A metallic pulley is in the shape of a disc of radius a. It can rotate freely about a horizontal axis passing through its centre. The moment of inertia of the pulley about this axis is I. A light string is tightly wrapped around the pulley with its one end connected to a block of mass M. The centre of the pulley and its circumference are connected to a resistance R as shown. The contact of resistance at the circumference does not cause any friction. A uniform magnetic field B is switched on which is parallel to the axis of rotation of the pulley (see Figure). The mass M is allowed to fall. Assume that resistivity of the material of the pulley is negligible.
(a) Find the acceleration of the block of mass M at the instant its velocity becomes `v_(0)`.
(b) Assuming that the block can fall through a large distance, calculate the terminal speed `(v_(T))` that it will acquire.
(c) Find the rate of change of kinetic energy of the pulley at the instant speed of the falling block is `(v_(T))/(2)`.

Answer

(a) `(Mga^(2)-(B^(2)a^(4)v_(0))/(4R^(2)))/(I+Ma^(2))`
(b) `(4MgR^(2))/(B^(2)a^(2))`
(c) `(IM^(2)g^(2)R^(2))/(B^(2)a^(2)(I+Ma^(2)))`