A weight with massmis tied to a piece of thread wrapped around a pulley, which is suspended in such a way thatit can rotate freely and without friction about its centre of mass. Answer all of the following in terms of the givenvariables (m, M,R) and the acceleration due to gravity, g.

(a) Turn the drawing into a free-body diagram, clearly indicating yourx-ycoordinatesystem and positive sense of rotation for the pulley.

(b) The pulley is constructed from a thin cylinder of radiusRand massMwith 6 spokeseach of lengthRand mass12M. Determine:

i. the moment of inertia of the cylinder.
ii. the moment of inertia of one of the spokes.
iii. the total moment of inertia of the pulley.

(c) When the weight is released, it accelerates toward the floor as the thread unwinds. Find a relationship between the angular acceleration of the pulley and the tensionin the thread. If you didn’t get part (b)iii, useItotfor the moment of inertia of thepulley here and below.

(d) How is the angular acceleration of the pulley related to the acceleration of the block and radius of the pulley?

(e) Use the answers above and Newton’s Laws to find the acceleration of the weight as it falls.