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Question

A hemisphere of radius R and of mass 4m 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. Find the angular velocity of the particle relative to hemisphere at an angular displacement θ when velocity of hemisphere has become v.
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Solution

Let the hemisphere be moved by a distance of x through angle θ. CM remains same as there are no forces acting horizontally

m(Rsinθx)=MxmRsinθ=(M=m)xDifferentiating,mRωcosθ=(M+m)vwherev=xandω=θ.wkt,M=4mω=5vRcosθ

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