AP Physics C Mechanics 7.5: Simple and Physical Pendulums

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.

Using $T=2 \pi \sqrt{\frac{I}{\kappa}}$, derive an expression for T as a function of N. Express your answer in terms of $M, R, \kappa, N$, and physical constants, as appropriate.