Chapter 2.4, Problem 38E

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

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

where $\mu$ is a constant called the coefficient of friction.

(a) Find the rate of change of F with respect to $\theta$.

(b) When is this rate of change equal to 0?

(c) If $W=50\text{ lb}$ and $\mu =0.6$, draw the graph of F as a function of $\theta$ and use it to locate the value of $\theta$ for which $dF/d\theta =0$. Is the value consistent with your answer to part (b)?