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Problem 58

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 positive constant called the coefficient of friction and where 0$\leqslant \theta \leqslant \pi / 2 .$ Show that $F$ is minimized when $\tan \theta=\mu$

Answer

Therefore $F^{\prime}(\theta)$ changes from decreasing to increasing when $\tan \theta=\mu$ Therefore there is local minima when $\tan \theta=\mu$

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