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

Answer 1

The correct option is B $4m{r}^2$

Moment of inertia of the top sphere about an axis passing through its center: $I_{top}=\frac{2}{3}m{r}^2$
Using parallel axis theorem, MI of one of the bottom sphere about the axis XX is $I_{bottom}=\frac{2}{3}m{r}^2+m{r}^2=\frac{5}{3}m{r}^2$
MI of both bottom spheres is $2\times I_{bottom}=\frac{10}{3}m{r}^2$
Total MI of the system: $I=\frac{2}{3}m{r}^2+\frac{10}{3}m{r}^2=4m{r}^2$