Question #81187

A torsional pendulum consists of a uniform horizontal disk of mass $M$ (in kg) and radius $R$ (in m) suspended by a vertical wire of torsional constant $C$ (in N$\cdot$m/rad). At $t=0$, the disk is released from rest at an angular displacement $\theta_0$ (in rad). Immediately after release, sand begins to fall vertically onto the edge of the disk at a constant rate $\mu$ (in kg/s) and sticks to it. Assume $\mu$ is sufficiently small such that the adiabatic approximation holds, meaning the accumulated phase of oscillation is given by $\Phi(t) = \int_0^t \omega(t') dt'$, where $\omega(t)$ is the instantaneous natural frequency. The angular position of the disk at time $t$ can be locally expressed as $\theta(t) \approx A(t) \cos(\omega(t) t + \delta(t))$. Which of the following expressions gives the effective phase constant $\delta(\tau)$ at time $t=\tau$ (in s)?

(Let $I_0 = \frac{1}{2}MR^2$ and $I_\tau = I_0 + \mu R^2 \tau$, both in kg$\cdot$m$^2$)