Question

Three identical spherical shells, each of mass $m$ and radius $r$ are placed as shown in figure. Consider an axis $X{X}^{\prime }$ 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 $X{X}^{\prime }$ axis is

• A. $\frac{16}{5}m{r}^2$
• B. $4m{r}^2$
• C. $\frac{11}{5}m{r}^2$
• D. $3m{r}^2$

Answer 1

Answer: (B)

Net moment of inertia of the system,
$I=I_1+I_2+I_3$
The moment of inertia of a shell about its diameter,
$I_1=\frac{2}{3}m{r}^2$
The moment of inertia of a shell about its tangent is given by
$I_2=I_3=I_1+m{r}^2=\frac{2}{3}m{r}^2+m{r}^2=\frac{5}{3}m{r}^2$
$\therefore I=2\times \frac{5}{3}m{r}^2+\frac{2}{3}m{r}^2=\frac{12m{r}^2}{3}=4m{r}^2$