AP Physics C Mechanics 7.4 SHM Energy Overview

A student makes a torsional pendulum by suspending a uniform disk of mass M and radius R from a light wire with torsion constant $\kappa$ that is attached to the center of the disk as shown in Figure 1. The rotational inertia of the disk is given by $I=\frac{1}{2} M R^{2}$. The student conducts an investigation to determine the relationship between the period of oscillation T of the torsional pendulum and the number N of identical disks that are suspended from the wire.
The student starts with a single disk. Holding the disk at a small initial angular displacement $\theta_{0}$ from the untwisted position, the student releases the disk from rest and the pendulum oscillates. The student records the period of oscillation for a single disk. An additional identical disk is attached, as shown in Figure 2, and the procedure is repeated for N=2 disks. This procedure is repeated through N=10 identical disks. Assume the disks move together as one system.

The potential energy stored in the torsional pendulum when the disks are displaced is $U=\frac{1}{2} \kappa(\Delta \theta)^{2}$. On the following axes, sketch a graph of the maximum kinetic energy $K_{\text {max }}$ of the torsional pendulum as a function of N for $N \geq 1$.