# 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:

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.