A hemispherical bowl of radius R is rotated about its axis of symmetry which is kept vertical. A small block is kept in the bowl at a position where the radius makes an angle θ with the vertical. The block rotates with the bowl without any slipping. The friction coefficient between the block and the bowl surface is μ. Find the range of the angular speed for which the block will not slip.
If the bowl rotates at maximum angular speed, the block tends to slip upwards.
So the frictional force acts downwards.
Here, r=R sin θ
From the free body diagram-1,
R1−mg cos θ−mω21 (R sin θ)sin θ= 0 ...(i)
[because. r=R sin θ] and
μR1+mg sin θ−mω21 (R sin θ)cos θ= 0 ...(ii)
Substituting the value of R1 from equation (i) in equation (ii), it can be found out that,
ω1=[g (sin θ+μ cos θ)R sin θ (cos θ−μ sin θ)]1/2
again, for minimum speed, the frictional force μR2 acts upward. From free body diagram-2, it can be proved that,
ω2=[g (sin θ+μ cos θ)R sin θ (cos θ−μ sin θ)]1/2
∴ the range of speed is between ω2 and ω1.