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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.
Solution
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
\[Mg - mg = Ma + ma + \frac{2Ia}{R^2}\]
\[ \Rightarrow a = \frac{\left( M - m \right)g}{\left( M + m + \frac{2I}{R^2} \right)}\]
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