Problem 38E

38. 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 $W=50 \mathrm{lb}$ 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 $d F / d \theta=0 .$ Is the value consistent with your answer to part (b)?

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