Question

A mass of 3kg hangs from a pulley (I=3/4 MR^(2), M=5kg, R=0.5m) as shown. What is the tension in the cable?

What is the acceleration and final velocity when h=1.5m?

Answer 1

Consider the angular acceleration of the pulley is as shown in the figure above.

Then the acceleration of the rope in downward direction is

This is the acceleration with which the block of mass 3kg will goes down.

FBD-1:

From the above FBD, at equilibrium, algebraic sum of forces in vertical direction must be zero.

Therefore,

     

     

                                           ...... (1)

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Since the pulley is having mass , the torque acted at centre O due to angular acceleration is given by

     

here is moment of inertia of pulley about O and from given data,

                              

     (since )

                                ...... (2)

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In order to avoid slippage of rope over pulley, this torque must be balanced by the torque generated due to tension in cable. That means,

       

       

From (1) and (2)

       

       

Substitute ; and

       

       

       

Thus, the acceleration of the mass 3kg is

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Assume the block starts from rest. Then, the initial velocity of the block is

Final velocity is given by

substitute and ,

      

      

Thus, the final velocity is