A hollow cylinder of radius r is rotating about its own vertical axis. Both ends of the cylinder are open. A piece of stone of mass m remains fixed on the inner side of the cylinder. What should be the minimum velocity of rotation of the cylinder so that the stone remains fixed on the surface of the cylinder without falling down? Coefficient of static friction between the wall of the cylinder and the stone = `mu`, acceleration due to gravity = g. Also, prove that this velocity is independent of the mass of the stone.

Explanation

The forces acting on the piece of stone are-
(1) weight of the stone, W = mg
(2) limiting frictional force, F = `mu`R
[ `mu` = coefficient of friction,
R = normal force of the wall ,]
Minimum velocity of rotation = v,
Radius of the cylinder = r.
In equilibrium (i.e., when the stone remains fixed to the wall),
W = mg = F
or, mg = `mu`R
Again, R = `(mv^(2))/(r)`
`therefore " " mg = mu cdot (mv^(2))/(r) " " or, v^(2) = (gr)/(mu) " " or, v = sqrt((gr)/(mu))`
The above expression is independent of mass.