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 is $\mu \ge \frac{a\sin \theta }{b+\cos \theta }.$ then the value of a+b is _____
 
 

For cylinder
$f_1+f_2\cos \theta +N_2\sin \theta \mathrm{...........}\left(1\right)$ 
And $N_2\cos \theta +f_2\sin \theta =mg+N_1\mathrm{.........}\left(2\right)$  
Rotational equilibrium: $f_{1}R=f_{2}R\Rightarrow f_1=f_2=f$  
Equation (1) and (2) become $f+f\cos \theta =N_2\sin \theta \mathrm{....}\left(3\right)$
and $N_2\cos \theta +f\sin \theta =mg+N_1\mathrm{.........}\left(4\right)$  
For stick $mg\frac{l}{2}=N_{1}l\Rightarrow N_1=\frac{mg}{2}\mathrm{........}\left(5\right)$  
From (3) $f=\left(\frac{\sin \theta }{1+\cos \theta }\right)N_2\mathrm{..........}\left(6\right)$  
$\therefore \left(\frac{\sin \theta }{1+\cos \theta }\right)N_2\le \mu N_2$ 
$\therefore \frac{\sin \theta }{1+\cos \theta }\le \mu \mathrm{.......}\left(A\right)$ 
Substituting for $N_2$ form (6) into (4)
$f\left(\frac{1+\cos \theta }{\sin \theta }\right)\cos \theta +f\sin \theta =mg+N_1$ 
$f\left[\frac{\cos \theta +{\cos }^2\theta +{\sin }^2\theta }{\sin \theta }\right]=3N_1[\therefore N_1=\frac{mg}{2}]$ 
$\therefore f=\left(\frac{3\sin \theta }{1+\cos \theta }\right)N_1$ 
$\therefore \frac{3\sin \theta }{1+\cos \theta }.N_1\le \mu N_1$ 
$\therefore \frac{3\sin \theta }{1+\cos \theta }\le \mu \mathrm{........}\left(B\right)$ 
Note: $f_1=f_2$ does not mean that $N_1=N_2$  
From A and B

 $\mu \ge \frac{3\sin \theta }{1+\cos \theta }$