A hemisphere of radius R and of mass 4m is free to Slide with its base on a smooth horizontal table. A particle mass m is placed on the top of the hemisphere. Find the angular velocity of the particle relative to hemisphere at an angular displacement theta when velocity of hemisphere has become v.

A hemisphere of radius R and of mass 4m is free to Slide with its base

1.	A hemisphere of radius R and of mass 4m is free to Slide with its base on a smooth horizontal table. A particle mass m is placed on the top of the hemisphere. Find the angular velocity of the particle relative to hemisphere at an angular displacement theta when velocity of hemisphere has become v.
Answer» Solution :Actual velocilty of ball is `v_(1)` is horizontal direction. ` :. v_(1)-(-v)=v_(rel)costheta` `:. v_(1)+v=v_(rel) costheta`……….i Applying conservation PRINCIPLE of momentum in horizotal direction. `P_(ix)=P_(fx)` `0=mv_(1)-4mv` `:. v_(1)=4V`.......ii Applying mechanic energy conservation principle. Loss in `PE=` gain in `KE` `mg(r-rcostheta)=1/2m[sqrt(v_(1)^(2)+(v_(rel)sitheta)^(2))]^(2)+1/24mv^(2)`.........IIII after solving eqn i , ii , iii `v_(rel)=(5v)/(costheta)` `:. omega_(rel)=v_(rel)/R=(5v)/(Rcostheta)`