JEE Physics 2025 Question with Solution

Given: Bodies A, B and C are a disc, solid sphere and spherical shell respectively, each of mass $M$ and radius $R$. The axis is PQ. Also, $I$ is the moment of inertia of the disc about its diameter.

Find: The value of $x$ if the total moment of inertia is $\frac{x}{15} I$.

For the disc,

$$I = I_{\text{disc}} = \frac{1}{4}MR^2$$

For the solid sphere about its own centre,

$$I_{\text{sphere}} = \frac{2}{5}MR^2$$

Using the parallel axis theorem with distance $d = R$,

$$I_{\text{sphere, PQ}} = I_{\text{sphere}} + Md^2 = \frac{2}{5}MR^2 + MR^2 = \frac{7}{5}MR^2$$

For the spherical shell about its own centre,

$$I_{\text{shell}} = \frac{2}{3}MR^2$$

Again using the parallel axis theorem with $d = R$,

$$I_{\text{shell, PQ}} = I_{\text{shell}} + Md^2 = \frac{2}{3}MR^2 + MR^2 = \frac{5}{3}MR^2$$

Hence, total moment of inertia about PQ is

$$I_{\text{total}} = I_{\text{disc, PQ}} + I_{\text{sphere, PQ}} + I_{\text{shell, PQ}}$$ $$I_{\text{total}} = \frac{1}{4}MR^2 + \frac{7}{5}MR^2 + \frac{5}{3}MR^2$$

Taking the LCM as $60$,

$$I_{\text{total}} = \left(\frac{15}{60} + \frac{84}{60} + \frac{100}{60}\right)MR^2 = \frac{199}{60}MR^2$$

Now given

$$\frac{x}{15}I = \frac{199}{60}MR^2$$

Substitute $I = \frac{1}{4}MR^2$:

$$\frac{x}{15}\cdot \frac{1}{4}MR^2 = \frac{199}{60}MR^2$$ $$\frac{x}{60} = \frac{199}{60}$$ $$x = 199$$

Therefore, the value of $x$ is $199$.

Given: $I = \frac{1}{4}MR^2$ for the disc about its diameter.

Find: The value of $x$ in $I_{PQ} = \frac{x}{15}I$.

Directly write the three contributions about PQ:

$$I_A = \frac{1}{4}MR^2, \qquad I_B = \frac{7}{5}MR^2, \qquad I_C = \frac{5}{3}MR^2$$

So,

$$I_{PQ} = MR^2\left(\frac{1}{4} + \frac{7}{5} + \frac{5}{3}\right) = \frac{199}{60}MR^2$$

But,

$$\frac{x}{15}I = \frac{x}{15}\cdot \frac{1}{4}MR^2 = \frac{x}{60}MR^2$$

Comparing coefficients of $MR^2$,

$$\frac{x}{60} = \frac{199}{60}$$ $$x = 199$$

This works because every term is proportional to $MR^2$, so comparing coefficients is enough. Thus, the value of $x$ is $199$.