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Tilted Gyroscope

A gyroscope consists of a flywheel of mass m, which has a moment of inertia I for rotation about its axis. It is mounted on a rod of negligible mass, which is supported at one end by a frictionless pivot attached to a vertical post, as shown in the diagram. The distance between the center of the wheel and the pivot is d. The wheel rotates about its axis with angular velocity omega, where positive omega refers to counterclockwise rotation as seen by an observer looking at the face of the wheel that is opposite the pivot. The rod is tilted upward, making an angle theta with respect to the horizontal. Gravity acts downward with a force of magnitude m g.

Adopt a coordinate system with the z axis pointing upward and the x and y axes in the horizontal plane. The gyroscope is moving, but at t = 0, the rod is in the yz plane.

a). Assuming that the only significant contribution to the angular momentum comes from the spinning of the flywheel about its center, what is the angular momentum vector Lvec about the pivot at t = 0?

Specify the components of Lvec with respect to the axes shown in the diagram. Write the components in order Lx, Ly, Lz separated by commas.

b). At t = 0, what is the torque acting on the wheel about the pivot? (Express your answer in terms of x,y and z components)

c). The gyroscope is observed to precess about the vertical axis, with an angular velocity of precession Omega, defined as positive for counterclockwise precession as seen from above. Find Omega in terms of the given quantities.

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