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SM_PDF_chapter11 - Gravity Planetary Orbits and the...

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297 Gravity, Planetary Orbits, and the Hydrogen Atom CHAPTER OUTLINE 11.1 Newton’s Law of Universal Gravitation Revisited 11.2 Structural Models 11.3 Kepler’s Laws 11.4 Energy Considerations in Planetary and Satellite Motion 11.5 Atomic Spectra and the Bohr Theory of Hydrogen 11.6 Context Connection Changing from a Circular to an Elliptical Orbit ANSWERS TO QUESTIONS Q11.1 The Earth creates the same gravitational field g for all objects near the Earth’s surface. The larger mass needs a larger force to give it just the same acceleration. Q11.2 To a good first approximation, your bathroom scale reading is unaffected because you, the Earth, and the scale are all in free fall in the Sun’s gravitational field, 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 field 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. Q11.3 Because both the Earth and Moon are moving in orbit about the Sun. As described by F ma gravitational centripetal = , the gravitational force of the Sun merely keeps the Moon (and Earth) in a nearly circular orbit of radius 150 million kilometers. Because of its velocity, the Moon is kept in its orbit about the Earth by the gravitational force of the Earth. There is no imbalance of these forces, at new moon or full moon. Q11.4 Air resistance causes a decrease in the energy of the satellite-Earth system. This reduces the diameter of the orbit, bringing the satellite closer to the surface of the Earth. A satellite in a smaller orbit, however, must travel faster. Thus, the effect of air resistance is to speed up the satellite! Q11.5 The escape speed from the Earth is 11.2 km/s and that from the Moon is 2.3 km/s, smaller by a factor of 5. The energy required—and fuel—would be proportional to v 2 , or 25 times more fuel is required to leave the Earth versus leaving the Moon. Q11.6 In a circular orbit each increment of displacement is perpendicular to the force applied. The dot product of force and displacement is zero. The work done by the gravitational force on a planet in an elliptical orbit speeds up the planet at closest approach, but negative work is done by gravity and the planet slows as it sweeps out to its farthest distance from the Sun. Therefore, net work in one complete orbit is zero.
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298 Gravity, Planetary Orbits, and the Hydrogen Atom Q11.7 For a satellite in orbit, one focus of an elliptical orbit, or the center of a circular orbit, must be located at the center of the Earth. If the satellite is over the northern hemisphere for half of its orbit, it must be over the southern hemisphere for the other half. We could share with Easter Island a satellite that
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