Question 4

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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