$\mathrm{A}, \mathrm{B}$ and C are disc, solid sphere and spherical shell respectively with same radii and masses. These masses are placed as shown in figure.

The moment of inertia of the given system about $P Q$ is $\frac{X}{15} I$, where $I$ is the moment of inertia of the disc about its diameter. The value of x is _______.

Solution

All bodies have same mass and same radius.
$\mathrm{A} \rightarrow$ Disc
$\mathrm{B} \rightarrow$ Solid sphere
$\mathrm{C} \rightarrow$ Spherical shell
and, $\mathrm{I}=\frac{\mathrm{MR}^2}{4}$
$\mathrm{I}_{\mathrm{PQ}}=\frac{\mathrm{MR}^2}{4}+\left(\frac{2}{5} \mathrm{MR}^2+\mathrm{MR}^2\right)+\left(\frac{2}{3} \mathrm{MR}^2+\mathrm{MR}^2\right)$
$\mathrm{I}_{\mathrm{PQ}}=\frac{15 \mathrm{MR}^2+24 \mathrm{MR}^2+60 \mathrm{MR}^2+40 \mathrm{MR}^2+60 \mathrm{MR}^2}{60}$
$\mathrm{I}_{\mathrm{PQ}}=\frac{199}{60} \mathrm{MR}^2=\frac{199}{15}\left(\frac{\mathrm{MR}^2}{4}\right)$
$=\frac{199}{15} \mathrm{I}$

Hence, the correct answer is 199.