Question

A thin cylindrical rod of uniform mass density, rests against a smooth wall as shown in figure. Find the coefficientof friction between ground and rod (if the minimum angle that the rod can make with the horizontal is theta)

Answer 1

Step 1: Finding cofficient of friction:

From above figure we can make free body diagram and weuations will be written as follows:

For vertical equilibrium of rod,

$mg=N_2$

For horizontal equilibrium of rod,

$N_1=f_m$

As , is the minimum angle that rod can make to remainin equilibrium, maximum friction,

$f_{max}=\mu N_2$ will be acting acting between rod and ground.

For rotational equilibrium about point A

$N_1\times Lsin\theta -mg\times \frac{1}{2}cos\theta =0$

$N_1=\frac{mgcot\theta }{2}$

$f_m=N_1=\mu N_2$

$\mu mg=\frac{mgcot\theta }{2}or$

$\mu =\frac{cot\theta }{2}$

Coefficientof friction between ground and rod,

$\mu =\frac{cot\theta }{2}$