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 X X' 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 X X' axis is

Moderate

A

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

$4 m r^{2}$
C

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

$3 m r^{2}$

Solution

Net moment of inertia of the system,
$I = I_{1} + I_{2} + I_{3}$
$The moment of inertia of a shell about its diameter,$
$I_{1} = \frac{2}{3} m r^{2}$
$T h e m o m e n t o f i n e r t i a o f a s h e l l a b o u t i t s \tan g e n t i s g i v e n b y$
$I_{2} = I_{3} = I_{1} + m r^{2} = \frac{2}{3} m r^{2} + m r^{2} = \frac{5}{3} m r^{2}$
$∴ I = 2 \times \frac{5}{3} m r^{2} + \frac{2}{3} m r^{2} = \frac{12 m r^{2}}{3} = 4 m r^{2}$

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