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 lower shells and passing through diameter of third shell. Moment of inertia of the system consisting of these three spherical shells about XX’ axis is

Easy

A

$\large 4mr^2$
B

$\large \frac {11}5mr^2$
C

$\large 3mr^2$
D

$\large \frac {16}5mr^2$

Solution

$\large I = {I_1} + {I_2} + {I_3}$
$\large Here,{I_1} = \frac{2}{3}m{r^2}$
$\large {I_2} = {I_3} = \frac{2}{3}m{r^2} + m{r^2}$
[from parallel axis theorem]
$\large = \frac{5}{3}m{r^2};$ From Eq.(i)
$\large I = \frac{2}{3}m{r^2} + 2 \times \frac{5}{3}m{r^2} = m{r^2}\left( {\frac{2}{3} + \frac{{10}}{3}} \right)$
$\large I = 4m{r^2}$

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