hw04soln - Ph-507 Homework 4(due Friday February 25 PROBLEM...

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Ph-507. Homework 4 (due: Friday, February 25). PROBLEM 4-1 (4 pts) Episode II: A long time ago, in a galaxy far, far away... In 2D space, the gravitational potential energy should have a logarithmic form. In particular, for a system of N particles, U ( r 1 , r 2 , ..., r N ) = γ 2 X j 6 = k m j m k log | r j r k | . Here the summation is performed over all pairs of di ff erent particles ( a, b = 1 , ..., N ), and the double—counting is cancelled by the 1/2 factor. γ is the "gravitational constant". a) What is the time-averaged kinetic energy of this system? (Hint: try to do "rescaling" of the coordinates, and follow the logic of virial theorem); b) 2D astronomer Relpek has found that the period of the radial motion of a planet depends on its minimal and maximal distances from the star ( R min , R max ), according to the following law: T = R α max F μ R min R max . What is the scaling exponent α ? How much does F vary when R min / R max changes from 1 (circular orbit), to 0. PROBLEM 4-2. (4 pts.) Two asteroids of the same mass collide inelastically and form a single object. At the moment of the collision, they are moving in the same direction. Originally, one of the asteroids had a circular orbit with period T 1 , while the period of the other was T 2 . Find the period T of the orbital motion of the newly formed object. PROBLEM 4-3 (3 pts) Consider a particle of mass m scattered by the following attractive potential: U ( r ) = k r β , a) Suppose you know the di ff erential cross-section, as a function of de fl ection angle α : dσ/d = f 0 ( α ) , for certain energy E 0 . Apply scaling ideas to fi nd the cross-section for the same particle with an arbitrary energy E .
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