A spherical ball of mass m and radius r rolls without slipping on a rough concave surface possessing a large radius R . It executes small oscillations about the lowest point. Determine the time period.

Step-by-step solution

Let be the angular displacement of the center of the ball from the vertical. The center of the ball moves in a circle of radius R - r . The velocity of the center of the ball is v = (R - r) . For pure rolling without slipping, the angular velocity of the ball about its center is = v r = (R - r) r . The total kinetic energy of the ball is: K = 1 2 mv^2 + 1 2 I ^2 Substituting I = 2 5 mr^2 for a solid sphere: K = 1 2 mv^2 + 1 2 ( 2 5 mr^2 ) ( v r )^2 = 7 10 mv^2 = 7 10 m(R - r)^2 ^2 The potential energy of the ball r