A hemispherical bowl of radius R is set rotating about its axis of symmetry which is kept vertical. A small block kept in the bowl rotates with the bowl without slipping on its surface. If the surface of bowl is smooth, and the angle made by the radius through the block with the vertical is $\theta$ the angular speed at which the bowl is rotating is

Step 1: Draw the free-body diagram of the system.
 

Step 2 : Balance the forces in the horizontal and vertical directions.

Balancing forces along the vertical direction.

Ncosθ = mg  .............(1)

Balancing forces along the horizontal direction.

ΣF_net​ = m(a_c​)

Nsinθ = m(ω²R′)   .............(2)

Here, r′ = Rsinθ ...................(3)

N is the normal reaction, m is the mass of the block and g is the acceleration due to gravity.

Step 3 : Solve the equations.

Dividing equations 1 and 2 and using equation 3, we get: 

ω = $\sqrt{\frac{g}{r\cos \theta }}$