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A uniform disc of radius $$R$$ and mass $$M$$ is free to rotate only about its axis. A string is wrapped over its rim and a body of mass $$m$$ is tied to the free end of the string as shown in the figure. The body is released from rest. Then the acceleration of the body is:
Two forces act on the mass $$m$$, weight downwards and tension upwards.
If a is the acceleration of the system in the downwards direction, then
$$mg-T=ma$$
For the disc,
$$τ=T\times R=I\times\alpha\ $$
$$I=\frac{1}{2}MR^2$$ & $$a=R\cdot\alpha\ $$
$$TR=\frac{1}{2}MR^2\frac{a}{R}$$
$$T=\frac{1}{2}Ma$$
$$\therefore\ mg-\frac{1}{2}Ma=ma$$
$$a\left(2m+M\right)=2mg$$
$$\therefore\ a=\frac{2mg}{2m+M}$$
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