An axially symmetric pulley of mass $M$ rotates about a fixed, horizontal axis, say the $x$-axis. A string of fixed length and negligible mass connects two blocks with masses $m_{1}=M$ and $m_{2}=2 M$. The string is hung over the pulley, with one mass on each side. The tensions in the string due to masses $m_{1}$ and $m_{2}$ can respectively be labelled $T_{1}$ and $T_{2}$. The moment of inertia of the pulley is $I=q M a^{2}$, where $q$ is a number and $a$ is the radius of the

The motion of the pulley is opposed by a frictional torque of magnitude $\lambda M \omega$, where $\omega$ is the angular velocity of the pulley and $\lambda$ is a real positive constant. Obtain a first-order differential equation for $\omega$ and, from it, find $\omega(t)$ given that the system is released from rest.

The surface of the pulley is defined by revolving the function $b(x)$ about the $x$-axis, with

$$b(x)= \begin{cases}a(1+|x|) & -1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise. }\end{cases}$$

Find a value for the constant $q$ given that the pulley has uniform mass density $\rho$.