A solid sphere is set into motion on a rough horizontal surface, possessing a forward linear speed v and an anticlockwise angular speed v/R . Determine the linear speed of the sphere at the moment slipping finally ceases and pure rolling begins.

Step-by-step solution

Let the mass of the solid sphere be m and its radius be R . The initial linear velocity of the sphere is v (forward) and its initial angular velocity is ₀ = v R (anticlockwise). Taking the clockwise direction as positive, the initial angular momentum of the sphere about a fixed point on the ground is: L_i = m v R - I ₀ Since the moment of inertia of a solid sphere is I = 2 5 m R^2 , we have: L_i = m v R - ( 2 5 m R^2 ) ( v R ) = m v R - 2 5 m v R = 3 5 m v R Let the final linear velocity of the sphere when pure rol