A, B and C are disc, solid sphere and spherical shell respectivel… - Vidyadip

Question

$A, B$ 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 __________ .

✓

Answer

199

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

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