(a) and (b)

A solid cylinder is attached to a horizontal massless spring so that it can ‘roll without slipping’.

Force constant k of the spring is

If the system is released from rest at a position when the spring is stretched by a = 23.9 cm, the total mechanical energy of the system is equal to the stored potential energy of the spring in that position, . If we assume that the radius of the cylinder is R, its rotational inertia about the axis perpendicular to the plane of the diagram will be

.

At the equilibrium position the spring will be relaxed and so its stored potential energy will be zero. Therefore, when the cylinder passes through the equilibrium position of its SHM, its mechanical energy will comprise of the translational kinetic energy and the rotational kinetic energy. Let v be the translational speed and be the angular speed of the cylinder when it passes through the equilibrium position of the spring. As the cylinder is rolling without slipping, and v are related as

Energy conservation equation is

Substituting expressions for I and , and after making algebraic simplifications, we get

Therefore, the translational kinetic energy of the cylinder at the equilibrium position of the spring is

And, the rotational energy of the cylinder

(c)

At a general displacement x of the spring from its equilibrium position the equation of energy conservation is

Differentiating this equation with respect to time, t, we get

This is an equation of SHM, with period T given by