Question

A particle slides on the surface of a fixed smooth sphere starting from the topmost point. Find the angle rotated by the radius through the particle, when it leaves contact with the sphere.

Answer 1

When the body leaves surface ,Assume that the velocity = v.

If the Normal force on the block is N and weight is mg

Centripetal force = mg cosθ -N

Here , velocity
mgcosθ -N =mv² /R
so when the particle leaves surface the Normal force = 0
mg cosθ = mv²/R
v² =Rg cosθ

Therefore change in height particle ,
R-R cosθ

Change in Potential energy 
mg (R-R cosθ)

Kinetic energy will increase in result of change in potential energy
So,  
mg(R-R cosθ)=1/2 mv² =1/2 mgr.cosθ
1-cosθ=1/2.cosθ
3/2 × cosθ = 1
cosθ = 2/3
____________

θ = cos⁻¹(2/3)   
____________ 


Hope it Helps. :-)

Answer 2

Explanation:

See the attachment.

Hope it helps you.