A keen cyclist wishes to analyse her performance on training rollers. She decides that the key components are her bicycle's rear wheel and the roller on which the wheel sits. The wheel, of radius $R$, has its mass $M$ entirely at its outer edge. The roller, which is driven by the wheel without any slippage, is a solid cylinder of radius $S$ and mass $M / 2$. The angular velocities of the wheel and roller are $\omega$ and $\sigma$, respectively.

Determine $I$ and $J$, the moments of inertia of the wheel and roller, respectively. Find the ratio of the angular velocities of the wheel and roller. Show that the combined total kinetic energy of the wheel and roller is $\frac{1}{2} K \omega^{2}$, where

$$K=\frac{5}{4} M R^{2}$$

is the effective combined moment of inertia of the wheel and roller.

Why should $K$ be used instead of just $I$ or $J$ in the equation connecting torque with angular acceleration? The cyclist believes the torque she can produce at the back wheel is $T=Q(1-\omega / \Omega)$ where $Q$ and $\Omega$ are dimensional constants. Determine the angular velocity of the wheel, starting from rest, as a function of time.

In an attempt to make the ride more realistic, the cyclist adds a fan (of negligible mass) to the roller. The fan imposes a frictional torque $-\gamma \sigma^{2}$ on the roller, where $\gamma$ is a dimensional constant. Determine the new maximum speed for the wheel.