A hollow sphere of mass `M` lies on a rough horizontal plane when a particle of mass `m` traveling with speed `v` collides and sticks with it. If line of motion of the particle is at height `h` above the center of sphere, find `h` if the body rolls without slipping after collision (`m lt lt M`).

Explanation

Conservation of linear momentum
`mv=(M+m)v_(c.m.)implies v_(c.m.)=(mv)/(M+m)`
Conservation of angular momentum about `O`
`mvh=I_(0)omega=((2)/(3)MR^(2)+mR^(2))omega`
` implies omega=(mvh)/(((2)/(3)M+m)R^(2))`
For pure rolling
`v_(c.m.)=Romega`
`(mv)/(M+m)=(mvh)/(((2)/(3)M+m)R)`
`h=(((2)/(3)M+m)R)/(M+m)=(2R)/(3)`(sincem`lt lt M`)