Chapter 2.2, Problem 43ES

It is a fact that when a flexible rope is wrapped around a rough cylinder, a small force of magnitude $F_0$ at one end can resist a large force of magnitude $F$ at the other end. The size of $F$ depends on the angle $\theta$ through which the rope is wrapped around the cylinder (see the accompanying figure). The figure shows the graph of $F$ (in pounds) versus $\theta$ (in radians), where $F$ is the magnitude of the force that can be resisted by a force with magnitude $F_0=10$ lb for a certain rope and cylinder.

(a) Estimate the values of $F$ and $dF/d\theta$ when the angle $\theta =10$ radians.

(b) It can be shown that the force $F$ satisfies the equation $dF/d\theta =\mu F$ , where the constant $\mu$ is called the coefficient of friction. Use the results in part (a) to estimate the value of $\mu$ .