.A horizontal stick of mass m has its right end attached to a pivot on a wall, while its left end rests on the top of a cylinder of mass m which in 
turn rests on an incline plane inclined at an angle $\theta$. The stick remains horizontal. The coefficient of friction between the cylinder and both the plane and the stick is $\mu$Find the minimum value of $\mu$ as function of  $\theta$  for which the system stays in equilibrium
 

For cylinder 

${\mathrm{f}}_1+{\mathrm{f}}_2\cos \mathrm{\theta }={\mathrm{N}}_2\sin \mathrm{\theta }$…………………..$\left(1\right)$
and ${\mathrm{N}}_2\cos \mathrm{\theta }+{\mathrm{f}}_2\sin \mathrm{\theta }=\mathrm{mg}+{\mathrm{N}}_1\dots \dots \dots \dots \dots \text{ (2) }$

Rotational equilibrium : ${\mathrm{f}}_1\mathrm{R}={\mathrm{f}}_2\mathrm{R}$
$\Rightarrow {\mathrm{f}}_1={\mathrm{f}}_2=\mathrm{f}\text{ (say) }$
Equation (1) and (2) become
$\mathrm{f}+\mathrm{fcos}\mathrm{\theta }={\mathrm{N}}_2\sin \mathrm{\theta }$ …..(3)
and ${\mathrm{N}}_2\cos \mathrm{\theta }+\mathrm{fsin}\mathrm{\theta }=\mathrm{mg}+{\mathrm{N}}_1$  …..(4)
For stock $\mathrm{mg}\frac{\mathrm{l}}{2}={\mathrm{N}}_1\mathrm{l}$
$\Rightarrow {\mathrm{N}}_1=\frac{\mathrm{mg}}{2}$ …….(5)

From (3) $\mathrm{f}=\left(\frac{\sin \mathrm{\theta }}{1+\cos \mathrm{\theta }}\right){\mathrm{N}}_2\dots \dots \dots \left(6\right)$
$\therefore \left(\frac{\sin \mathrm{\theta }}{1+\cos \mathrm{\theta }}\right){\mathrm{N}}_2\le {\mathrm{\mu N}}_2$
$\therefore \frac{\sin \mathrm{\theta }}{1+\cos \mathrm{\theta }}\le \mathrm{\mu }. ......\left(7\right)$
Substituting for N2 from (6) into (4)

$$\begin{gathered} \mathrm{f}\left(\frac{1+\cos \mathrm{\theta }}{\sin \mathrm{\theta }}\right)\cos \mathrm{\theta }+\mathrm{fsin}\mathrm{\theta }=\mathrm{mg}+{\mathrm{N}}_1 \\ \mathrm{f}\left[\frac{\cos \mathrm{\theta }+{\cos }^2\mathrm{\theta }+{\sin }^2\mathrm{\theta }}{\sin \mathrm{\theta }}\right]=3{\mathrm{N}}_1\left[\because {\mathrm{N}}_1=\frac{\mathrm{mg}}{2}\right] \\ \therefore \mathrm{f}=\left(\frac{3\sin \mathrm{\theta }}{1+\cos \mathrm{\theta }}\right){\mathrm{N}}_1 \\ \therefore \frac{3\sin \mathrm{\theta }}{1+\cos \mathrm{\theta }}\cdot {\mathrm{N}}_1\le {\mathrm{\mu N}}_1 \\ \therefore \frac{3\sin \mathrm{\theta }}{1+\cos \mathrm{\theta }}\le \mathrm{\mu }. .........\left(8\right) \\ \mathrm{Note} :{\mathrm{f}}_1={\mathrm{f}}_2\text{ does not mean that }{\mathrm{N}}_1={\mathrm{N}}_2 \end{gathered}$$
From $\left(7\right)$ and $\left(8\right)$
$\mathrm{\mu }\ge \frac{3\sin \mathrm{\theta }}{1+\cos \mathrm{\theta }}$ $\Rightarrow {\mu }_{min}=3 \tan \frac{\theta }{2}$