# Block under an overhang

A block of mass $M$ is positioned underneath an overhang that makes an angle $\beta$ with the horizontal. You apply a horizontal force of $Mg$ on the block. Assume that the friction force between the block and the overhang is large enough to keep the block at rest. What are the normal and friction forces (call them $N$ and $F_f$ ) that the overhang exerts on the block? If the coefficient of static friction is $\mu$, for what range of angles $\beta$ does the block in fact remain at rest?

Problem Source : Introduction to Classical Mechanics

$N = Mg\sin \beta - Mg\cos \beta$

$F_f = \mu (Mg\sin \beta - Mg\cos \beta)$

To achieve a static state,

\begin{aligned} F_f &\geq Mg \cos \beta + Mg \sin \beta \\ \mu (Mg\sin \beta - Mg\cos \beta) &\geq Mg \cos \beta + Mg \sin \beta \\ \mu \sin \beta - \sin \beta &\geq \mu \cos \beta + \cos \beta \\ \tan \beta &\geq \frac{\mu + 1}{\mu - 1} \end{aligned}