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 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 :

• A. 4 mr²
• B. (11/5) mr²
• C. 3 mr²
• D. (16/5) mr²

Answer 1

The correct option is A

4 mr²

See figure alongside

A is a spherical shell whose mass is m and radius is r.

Its moment of inertia about the axis is XX’ axis is $I_A=\frac{2}{3}m{r}^2$

B is a spherical shell whose mass is m and radius is r.

Its moment of inertia about its own axis is $I_B=\frac{2}{3}m{r}^2$

Its moment of inertia about XX’ axis is

$I_B^{\prime }=I_B+m{r}^2=\frac{5}{3}m{r}^2$

Similarly the moment of inertia of the spherical shell C about the XX’ axis is

$I_c=\frac{5}{3}m{r}^2$

Total moment of inertia is

$I=I_A+I_B‘+I_C‘$
$=\frac{2}{3}m{r}^2+\frac{5}{3}m{r}^2+\frac{5}{3}m{r}^2=4m{r}^2$