Mechanics....not so easy one
A hemisphere of radius R and of mass 4m is free to slide with its base on smooth horizontal table.
A particle of mass m is placed on the top of the hemisphere.
The angular velocity of the particle relative to the hemisphere at an angular displacement
$ when the velocity of hemisphere has become v is:
Note : $=angle = theta.
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1 solution
When the hemisphere has moved right by distance x the particle has slid down through angle θ. There are no external forces acting horizontally so the CM remains in the same position horizontally : m(Rsinθ−x)=Mx mRsinθ=(M+m)x Differentiate wrt time : mRωcosθ=(M+m)v where v=x˙ and ω=θ˙. We are given that M=4m so ω=5v/Rcosθ.