A small body of mass m slides down from the top of a hemisphere or radius r (Fig. 2). The surface of block and hemisphere are frictionless. The height at which the body lose contact with the surface of the sphere is

See fig. (8). The body will lose contact when centripetal acceleration becomes equal to the component of acceleration due to gravity along the radius.Velocity at P,       v=2g(r−h)∵        v2−u2=2gxCentripetal acceleration will be v2/r. It should be equal to the component of g along PO. Hencev2r=gcos⁡θor             2g(r−h)r=g×hrSolving, we get h=2r3