PHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013

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1 PHYSICS 111 HOMEWORK SOLUTION #13 May 1, 2013

2 0.1 In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 2.10 kg and 21.0 g whose centes ae sepaated by about 3.90 cm. Calculate the gavitational foce between these sphees, teating each as a paticle located at the cente of the sphee. The gavitational foce between the two masses is: F G mm N ( ) 2 2

3 Mianda, a satellite of Uanus, is shown in pat a of the figue below. It can be modeled as a sphee of adius 242 km and mass kg. a) Find the fee-fall acceleation on its suface. b) A cliff on Mianda is 5.00 km high. It appeas on the limb at the 11 o clock position in pat a of the figue above and is magnified in pat b of the figue above. A devotee of exteme spots uns hoizontally off the top of the cliff at 7.70 m/s. Fo what time inteval is he in flight? c) How fa fom the base of the vetical cliff does he stike the icy suface of Mianda? d) What is his vecto impact velocity? a) The fee-fall acceleation on Mianda s suface can be deived by equating the gavitational foce F G mm 2 and the fee-fall foce mg : mg G mm 2 g G M m/s ( ) 2 3

4 b) We can use the fee-fall equation of motion unde the above-calculated acceleation h 1 2 gt2 to get the time in flight: 2 h t g s c) How fa fom the base cliff will he stike can be evaluated by looking at the hoizontal component of the equations, x v x t: x m d) The hoizontal component of his velocity is being constant thoughout the motion v x 7.70m/s, we can evaluate the vetical component at the impact by using the time-independent equation: and v 2 y v 2 y0 2g h The diection of his impact is : v y 2g h m/s v m/s θ actan v y v x actan

5 A comet (see figue below) appoaches the Sun to within AU, and its obital peiod is 90.6 yeas. (AU is the symbol fo astonomical unit, whee 1 AU m is the mean EathSun distance.) How fa fom the Sun will the comet tavel befoe it stats its etun jouney. Keple s Law elates the squae of the obital peiod of a planet to the cube of the semi-majo axis (distance a in the figue), the popotionality constant is GM 4π 2 : a 3 GM 4π 2 T π 2 T T ( ) m 3 5

6 and a m AU x 2a AU 0.4 Neuton stas ae extemely dense objects fomed fom the emnants of supenova explosions. Many otate vey apidly. Suppose the mass of a cetain spheical neuton sta is twice the mass of the Sun and its adius is 11.0 km. Detemine the geatest possible angula speed it can have so that the matte at the suface of the sta on its equato is just held in obit by the gavitational foce. The matte at the suface of the neuton stas is subject to the gavitational foce G Mnm which is balanced by the centipetal mv2 2 foce that keeps the sta in its obit: G M nm 2 mv2 m(ω)2 GMn ω ad/s

7 How much wok is done by the Moon s gavitational field as a 995 kg meteo comes in fom oute space and impacts on the Moon s suface? The wok done by the Moon s gavitational field can be evaluated though the change in potential enegy: W U ( GMm R GMm ) ( GMm R 0) J 0.6 Afte the Sun exhausts its nuclea fuel, its ultimate fate will be to collapse to a white dwaf state. In this state, it would have appoximately the same mass as it has now, but its adius would be equal to the adius of the Eath. a) Calculate the aveage density of the white dwaf. b) Calculate the suface fee-fall acceleation. c) Calculate the gavitational potential enegy associated with a 4.93-kg object at the suface of the white dwaf. a) The white dwaf will have a mass almost the same mass as the sun M s kg but will have a smalle adius compaable to eath s m. 7

8 The density is thus: b) ρ M s 4π π ( ) kg/m 3 Following the same steps in poblem 2-a), the fee fall acceleation is : c) g G M m/s ( ) 2 Potential enegy of the 4.93-kg object at the suface of the white dwaf is : 0.7 U g GM sm J a) What is the minimum speed, elative to the Sun, necessay fo a spacecaft to escape the sola system if it stats at the Eath s obit? b) Voyage 1 achieved a maximum speed of 125,000 km/h on its way to photogaph Jupite. Beyond what distance fom the Sun is this speed sufficient to escape the sola system? The value of GMm is the potential enegy that keeps the spacecaft in the sola system and to ovecome this baie, the spacecaft will need a minimum kinetic enegy 1 2 mv2 compaable to the fist amount: 8

9 mv2 GMm 2GM v km/s b) Again the same equation applies, knowing the acquied speed we can compute the sufficient distance fo the escape: 1 2 mv2 GMm 2GM v ( ) m 0.8 A satellite of mass 190 kg is placed into Eath obit at a height of 700 km above the suface. a) Assuming a cicula obit, how long does the satellite take to complete one obit? b) What is the satellite s speed?. c) Stating fom the satellite on the Eath s suface, what is the minimum enegy input necessay to place this satellite in obit? Ignoe ai esistance but include the effect of the planet s daily otation. a) In poblem 3), using Keple s Law fo elliptic motion, we wee able to get the semi-majo axis distance in tems of the peiod of motion. Hee we assume a simple cicula motion and we will use the adius instead (we should also 9

10 include the satellite height above the suface ): b) R 3 GM 4π 2 T 2 ( ) π 2 T T 2 T s 1.64 h Knowing the peiod and the adius we can easily deive the speed: c) v 2πR T 2π( ) m/s The total enegy of the {Eath+Satellite} system is to be conseved: Initially, on eath suface and by taking eath s speed as v e 2π m/s E i 1 2 mv2 e G Mm On obit, E f 1 2 mv2 G Mm R The change in enegy is the minimum enegy equied to put this satellite on obit: E 1 2 m(v2 v 2 e) + GMm( 1 1 R ) ( ) ( ) J 10

11 Studies of the elationship of the Sun to ou galaxythe Milky Way have evealed that the Sun is located nea the oute edge of the galactic disc, about ly (1 ly m) fom the cente. The Sun has an obital speed of appoximately 250 km/s aound the galactic cente. a) What is the peiod of the Sun s galactic motion? b)what is the ode of magnitude of the mass of the Milky Way galaxy? c) Suppose the galaxy is made mostly of stas of which the Sun is typical. What is the ode of magnitude of the numbe of stas in the Milky Way? a) The peiod can be easily evaluated assuming a cicula motion: b) v 2πR T T 2πR v 2π s ys Using the same pocedue as in poblem 8-a): the peiod, the adius and the mass of the object giving ise to gavity ae elated with Keple/Newton law: a ode of magnitude R 3 GM 4π 2 T 2 M 4π2 R 3 GT 2 4π 2 ( ) ( ) kg 11

12 c) Such a huge mass will oughly include a numbe of Suns about kg each: an ode of magnitude of Suns. 12

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