Question

The pulleys shown in the following figure are identical, each having a radius R and moment of inertia I. Find the acceleration of the block M.

Answer 1

Free the body diagram of the system,

For block of mass M,

\[Mg -  T_1  = Ma ..........(1)\]

\[\left( T_1 - T_2 \right)  R = I\alpha\text{ using, }a =\alpha r\] 

\[ \Rightarrow \left( T_1 - T_2 \right) = I\frac{a}{R^2}......(2)\left(\text{For pully 1} \right)\] 

\[\text{Similarly, }\left( T_2 - T_3 \right) = I\frac{a}{R^2}..........(3)\left(\text{For pully 2}\right)\]

For block of mass m,

\[T_3  - mg = ma.........(4)\left(\text{For block m} \right)\]

Adding equations (2) and (3), we get

\[\left( T_1 - T_3 \right) = \frac{2Ia}{R^2}..........(5)\]

Adding equations (1) and (4), we get

\[- mg + Mg + \left( T_3 - T_1 \right) = Ma + ma..........(6)\]

Using equations (5) and (6), we get

\[Mg - mg   =   Ma + ma + \frac{2Ia}{R^2}\]

\[ \Rightarrow a = \frac{\left( M - m \right)g}{\left( M + m + \frac{2I}{R^2} \right)}\]