First slide

Moment of intertia

Question

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 diameter of third shell. Moment of inertia of the system consisting of these three spherical shells about XX' axis is [CBSE AIPMT 2015]

Moderate

A

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

$3 m r^{2}$
C

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

$4 m r^{2}$

Solution

The total moment of inertia of the system,
$I = I_{1} + I_{2} + I_{3}$ …(i)
Here, $I_{1} = \frac{2}{3} m r^{2}$
$I_{2} = I_{3} = \frac{2}{3} m r^{2} + m r^{2}$ (From parallel axes theorem)
$= \frac{5}{3} m r^{2}$
From Eq. (i), we get
$I = \frac{2}{3} m r^{2} + 2 \times \frac{5}{3} m r^{2} = m r^{2} (\frac{2}{3} + \frac{10}{3}) \Rightarrow I = 4 m r^{2}$

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