final - Physics 821 Dr. Herbst Spring Quarter 2005 FINAL...

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Physics 821 Spring Quarter 2005 Dr. Herbst FINAL EXAMINATION WEDNESDAY, JUNE 8 9:30 – 11:18 AM 100 POINTS 1) (25 points) The Lagrangian for the torque-free rotation of a rigid symmetric top around its center of mass is given, in terms of Eulerian angles, by the expression L = 1 2 I 1 ( ˙ f 2 sin 2 q + ˙ q 2 ) + 1 2 I 3 ( ˙ cos + ˙ y ) 2 where the two equal moments of inertia are designated I 1 . a) (15 pts) Determine decoupled expressions for the rates of precession ( ˙ ) and spinning ( ˙ ) in terms of constants of the motion and the nutation angle q . Is either of these rates constant? If so, under what conditions? b) (10 pts) Use your results to a) to get expressions for w 3 and w 1 2 + w 2 2 . Discuss the meaning of your answers. In particular, try to answer whether or not the top nutates. Remember that w 1 2 + w 2 2 was shown to equal a constant A 2 when we used Euler’s equations. 2) (35 points) a) (15 pts) Find the eigenvalues and normalized eigenvectors for the problem: L = 1 2 m ( ˙ x 1 2 + ˙ x 2 2 ) - 1 2 kx 1 2 - 1 2 kx 2 2 - 1 2 k ( x 2 - x 1 ) 2 where the x
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This note was uploaded on 06/13/2011 for the course PHYSICS 821 taught by Professor Eric during the Spring '07 term at Ohio State.

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final - Physics 821 Dr. Herbst Spring Quarter 2005 FINAL...

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