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

Explanation

For equilibrium
`SigmaF_(x)=0, SigmaF_(y)=0`

and `Sigmatau=0`
For `SigmaF_(x)=0`
`f=N_(b)sintheta`………..i
for `SigmaF_(y)=0`
`N_(A)+N_(B)costheta=mg`…………….ii
Taking torque of forces about `A`
`N_(A) h/(sintheta)=mgl/2 cos theta`
that gives
`N_(A)=(mgl)/(2h)sinthetacostheta` .............iii
From eqn ii and iii we get
`(mgl)/(2h)sinthetacostheta+N_(B)costheta=mg`
`N_(B)costheta=mg(1-1/(2h)sinthetacostheta)` ...........iv
From eqn i and iv we get
`f=mgtantheta(1-1/(2h)sinthetacostheta)`........v
But `flemuN_(A)`
`muge(Lcosthetasin^(2)theta)/(2h-Lcos^(2)thetasintheta)`