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

Explanation

Let it rotates by angle `alpha`

`mgcosalpha-N=(mv^(2))/(R)`

It looses contact i.e. N = 0
`impliesmgcosalpha=(mv^(2))/(R)`
`=v^(2)=gRcosalpha" "....(i)`
Also applying work energy theorem
`mgh=(1)/(2)m(v^(2)-0)`
`impliesmgR(1-cosalpha)=(1)/(2)m(gRcosalpha-0)`

`impliescosalpha=(1)/(3)impliesalpha=cos^(-1)((1)/(3))`
Also from equation (i), `v=sqrt((gR)/(3))`