AP Physics C Mechanics 7.5 Physical Pendulums

[Maximum number: 2]

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.

For indicating the rotational inertia is the sum of the rotational inertia for all the stacked disks 1 point
Example Response

\begin{aligned} I_{\mathrm{eq}} & =\sum_{i=1}^{N} I_{i} \\ I_{\mathrm{eq}} & =N\left(\frac{1}{2} M R^{2}\right) \end{aligned}

For an expression for the period consistent with the previous rotational inertia expression 1 point
Example Response

T=2 \pi \sqrt{\frac{N\left(\frac{1}{2} M R^{2}\right)}{\kappa}}

Example Solution

\begin{aligned} & I_{\mathrm{eq}}=\sum_{i=1}^{N} I_{i} \\ & I_{\mathrm{eq}}=N\left(\frac{1}{2} M R^{2}\right) \\ & T=2 \pi \sqrt{\frac{I}{\kappa}} \\ & T=2 \pi \sqrt{\frac{N\left(M R^{2}\right)}{2 \kappa}} \end{aligned}

AP Physics C: Mechanics 7.5 Simple and Physical Pendulums Questions

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Question 1