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Unformatted text preview: Solution for Homework 27 Waves I, Gravitation problems Solution to Homework Problem 27.1() Problem: How do the speed of a wave and the velocity of the particles in the transmitting medium compare? Solution They are different quantities. The wave always moves in the same direction, but the particles displaced by this movement simply move back and forth when displaced by the wave (in other words, they vibrate). Total Points for Problem: 2 Points Solution to Homework Problem 27.2() Problem: How do overlapping waves behave? What is the name of this unique characteristic? Solution When waves overlap, the resulting wave function is simply the sum of the individual wave functions. This is called the principle of linear superposition . Total Points for Problem: 2 Points Solution to Homework Problem 27.3() Problem: If you are in a swimming pool that has a wave machine, sometimes the waves exhibit standing wave patterns. What would happen to you if you were in a float at a position of constructive interference? Destructive interference? Solution In a position of constructive interference (an antinode) you would move up and down at twice the amplitude of the normal, non-interfering wave. In a position of destructive interference (a node) you would not move at all; it would feel as if you were sitting on calm water. Total Points for Problem: 2 Points Solution to Homework Problem 27.4() Problem: Escape velocity is the velocity an object at a certain distace from a massive body must have, in order to escape the gravitational influence of the massive body. If the body is dense enough, then at some distance from its center of mass (but outside the body itself), the escape velocity will be that of light, c = 3 10 8 m s . This distance is called the Schwarzchild radius, and it describes an imaginary sphere around the body, inside of which not even light can escape (hence the term black hole). (a)Calculate the Schwarzchild radius for any object of inertia M (its okay to neglect relativity). (b)Calculate the Schwarzchild radius for the sun. (c)Calculate the Schwarzchild radius for the earth. (d)Calculate the Schwarzchild radius for a globe with a mass of 10kg . Solution to Part (a) Again, by equation 14.36 in the book, 1 2 mc 2 = GM m R s So R s = 2 GM c 2 = parenleftbigg 1 . 485 10 27 m kg parenrightbigg M Grading Key: Part (a) 2 Points Solution to Part (b) M = 2 10 30 kg , so R s = parenleftbigg 1 . 485 10 27 m kg parenrightbigg (2 10 30 kg) = 2 . 97km Grading Key: Part (b) 1 Points Solution to Part (c) M = 5 . 97 10 24 kg , so R s = parenleftbigg 1 . 485 10 27 m kg parenrightbigg (5 . 97 10 24 kg) = 8 . 85mm So, neglecting relativity, the mass of the earth would have to be concentrated in a volume about the size of a marble to create a black hole....
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