Physics 422 -- Fall 2006

Homework #6, due Wednesday Oct. 18 at beginning of class

1. [4pts]    Marion & Thornton problem 11.14. Hint: a little thought can save a lot of integration.

2. [4pts]    Part (a): Marion & Thornton problem 11.17. Note the shape of the mass distribution is completely arbitrary. You don't actually need to assume that it is homogeneous either. The only thing you need to assume is that the mass distribution is two-dimensional, i.e., that it has negligible thickness.

Part (b): Find the principal moments of inertia in terms of A, B, and C.

3. [4pts]    Marion & Thornton problem 11.18

4. [4pts]    Marion & Thornton problem 11.24. Do part (b) only.

5. [4pts]    Marion & Thornton problem 11.25. Again a little thought can save a lot of integration in calculating the moment of intertia.
Use the angle theta of the wheel as the generalized coordinate in the Lagrangian.

Part (b): After you find the Lagrangian, use it to obtain the equation of motion.

6. [4pts]    Find the frequency of oscillation of a simple pendulum that is released from rest at angles of 30, 60, 90, 120, 150 degrees relative to straight down, in units where the frequency of small oscillations is equal to 1. (You will need to use some numerical methods, e.g. Mathematica or one of its rivals, to do this problem.)