JEE Physics 2025 Question with Solution

Given: A uniform circular disc has radius $R = 20 \, \text{cm}$ and its center is at the origin. A circular hole of radius $r = 5 \, \text{cm}$ is removed such that the edge of the hole touches the edge of the disc.

Find: The distance of the center of mass of the remaining disc from the origin.

Treat the removed hole as a negative-mass region. Since the disc is uniform, mass is proportional to area.

The center of the hole is at a distance

$$R - r = 20 - 5 = 15 \, \text{cm}$$

from the origin along the $+x$-axis.

Using the center of mass relation for composite bodies,

$$X_{\text{cm}} = \frac{A_1 X_1 + A_2 X_2}{A_1 + A_2}$$

For the remaining disc, the removed area is taken as negative:

$$X_{\text{cm}} = \frac{A_1 x_1 + (-A_2)x_2}{A_1 - A_2}$$

where

$$A_1 = \pi R^2 = \pi (20)^2 = 400\pi \, \text{cm}^2$$ $$A_2 = \pi r^2 = \pi (5)^2 = 25\pi \, \text{cm}^2$$ $$x_1 = 0, \qquad x_2 = 15 \, \text{cm}$$

Substituting,

$$X_{\text{cm}} = \frac{(400\pi)(0) + (-25\pi)(15)}{400\pi - 25\pi}$$ $$X_{\text{cm}} = \frac{-375\pi}{375\pi} = -1 \, \text{cm}$$

The negative sign shows that the center of mass shifts towards the side opposite to the hole. Therefore, its distance from the origin is $1.0 \, \text{cm}$.

The correct option is D.

Given: Original disc radius $20 \, \text{cm}$, removed disc radius $5 \, \text{cm}$.

Find: Distance of the center of mass of the remaining part from the origin.

Because the smaller disc touches the boundary of the larger disc, the center of the removed disc is at

$$15 \, \text{cm}$$

from the origin.

For a uniform lamina, area can be used in place of mass. The original disc has area $400\pi$, while the removed disc has area $25\pi$. Thus the removed part is

$$\frac{25\pi}{400\pi} = \frac{1}{16}$$

of the full disc.

So the shift of the center of mass from the origin is

$$\frac{\left(\frac{1}{16}\right)(15)}{1 - \frac{1}{16}} = \frac{15/16}{15/16} = 1 \, \text{cm}$$

The direction is opposite to the removed hole, but the required distance is $1 \, \text{cm}$.

Hence, the correct option is D.