# 2005exam3 - 3 4. [5 pts] Derive the conditions under which...

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PHYSICS 321 EXAM 3 April 18, 2005 NAME 1. [5 pts] Imagine a peculiar cloud of cosmic dust whose mass distribution is spherically symmetric with a density (mass per unit volume) given by ρ ( r ) = k r 2 where r is the distance from the center and k is a constant. Find the gravitational force it produces on a “test mass” m at radius R which lies inside the dust cloud. 1

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2. [5 pts] A particle of mass m moves in a circular orbit of radius R under the inFuence of a potential U ( r ) = k r 3 where r = p x 2 + y 2 and k is a positive constant. ±ind its kinetic energy. (Hint: you can use the E²ective Potential, or you can ³nd the force and use the methods of ³rst year physics – your choice.) 2
3. [5 pts] Find the gravitational potential energy of a point mass m that is located a distance a from a half-circle of wire mass M as shown. The radius of the half-circle is R . You may leave your answer in the form of a well-de±ned integral— you do not need to evaluate the integral.

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Unformatted text preview: 3 4. [5 pts] Derive the conditions under which h ˙ S i = 0, where S = ~ p · ~ r . (We used this in connection with the Virial Theorem.) 4 5. [5 pts] The equation for an elliptical orbit can be written in the form 1 r = C 1 + C 2 cos θ . Express this equation in terms of Cartesian ( x and y ) coordinates and use your result to derive the condition on C 1 and C 2 for the orbit to be elliptical. 5 6. [5 pts] A satellite of small mass m is in a cir-cular orbit of radius R about a planet of large mass M . At a certain instant of time, a power-ful rocket motor on the satellite is Fred brie±y, so as to increase the velocity without changing its direction. As a result, the satellite goes into an elliptical orbit as shown. ²ind the total en-ergy (kinetic + potential) in the elliptical orbit in terms of G , m , M , and R . 6...
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## This note was uploaded on 07/25/2008 for the course PHY 321 taught by Professor B.pope during the Spring '08 term at Michigan State University.

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2005exam3 - 3 4. [5 pts] Derive the conditions under which...

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