# hw05soln - Ph-507. Homework 5 (due: Friday, March 11)....

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Ph-507. Homework 5 (due: Friday, March 11). PROBLEM 5-1 (3 pts) The distribution of dust in Solar system is roughly uniform. This results in the following small correction to regular gravitational potential energy of a planet of mass m : δU ( r )= 2 π 3 Gmρr 2 Here ρ is the average density of the dust. Find the precession rate of the planet due to this correction. Assume that the period of the orbital motion T , and eccentricity of the orbit ± , are known. Hint: use perturbative techniques. PROBLEM 5-2 (3 pts) A gas cloud in space has mass M , initial radius R , and angular velocity ' . The initial kinetic energy associated with the rotation and other types motion is negligible compared to the potential energy. Find the angular velocity after the cloud collapses due to gravitational attraction. (hint: the virial theorem might help). PROBLEM 5-3 (3 pts) Episode V: Empire strikes back. A long time ago, in a galaxy far, far away. .. A rebel space station has a disk shape of radius R , with uniform distribution of total mass M . The station is spinning around its symmetry axis z fast enough for the centrifugal force at distance R from the z -axis to be equal to the gravitational force at the surface of Earth. A missile from the Empire battleship is lunched parallel to the z -axis at radial distance R from it. The missile totally disintegrates after hitting the space station, without causing it any serious structural damage. However, the

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## hw05soln - Ph-507. Homework 5 (due: Friday, March 11)....

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