A flat horizontal belt is running at a constant speed V. There is a uniform solid cylinder of mass M which can rotate freely about an axle passing through its centre and parallel to its length. Holding the axle parallel to the width of the belt, the cylinder is lowered on to the belt. The cylinder begins to rotate about its axle and eventually stops slipping. The cylinder is, however, not allowed to move forward by keeping its axle fixed. Assume that the moment of inertia of the cylinder about its axle is `(1)/(2) MR^(2)` where M is its mass and R its radius and also assume that the belt continues to move at constant speed. No vertical force is applied on the axle of the cylinder while holding it.
(a) Calculate the extra power that the motor driving the belt has to spend while the cylinder gains rotational speed. Assume coefficient of friction `= mu`.
(b) Prove that `50%` of the extra work done by the motor after the cylinder is placed over it, is dissipated as heat due to friction between the belt and the cylinder

Answer

(a) `P = mu m g. V`