AP Physics C Mechanics 5.3: Torque on Rigid Systems

A solid uniform disk is supported by a vertical stand. The disk is able to rotate with negligible friction about an axle that passes through the center of the disk. The mass and radius of the disk are given by $M_{\mathrm{d}}$ and R, respectively. The rotational inertia of the disk is $I_{\mathrm{d}}=\frac{1}{2} M_{\mathrm{d}} R^{2}$. A string of negligible mass is draped over the disk and attached to the top of the disk at point P. One end of the string is connected to an unstretched ideal spring of spring constant k, which is fixed to the ground as shown in Figure 1.
A block of mass $m_{\mathrm{B}}$ is then attached to the string on the right side of the disk. The block is slowly lowered until the spring-disk-block system reaches equilibrium, as shown in Figure 2. In this equilibrium position, the disk has rotated clockwise through a small angle $\theta$.
Give all algebraic answers in terms of $M_{\mathrm{d}}, R, k, \theta$, and physical constants, as appropriate.

At time t=0, the string on the right side of the disk is cut and the block falls to the ground. On the circle below, which represents the disk, draw and label the forces (not components) that act on the disk immediately after the string is cut and the block is falling to the ground. Each force should be represented by an arrow that starts on and is directed away from the point of application.