ch13 - Chapter 13 1. The gravitational force between the...

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565 Chapter 13 1. The gravitational force between the two parts is     2 2 2 = = Gm M m G F mM m r r which we differentiate with respect to m and set equal to zero:   2 = 0 = 2 = 2 dF G M m M m dm r . This leads to the result m / M = 1/2. 2. The gravitational force between you and the moon at its initial position (directly opposite of Earth from you) is 0 2 ( ) m ME E GM m F R R where m M is the mass of the moon, ME R is the distance between the moon and the Earth, and E R is the radius of the Earth. At its final position (directly above you), the gravitational force between you and the moon is 1 2 ( ) m ME E GM m F R R . (a) The ratio of the moon’s gravitational pulls at the two different positions is 2 2 2 8 6 1 2 8 6 0 /( ) 3.82 10 m 6.37 10 m 1.06898. /( ) 3.82 10 m 6.37 10 m m ME E ME E m ME E ME E GM m R R F R R F GM m R R R R Therefore, the increase is 0.06898, or approximately 6.9%. (b) The change of the gravitational pull may be approximated as 1 0 2 2 2 2 3 1 2 1 2 ( ) ( ) 4 . m m m m E E ME E ME E ME ME ME ME m E ME GM m GM m GM m GM m R R F F R R R R R R R R GM mR R
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CHAPTER 13 566 On the other hand, your weight, as measured on a scale on Earth, is 2 E g E E GM m F mg R . Since the moon pulls you “up,” the percentage decrease of weight is 3 3 22 6 7 5 1 0 24 8 7.36 10 kg 6.37 10 m 4 4 2.27 10 (2.3 10 )%. 5.98 10 kg 3.82 10 m m E g E ME F F M R F M R       3. The magnitude of the force of one particle on the other is given by F = Gm 1 m 2 / r 2 , where m 1 and m 2 are the masses, r is their separation, and G is the universal gravitational constant. We solve for r :     11 2 2 1 2 12 6.67 10 N m / kg 5.2kg 2.4kg 19m 2.3 10 N Gm m r F . 4. We use subscripts s , e , and m for the Sun, Earth and Moon, respectively. Plugging in the numerical values (say, from Appendix C) we find 2 2 2 30 8 2 24 11 / 1.99 10 kg 3.82 10 m 2.16. / 5.98 10 kg 1.50 10 m sm s m sm s em em e m em e sm F Gm m r m r F Gm m r m r 5. The gravitational force from Earth on you (with mass m ) is 2 E g E GM m F mg R where 2 2 / 9.8 m/s . E E g GM R If r is the distance between you and a tiny black hole of mass 11 1 10 kg b M   that has the same gravitational pull on you as the Earth, then 2 . b
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ch13 - Chapter 13 1. The gravitational force between the...

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