Solve the following :
Question:

A uniform rod of length $L$ sets against a smooth roller as shown in figure. Find the friction coefficient between the ground and the lower end if the minimum angle that the rod can make with the horizontal is $\theta$.

Solution:

Translatory equilibrium

$N_{1} \cos \theta+N_{2}=m g g_{-(i)}$

$N_{1} \sin \theta=f f$-(ii)

Rotational Equilibrium about bottom

$N_{1}\left(\frac{\hbar}{\sin \theta}\right)=m g\left(\frac{L}{2} \cos \theta\right)$

$f f=\mu N_{2}$-(iv)

Solving (i),(ii),(iii) and (iv)

$\mu=\frac{L \cos ^{2} \theta \sin ^{2} \theta}{2 h-L \cos ^{2} \theta \sin ^{2} \theta}$