Halliday, Resnick & Walker — Oscillations: Problem Problem_97

A torsion pendulum consists of a metal disk with a wire running through its center and soldered in place. The wire is mounted vertically on clamps and pulled taut. Figure 15.39a gives the magnitude \(\tau\) of the torque needed to rotate the disk about its center (and thus twist the wire) versus the rotation angle \(\theta\). The vertical axis scale is set by \(\tau_s = 4.0 \times 10^{-3} \text{ N}\cdot\text{m}\). The disk is rotated to \(\theta = 0.200 \text{ rad}\) and then released. Figure 15.39b shows the resulting oscillation in terms of angular position \(\theta\) versus time \(t\). The horizontal axis scale is set by \(t_s = 0.40 \text{ s}\). (a) What is the rotational inertia of the disk about its center? (b) What is the maximum angular speed \(d\theta/dt\) of the disk? (Caution: Do not confuse the (constant) angular frequency of the SHM with the (varying) angular speed of the rotating disk, even though they usually have the same symbol \(\omega\). Hint: The potential energy \(U\) of a torsion pendulum is equal to \(\frac{1}{2}\kappa\theta^2\), analogous to \(U = \frac{1}{2}kx^2\) for a spring.)