically symmetical. 19. (a) What was the acceleation of Eath caused by asteoid Toutatis (see pevious poblem) at its closest appoach? (b) What was the acceleation of Toutatis at this point? 15 4 a m/s ; b m/s 1. (a) What is the acceleation due to gavity on the suface of the Moon? (b) On the suface of 6 Mas? The mass of Mas is kg and its adius is.8 m. a. 1.6 m/s ; b..75 m/s. The mass of a paticle is 15 kg. (a) What is its weight on Eath? (b) What is its weight on the Moon? (c) What is its mass on the Moon? (d) What is its weight in oute space fa fom any celestial body? (e) What is its mass at this point? a. 147 N; b. 5.5 N; c. 15 kg; d. 0; e. 15 kg Page of 8

4 OpenStax Univesity Physics Volume I 5. The mean diamete of the planet Satun is 8 1. m, and its mean mass density is 0.69 g/cm. Find the acceleation due to gavity at Satun s suface. 1 m/s 7. The acceleation due to gavity on the suface of a planet is thee times as lage as it is on the suface of Eath. The mass density of the planet is known to be twice that of Eath. What is the adius of this planet in tems of Eath s adius? (/ )R E 9. Find the escape speed of a pojectile fom the suface of Mas m/s 1. What is the escape speed of a satellite located at the Moon s obit about Eath? Assume the Moon is not neaby m/s 7 1. An aveage-sized asteoid located 5.0 km fom Eath with mass.0 kg is detected headed diectly towad Eath with speed of.0 km/s. What will its speed be just befoe it hits ou atmosphee? (You may ignoe the size of the asteoid.) 11 km/s 5. (a) What is the change in enegy of a 00-kg payload taken fom est at the suface of Eath and placed at est on the suface of the Moon? (b) What would be the answe if the payload wee taken fom the Moon s suface to Eath? Is this a easonable calculation of the enegy needed to move a payload back and foth? a J ; b J ; No. It assumes the kinetic enegy is ecoveable. This would not even be easonable if we had an elevato between Eath and the Moon. 7. Two planets in cicula obits aound a sta have speeds of v and v. (a) What is the atio of the obital adii of the planets? (b) What is the atio of thei peiods? a. 0.5; b What is the obital adius of an Eath satellite having a peiod of 1.00 h? (b) What is uneasonable about this esult? a km ; b. This less than the adius of Eath. 41. Find the mass of Jupite based on the fact that Io, its innemost moon, has an aveage obital adius of 41,700 km and a peiod of 1.77 days kg 4. (a) In ode to keep a small satellite fom difting into a neaby asteoid, it is placed in obit with a peiod of.0 hous and adius of.0 km. What is the mass of the asteoid? (b) Does this mass seem easonable fo the size of the obit? Page 4 of 8

5 OpenStax Univesity Physics Volume I a kg ; b. The satellite must be outside the adius of the asteoid, so it can t be lage than this. If it wee this size, then its density would be about 0 kg/m. This is just above that of wate, so this seems quite easonable The Sun obits the Milky Way galaxy once each.60 yeas, with a oughly cicula obit aveaging a adius of light-yeas. (A light-yea is the distance taveled by light in 1 yea.) Calculate the centipetal acceleation of the Sun in its galactic obit. Does you esult suppot the contention that a nealy inetial fame of efeence can be located at the Sun? (b) Calculate the aveage speed of the Sun in its galactic obit. Does the answe supise you? a m/s ; Yes, the centipetal acceleation is so small it suppots the contention that a 5 nealy inetial fame of efeence can be located at the Sun. b..17 m s 47. Calculate the mass of the Sun based on data fo aveage Eath s obit and compae the value obtained with the Sun s commonly listed value of kg kg; The values ae the same within 0.05%. 49. The mean obital adius listed fo astonomical objects obiting the Sun is typically not an integated aveage but is calculated such that it gives the coect peiod when applied to the equation fo cicula obits. Given that, what is the mean obital adius in tems of aphelion and peihelion?.00 Compae T = and T E 4 4 = a to see that they diffe only in that the cicula adius,, is eplaced by the semi-majo axis, a. Theefoe, the mean adius is one-half the sum of the aphelion and peihelion, the same as the semi-majo axis. 51. The peihelion of the comet Lagekvist is.61 AU and it has a peiod of 7.6 yeas. Show that the aphelion fo this comet is 4.95 AU. The semi-majo axis,.78 AU is found fom the equation fo the peiod. This is one-half the sum of the aphelion and peihelion, giving an aphelion distance of 4.95 AU. 5. Eos has an elliptical obit about the Sun, with a peihelion distance of 1.1 AU and aphelion distance of 1.78 AU. What is the peiod of its obit? 1.75 yeas 55. If the Sun wee to collapse into a black hole, the point of no etun fo an investigato would be appoximately km fom the cente singulaity. Would the investigato be able to suvive visiting even 00 km fom the cente? Answe this by finding the diffeence in the gavitational attaction the black holes exets on a 1.0-kg mass at the head and at the feet of the investigato. 19,800 N; this is clealy not suvivable 57. What is the Schwazschild adius fo the black hole at the cente of ou galaxy if it has the mass of 4 million sola masses? Page 5 of 8

6 OpenStax Univesity Physics Volume I km Additional Poblems 59. A neuton sta is a cold, collapsed sta with nuclea density. A paticula neuton sta has a mass twice that of ou Sun with a adius of 1.0 km. (a) What would be the weight of a 0-kg astonaut on standing on its suface? (b) What does this tell us about landing on a neuton sta? 14 a N ; b. Don t do it! 61. How fa fom the cente of the Sun would the net gavitational foce of Eath and the Sun on a spaceship be zeo? km 6. Suppose you can communicate with the inhabitants of a planet in anothe sola system. They tell you that on thei planet, whose diamete and mass ae 5.0 km and.6 kg, espectively, the ecod fo the high jump is.0 m. Given that this ecod is close to.4 m on Eath, what would you conclude about you extateestial fiends jumping ability? The value of g fo this planet is.8 m/s, which is about one-fouth that of Eath. So they ae weak high jumpes. 65. A body of mass 0 kg is weighed at the Noth Pole and at the equato with a sping scale. What is the scale eading at these two points? Assume that g = 9.8 m/s at the pole. At the Noth Pole, 98 N; at the equato, 980 N 67. Conside the pevious poblem and include the fact that Eath has an obital speed about the Sun of 9.8 km/s. (a) What speed elative to Eath would be needed and in what diection should you leave Eath? (b) What will be the shape of the tajectoy? a. The escape velocity is still 4.6 km/s. By launching fom Eath in the diection of Eath s tangential velocity, you need = 1.8 km/s elative to Eath. b. The total enegy is zeo and the tajectoy is a paabola. 69. An asteoid has speed 15.5 km/s when it is located.00 AU fom the sun. At its closest appoach, it is AU fom the Sun. What is its speed at that point? 61.5 km/s 71. A satellite of mass 00 kg is in cicula obit about Eath. The adius of the obit of the satellite is equal to two times the adius of Eath. (a) How fa away is the satellite? (b) Find the kinetic, potential, and total enegies of the satellite. 7 a. 1. m; b J;.1 J; 1.56 J 0 7. (a) (a) Given the asteoid Vesta which has a diamete of 50 km and mass of.67 kg, what would be the obital peiod fo a space pobe in a cicula obit of.0 km fom its suface? (b) Why is this calculation maginally useful at best? a. 6.4 s o about 1.8 hous. This was using the 50 km aveage diamete. b. Vesta is clealy not vey spheical, so you would need to be above the lagest dimension, nealy 580 km. Page 6 of 8

7 OpenStax Univesity Physics Volume I Moe impotantly, the nonspheical natue would distub the obit vey quickly, so this calculation would not be vey accuate even fo one obit. 75. (a) Using the infomation in the pevious poblem, what velocity do you need to escape the Milky Way galaxy fom ou pesent position? (b) Would you need to acceleate a spaceship to this speed elative to Eath? a. km/s; b. No, you need only the diffeence between the sola system s obital speed and escape speed, so about 8 = 95 km/s. 77. Show that fo eccenticity equal to one in = 1+ ecos fo conic sections, the path is a paabola. Do this by substituting Catesian coodinates, x and y, fo the pola coodinates, and and showing that it has the geneal fom fo a paabola, x = ay + by + c. Setting e = 1 in = 1+ ecos, we have = 1+ cos = + cos = + x ; hence, 1 = x + y = ( x). Expand and collect to show x= y Given the peihelion distance, p, and aphelion distance, q, fo an elliptical obit, show that the Sun q velocity at peihelion, v p, is given by vp =. (Hint: Use consevation of angula ( q + p) p, momentum to elate v p and v q, and then substitute into the consevation of enegy equation.) Substitute diectly into the enegy equation using pv p = qv q fom consevation of angula momentum, and solve fo v p Challenge Poblems 81. A tunnel is dug though the cente of a pefectly spheical and ailess planet of adius R. Using the expession fo g deived in gavitation Nea Eath s Suface fo a unifom density, show that a paticle of mass m dopped in the tunnel will execute simple hamonic motion. Deduce the peiod of oscillation of m and show that it has the same peiod as an obit at the suface. 4 4 Fom Gavitation Nea Eath s Suface, we have g = G F = mg = Gm, and fom d F = m, we get dt d dt 4 = G whee the fist tem is. Then T = = 4G and if we substitute M =, we get the same expession as fo the 4 / R peiod of obit R. Page 7 of 8

8 OpenStax Univesity Physics Volume I A 1 8. Show that the aeal velocity fo a cicula obit of adius about a mass M is =. t Does you expession give the coect value fo Eath s aeal velocity about the Sun? 15 Using the mass of the Sun and Eath s obital adius, the equation gives.4 m /s. The value of / (1 yea) gives the same value. h =. f i R ES 85. Show that fo small changes in height h, such that U m expession U = mg. h Em Em Em f i We stat with U = U = U f Ui = + = Em whee f i f i If, then, and upon substitution, we have h R E f i R E h R E, E = educes to the h E U = Em m h = whee we ecognize the expession with the paenthesis as RE RE g = G M E as the definition of g. 87. (a) Show that tidal foce on a small object of mass m, defined as the diffeence in the gavitational foce that would be exeted on m at a distance at the nea and the fa side of the m object, due to the gavitation at a distance R fom M, is given by Ftidal = whee is R the distance between the nea and fa side and R. (b) Assume you ae falling feet fist into the black hole at the cente of ou galaxy. It has mass of 4 million sola masses. What would be the diffeence between the foce at you head and you feet at the Schwazschild adius (event hoizon)? Assume you feet and head each have mass 5.0 kg and ae.0 m apat. Would you suvive passing though the event hoizon? m m Ftidal = = m = m a. Find the diffeence in foce, ; ( 1 )( + 1 ) ( R) m = m m 4 = 1 R R b. Fo the case given, using the Schwazschild adius fom a pevious poblem, we have a tidal foce of 9.5 N. This won t even be noticed! This file is copyight 016, Rice Univesity. All Rights Reseved. Page 8 of 8