Question

A ball of radius r rolls inside a heavy spherical shell of radius R.It is released from rest from point A.What is the angular velocity of the centre of the ball in position B about the center of shell

Answer 1

see the diagram. 
Initial position  A and final position B.

OC = (R - r) Sin α
OD = (R - r) Sin (α+ Ф)
CD = h = (R -r) 2 SinФ/2 Cos(α+Ф/2)

moment of inertia of the ball about diameter = 2/5 * m r²
v = r ω

we apply the law of conservation of energy.

m g h = 1/2 m v² + 1/2 I ω²
=> 2 g h =  r² ω² + 1/5 * r² ω²
     ω² = (5/3) g h / r²
     ω = √ [10 g (R - r) SinФ/2 Cos (α+Ф/2) / r²]

If  α = 0, then    ω = (1/r) * √ [ 5 g (R - r) SinФ ]