final - Physics 210 Classical Mechanics Take-Home Final...

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Physics 210 Classical Mechanics Autumn, 2003 Take-Home Final Exam Due by noon Thursday, December 11 Open-book, but no collaboration allowed. Please address all questions to me. 1. (Exercise 3-31) By solving the orbital equation, show that for a repulsive central force inversely proportional to the cube of the distance, 3 ) ( r k r f = 0 k , the orbits are of the form )] ( cos[ 1 0 f f b - = A r . Express b in terms of k, E, l , and the mass m of the incident particle. Show that the cross section for scattering of a particle subject to this force is ( 29 Θ Θ - Θ Θ - = sin 1 2 2 2 2 2 p p p s E k d d . 2. A double pendulum consists of two simple pendula, with the second pendulum suspended from the mass of the first. The pendula both are of the same length l but different masses m and m 2 . Find the normal modes and frequencies of small oscillations about equilibrium for this system, assuming the lower pendulum to have the larger mass. If you wish, you may start with the equations of motion that you
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