Use the figure to answer the problem. A spring of negligible mass and spring constant k is attached to the wall on one end, and the other end is attached to the center of mass axis of a solid sphere of mass m and radius r. The spring is stretched a distance A from equilibrium and let go. The sphere rolls freely on a smooth surface.
-What is the linear kinetic energy when the sphere is at the equilibrium position?
E) undetermined

Question

Use the figure to answer the problem. A spring of negligible mass and spring constant k is attached to the wall on one end, and the other end is attached to the center of mass axis of a solid sphere of mass m and radius r. The spring is stretched a distance A from equilibrium and let go. The sphere rolls freely on a smooth surface. 
-What is the linear kinetic energy when the sphere is at the equilibrium position?
E) undetermined

Quizplus · Accepted Answer

At equilibrium position, the spring is unstretched and all the potential energy stored in the spring during the displacement gets converted to kinetic energy. The potential energy stored in the spring is given by (1/2)kA^2. The kinetic energy of rolling motion of the sphere is (1/2)(2/5)mr^2(omega)^2, where omega is the angular velocity of the sphere. As the sphere is rolling freely without slipping, the linear velocity of the sphere is related to the angular velocity as v = r*omega. 
At equilibrium position, the sphere has only linear velocity(v) and no angular velocity. So, the kinetic energy of rolling motion is zero. Therefore, the total kinetic energy of the sphere when it reaches equilibrium position is entirely due to the potential energy stored in the spring. 
Thus, Total Kinetic Energy = (1/2)kA^2. As there is no other information given about the values of k and A, the answer cannot be determined numerically. But we can conclude that B is the best choice as it is the only option that correctly identifies the situation that the kinetic energy is entirely due to the potential energy stored in the spring, and not due to the rolling motion of the sphere.

Use the Figure to Answer the Problem