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Unformatted text preview: 13 Universal Gravitation CHAPTER OUTLINE 13.1 Newton’s Law of Universal Gravitation 13.2 Free-Fall Acceleration and the Gravitational Force 13.3 Kepler’s Laws and the Motion of Planets 13.4 The Gravitational Field 13.5 Gravitational Potential Energy 13.6 Energy Considerations in Planetary and Satellite Motion ANSWERS TO QUESTIONS *Q13.1 The force is proportional to the product of the masses and inversely proportional to the square of the separation distance, so we compute m 1 m 2 / r 2 for each case: (a) 2 ⋅ 3 / 1 2 = 6 (b) 18 (c) 18 / 4 = 4.5 (d) 4.5 (e) 16 / 4 = 4. The ranking is then b > a > c = d > e. *Q13.2 Answer (d). The International Space Station orbits just above the atmosphere, only a few hundred kilometers above the ground. This distance is small compared to the radius of the Earth, so the gravitational force on the astronaut is only slightly less than on the ground. We think of it as having a very different effect than it does on the ground, just because the normal force on the orbiting astronaut is zero. *Q13.3 Answer (b). Switching off gravity would let the atmosphere evaporate away, but switching off the atmosphere has no effect on the planet’s gravitational fi eld. Q13.4 To a good fi rst approximation, your bathroom scale reading is unaffected because you, the Earth, and the scale are all in free fall in the Sun’s gravitational fi eld, in orbit around the Sun. To a precise second approximation, you weigh slightly less at noon and at midnight than you do at sunrise or sunset. The Sun’s gravitational fi eld is a little weaker at the center of the Earth than at the surface subsolar point, and a little weaker still on the far side of the planet. When the Sun is high in your sky, its gravity pulls up on you a little more strongly than on the Earth as a whole. At midnight the Sun pulls down on you a little less strongly than it does on the Earth below you. So you can have another doughnut with lunch, and your bedsprings will still last a little longer. *Q13.5 Having twice the mass would make the surface gravitational fi eld two times larger. But the inverse square law says that having twice the radius would make the surface acceleration due to gravitation four times smaller. Altogether, g at the surface of B becomes (2 m / s 2 )(2) / 4 = 1 m / s 2 , answer (e). *Q13.6 (i) 4 2 = 16 times smaller: Answer (i), according to the inverse square law. (ii) m v 2 / r = GMm / r 2 predicts that v is proportional to (1 / r ) 1 / 2 , so it becomes (1 / 4) 1 / 2 = 1 / 2 as large: Answer (f ). (iii) (4 3 ) 1 / 2 = 8 times larger: Answer (b), according to Kepler’s third law. 337 ISMV1_5103_13.indd 337 ISMV1_5103_13.indd 337 12/5/06 12:09:27 PM 12/5/06 12:09:27 PM *Q13.7 Answer (b). The Earth is farthest from the sun around July 4 every year, when it is summer in the northern hemisphere and winter in the southern hemisphere. As described by Kepler’s second law, this is when the planet is moving slowest in its orbit. Thus it takes more time for the planet to law, this is when the planet is moving slowest in its orbit....
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