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 inertia of the given system about PQ is , where is the moment of inertia of the disc ab

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Solution

Q.
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 PQ is $\frac{x}{15} I$ , where $I$ is the moment of inertia of the disc about its diameter. The value of x is _________. [2025]

Ans.
(199)

All bodies have same mass and same radius.

Disk, $I_{A} = \frac{m R^{2}}{4} = I$

Solid sphere, $I_{B} = \frac{7}{5} m R^{2}$

Spherical shell, $I_{C} = \frac{5}{3} m R^{2}$

$I_{P Q} = \frac{m R^{2}}{4} + (\frac{2}{5} m R^{2} + m R^{2}) + (\frac{2}{3} m R^{2} + m R^{2})$

$I_{P Q} = \frac{15 m R^{2} + 24 m R^{2} + 60 m R^{2} + 40 m R^{2} + 60 m R^{2}}{60}$

$I_{P Q} = \frac{199}{60} m R^{2} = \frac{199}{15} (\frac{m R^{2}}{4}) = \frac{199}{15} I$

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