AP Physics C Mechanics 6.3: Angular Momentum

Mech.3.
A uniform rod of length d has one end fixed to the central axis of a horizontal, frictionless circular platform of radius R=2 d. Fixed at the other end of the rod is an ideal spring of negligible mass to which a block is attached. The block is set in frictionless grooves so that it can only move along a radius of the platform, as shown in Figure 1 above. The equilibrium length of the spring is d / 2. Below is a table showing the mass of the block and the masses and rotational inertias of the rod and platform.

A motor begins to slowly rotate the platform counterclockwise as viewed from above until the platform reaches a constant angular speed $\omega$. Under these conditions, the spring has stretched by an additional length d / 2, as shown in Figure 2.

Answer the following questions for the platform rotating at constant angular speed $\omega$. Express all algebraic answers in terms of $m, d, \omega$, and physical constants, as appropriate.

Determine an expression for the angular momentum of the entire system about the axis of the platform.

While the system continues to rotate, a small mechanism in the pivot moves the rod slowly until the center of the rod is positioned on the axis, as shown in Figure 3 above. The same constant angular speed $\omega$ is maintained by the motor driving the platform.