A hemispherical bowl of radius R si set rotating abouv its axis of symmetry whichis kept vertical. A small blcok kept in the bowl rotates with the bowl without slippingn on its surface. If the surfaces of the bowl is mooth, and the abgel made by the radius through the block with the vertical is `theta`, find the angular speed at which the bowl is rotating.

Explanation

O is center of bowl of radius OA = R
The block is at A and ` angleAOC = theta`
The block is moving in a horizontal circle with centre C and radius AC = r
As `sin theta = (AC)/(OA) = (r)/(R) ,r = R sin theta`
In equilibrium, ` N cos theta = mg` and
`N sin theta = m r omega^(2)`
`= m (R sin theta) omega^(2)`
` :. N = m R omega^(2)`
`:. mg = N cos theta = m R omega^(2) cos theta`
`omega = sqrt((g)/(Rcos theta))`