The pulley in the figure (Figure 1) has radius R and a moment of inertia I . The

Question

The pulley in the figure (Figure 1) has radius R and a moment of inertia I . The rope does not slip over the pulley, and the pulley spins on a frictionless axle. The coefficient of kinetic friction between block A and the tabletop is k . The system is released from rest, and block B descends. Block A has mass mA and block B has mass mB .

Part A

Use energy methods to calculate the speed of block B as a function of the distance d that it has descended.

Express your answer in terms of the variables mA , mB , R , I , k , d and appropriate constants.

Explanation / Answer

Take the zero point for potential energy to be the starting lowest height for block (B). As (B)
falls, its potential energy will decrease. This choice is arbitrary. You could
also have choosen the highest point to be U = 0 in which case Ufinal = -mgd

Recall, work done by friction changes the energy of the system:
Wf = E - E0
=> -uk*mA*g*d = (I*w^2)- mB*g*d

We need one more relation connecting the angular speed of the pulley. If the string does not slip
then
v = Rw => mB*g*d - uk*mA*g*d = 1/2(mA+mB+I/R^2)v^2
Solving for v
v = (2*(mBg*d-uk*mA*g*d)/(mA+mB+I/R^2))^1/2

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