A uniform disc with mass $$M = 4$$ kg and radius $$R = 10$$ cm is mounted on a fixed horizontal axle as shown in figure. A block with mass $$m = 2$$ kg hangs from a massless cord that is wrapped around the rim of the disc. During the fall of the block, the cord does not slip and there is no friction at the axle. The tension in the cord is ______ N.
(Take $$g = 10$$ ms$$^{-2}$$)

For the hanging block of mass $$m$$ moving downward with linear acceleration $$a$$:

$$mg - T = ma \quad \text{--- (1)}$$

For the rotating uniform disc of mass $$M$$ and radius $$R$$:

$$\tau = I\alpha \implies T \cdot R = \left(\frac{1}{2}MR^2\right)\alpha$$

Since the cord does not slip, the linear acceleration is related to the angular acceleration by $$a = R\alpha$$:

$$T \cdot R = \frac{1}{2}MR^2 \left(\frac{a}{R}\right)$$

$$\implies T = \frac{1}{2}Ma \implies a = \frac{2T}{M} \quad \text{--- (2)}$$

Substituting equation (2) into equation (1):

$$mg - T = m\left(\frac{2T}{M}\right)$$ $$\implies mg = T\left(1 + \frac{2m}{M}\right)$$ $$\implies T = \frac{mg}{1 + \frac{2m}{M}}$$

$$T = \frac{2 \times 10}{1 + \frac{2 \times 2}{4}} = \frac{20}{1 + 1} = 10\ \text{N}$$