question_answer

Three identical spherical shells, each of mass m and radius r are placed as shown in figure.

Consider an axis XX’, which is touching to two shells and passing through diameter of third shell.

Moment of inertia of the system consisting of these three spherical shells about XX’ axis is [NEET 2015 ]

A) \[\frac{11}{5}m{{r}^{2}}\]

B) \[3\,m{{r}^{2}}\]

C) \[\frac{16}{5}m{{r}^{2}}\]

D) \[4\,m{{r}^{2}}\]

Correct Answer: D

Solution :

The total moment of inertia of the system is

\[I={{I}_{1}}+{{I}_{2}}+{{I}_{3}}\]                          (i)

Hero    \[{{I}_{1}}=\frac{2}{3}m{{r}^{2}}\]

\[{{I}_{2}}={{I}_{3}}=\frac{2}{3}\,m{{r}^{2}}+m{{r}^{2}}\]

(From parallel axis theorem]

\[=\frac{5}{3}m{{r}^{2}}\]

From Eq. (i),

\[I=\frac{2}{3}m{{r}^{2}}+2\times \frac{5}{3}m{{r}^{2}}=m{{r}^{2}}\left( \frac{2}{3}+\frac{10}{3} \right)\]

\[I=4m{{r}^{2}}\]