A hollow sphere of mass `M` and radius `R` 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

Mass m moving with velocity v creating rotation in sphere

Conservation of linear momentum
`mv=(M+m)v_(CM)rArrv_(CM)=(mv)/((M+m))`
Conservation of angular momentum about O
`mvh=I_(0)omega=((2)/(3)MR^(2)+mR^(2))omega`
`omega=(mvh)/(((2)/(3)M+m)R^(2))`
For pure rolling, `v_(CM) = omega R`
`(mv)/(M+m)=(mvh)/(((2)/(3)M+m)R)`
`h=(((2)/(3)M+m)R)/((M+m))=(2R)/(3)" "("Since", m lt ltM)`.