Serway & Jewett — Rotation of a Rigid Object about a Fixed Axis: Problem 47

A uniform, hollow, cylindrical spool has inside radius \(R/2\), outside radius \(R\), and mass \(M\) (Fig. P10.47). It is mounted so that it rotates on a fixed, horizontal axle. A counterweight of mass \(m\) is connected to the end of a string wound around the spool. The counterweight falls from rest at \(t = 0\) to a position \(y\) at time \(t\). Show that the torque due to the friction forces between spool and axle is \[ \tau_f = R \left[ m \left( g - \frac{2y}{t^2} \right) - M \frac{5y}{4t^2} \right] \]