Chapter 10, Problem 45CP

A spool of thread consists of a cylinder of radius R₁ with end caps of radius R₂ 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 $\stackrel{\to }{T}$ 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

$f=\left(\frac{I+m{R}_1{R}_2}{I+m{R}_2{}^2}\right)T$

(b) Determine the direction of the force of friction.

Figure P10.45