A small spherical ball of radius ' r ' is rolling on a curved surface which is frictionless and has a radius of curvature ' R '. Its motion is simple harmonic. Then its time period of oscillation is proportional to ( g = acceleration due to gravity)

Step-by-step solution

The motion of the ball's center of mass along the frictionless curved surface is analogous to that of a simple pendulum. Since the surface has radius of curvature R , the center of mass oscillates at a distance L_ eff = R - r from the center of curvature. For small displacements, the time period is T = 2 L_ eff g = 2 R - r g . The time period is therefore proportional to R - r g .