A small spherical ball of mass m rolls down a loop track. The ball is released on the linear portion at a vertical height H from the lowest point. The circular part shown has a radius R . Determine the radial and the tangential accelerations of the centre when the ball is at a point A , where the radius makes an angle with the horizontal. (Take the direction of velocity at A as the positive tangential direction).

Step-by-step solution

Let the lowest point of the track be the reference level for potential energy. The height of point A from the lowest point is h = R + R . By conservation of mechanical energy, the loss in potential energy of the ball is equal to its gain in kinetic energy. For a solid sphere rolling without slipping, the total kinetic energy is the sum of translational and rotational kinetic energies: K = 1 2 mv^2 + 1 2 I ^2 Substituting I = 2 5 mr^2 and = v r : K = 1 2 mv^2 + 1 2 ( 2 5 mr^2 ) ( v r )^2 = 7 10 mv^2 Equating the los