Serway & Jewett — Rotation of a Rigid Object about a Fixed Axis: Problem 45

A spool of thread consists of a cylinder of radius \(R_1\) with end caps of radius \(R_2\) as depicted in the end view shown in Figure P10.45. The mass of the spool, including the thread, is \(m\), and its moment of inertia about an axis through its center is \(I\). The spool is placed on a rough, horizontal surface so that it rolls without slipping when a force \(\vec{T}\) acting to the right is applied to the free end of the thread. (a) Show that the magnitude of the friction force exerted by the surface on the spool is given by \[f = \left( \frac{I + mR_1R_2}{I + mR_2^2} \right) T\] (b) Determine the direction of the force of friction.