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?

Screenshot 2015-01-18 at 12.53.22

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}

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Block under an overhang

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