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

Explanation

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)`