Three identical spherical shells, each of mass and radius are placed as shown in figure. Consider an axis XX' which is touching the two shells and passing through the diameter of the third shell. Moment of inertia of the system consisting of these three

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Solution

Q.
Three identical spherical shells, each of mass $m$ and radius $r$ are placed as shown in figure. Consider an axis XX' which is touching the two shells and passing through the diameter of the third shell. Moment of inertia of the system consisting of these three spherical shells about XX' axis is [2015]

1 $\frac{16}{5} m r^{2}$

2 $4 m r^{2}$

3 $\frac{11}{5} m r^{2}$

4 $3 m r^{2}$

Ans.
(2)

Net moment of inertia of the system,

$I = I_{1} + I_{2} + I_{3}$

The moment of inertia of a shell about its diameter,

$I_{1} = \frac{2}{3} m r^{2}$

The moment of inertia of a shell about its tangent is given by

$I_{2} = I_{3} = I_{1} + m r^{2} = \frac{2}{3} m r^{2} + m r^{2} = \frac{5}{3} m r^{2}$

$∴$ $I = 2 \times \frac{5}{3} m r^{2} + \frac{2}{3} m r^{2} = \frac{12 m r^{2}}{3} = 4 m r^{2}$

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