Physics 422 -- Fall 2012

Homework #7, due Fri Oct. 26 at beginning of class

1. [4pts]    A DVD is rotating freely (in a demo in the International Space Station, for example, so gravitational torques play no role) such that its symmetry axis always makes an angle of 30 degrees with respect to the angular momentum direction. The symmetry axis precesses around the angular momentum direction at angular velocity W. You can describe the motion using Euler angles theta=30deg, phi= W*t, and psi = C*t.
(a) Find the three components of the rotation vector omega in the body frame using formulas derived in class that express (omega_1,omega_2,omega_3) in terms of the Euler angles. Then find the value of C that is required by the Euler equations of motion.
(b) Use your omega vector to find the components of the angular momentum vector in the Body frame.
(c) Apply the rotations R_z(phi) R_y(theta) R_z(psi) to the anglular momentum vector to find the components of the angular momentum vector in the Space frame, and see if the result makes sense.

You can model the DVD as a uniform disk with zero thickness.

2. [4pts]    Marion & Thornton problem 11.22

3. [4pts]    Marion & Thornton problem 11.29.   Hint: You can't assume theta=0, because you want to allow it to fall. But you want to assume theta=0 is *possible*, which means that you must assume p_phi = p_psi.

4. [4pts]    Marion & Thornton problem 11.31. Hints: Kinetic energy and the square of the angular momentum are constants of the motion. You can use the given initial conditions to evaluate those constants. One of them is pleasantly simple. Using these results, you can change one of the three Euler equations into an equation that involves only one of the three variables omega_1, omega_2, or omega_3; and you can integrate that equation. Note that the angle between omega and the plane of the object is given by alpha -- that means that the angle between omega and the body z direction is (pi/2 - alpha).