Chapter 3.1, Problem 64E

An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle $\theta$ with the plane, then the magnitude of the force is

$F=\frac{\text{μ}W}{\text{μ}\sin \theta +\cos \theta }$

where $\text{μ}$ is a positive constant called the coefficient of friction and where $0\le \theta \le \pi /2$ . Show that F is minimized when $\tan \theta =\text{μ}\text{.}$