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Problem 72 Hard Difficulty

An object with weight $W$ is dragged along a horizontal plane by 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 $.

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Video Transcript

Okay, let's take the derivative off. Prime of data is gonna be okay. We know have promised data is greater than 01 tan. They dies greater, and the F craft is less than zero in 10. They does less. Therefore, we know it changes from decreasing to increase in. So we know that this over here is gonna be where there is local minimum.