A solid sphere, a hollow sphere and a disc, all having the same mass and radius, are released from rest at the top of an incline. The friction coefficients are identical and not sufficient to allow pure rolling. The smallest kinetic energy at the bottom of the incline will be achieved by

Step-by-step solution

Since the friction is not sufficient to allow pure rolling, all the objects will slip as they move down the incline. The kinetic friction force acting on each object is f_k = M g . The linear acceleration of the center of mass for all objects is given by: a = g - g Since a is independent of the shape and moment of inertia, all three objects will take the same time t to reach the bottom and will have the same translational velocity v = at . Thus, their translational kinetic energies K_ trans = 1 2 M v^2 will be iden