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homework7 - Homework Assignment 7 Physics 302 Classical...

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Homework Assignment 7 Physics 302, Classical Mechanics Fall 2010 A. V. Kotwal Handed out: Monday, October 25, 2010 Due time: Monday, November 1, 2010 Problems Each problem is out of 20 points. 1. Consider the Kepler problem. The energy conservation equation is given by E = μ 2 dr dt 2 + l 2 2 μr 2 - α r , where α = GμM = Gm 1 m 2 , l is the angular momentum, and E is the total energy of the system. (a) Prove the Kepler’s third law for elliptical orbits T 2 = 4 π 2 G M a 3 , where M = m 1 + m 2 is the total mass, a = 1 2 ( r min + r max ) is the semi-major axis, and T is the period of the orbit. (b) Show that the semi-minor axis is given by b = a p 1 - 2 , where = q 1 + 2 El 2 μα 2 is the eccentricity of the orbit. We notice that when = 0, b = a , the orbit is a circle. 2. A uniform distribution of dust in the solar system adds to the gravitional attaction of the Sun on a planet an additional force ~ F = - mC~ r where m is the mass of the planet, C is a constant proportional to the gravitational constant and the density of the dust, and ~ r
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