Pulling a block

A person pulls on a block with a force F, at an angle \theta with respect to the
horizontal. The coefficient of friction between the block and the ground is \mu.
For what \theta is the F required to make the block slip a minimum?

Problem Source : Introduction to Classical Mechanics

The free body diagram can be drawn as below:

M1

What does slip a minimum means? It yields that the acceleration of the block a \rightarrow 0. Therefore, the friction force is a kinetic friction but the net force is 0. With this in mind, we can now construct our equation:

F \cos \theta = F_f

F \cos \theta = \mu (mg - F \sin \theta)

F \cos \theta = \mu mg - \mu F \sin \theta

Taking the derivatives of F respect to \theta,

\begin{aligned}  - F \sin \theta &= - \mu F \cos \theta \\ \tan \theta &= \mu  \end{aligned}

Hence, \theta = \tan ^{-1} \mu is the F required to make the block slip a minimum.

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Pulling a block

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