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.
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.
$$\frac{{11}}{5}m{r^2}$$
B.
$$3m{r^2}$$
C.
$$\frac{{16}}{5}m{r^2}$$
D.
$$4m{r^2}$$
Answer :
$$4m{r^2}$$
Solution :
The total moment of inertia of the system is
$$I = {I_1} + {I_2} + {I_3}\,......\left( {\text{i}} \right)$$
Here, $${I_1} = \frac{2}{3}m{r^2}$$
$$\eqalign{ & {I_2} = {I_3} = \frac{2}{3}m{r^2} + m{r^2}\,\,\left[ {{\text{From parallel axis theorem}}} \right] \cr & = \frac{5}{3}m{r^2} \cr} $$
From Eq. (i),
$$\eqalign{ & I = \frac{2}{3}m{r^2} + 2 \times \frac{5}{3}m{r^2} = m{r^2}\left( {\frac{2}{3} + \frac{{10}}{3}} \right) \cr & I = 4m{r^2} \cr} $$
The total moment of inertia of the system is
$$I = {I_1} + {I_2} + {I_3}\,......\left( {\text{i}} \right)$$
Here, $${I_1} = \frac{2}{3}m{r^2}$$
$$\eqalign{ & {I_2} = {I_3} = \frac{2}{3}m{r^2} + m{r^2}\,\,\left[ {{\text{From parallel axis theorem}}} \right] \cr & = \frac{5}{3}m{r^2} \cr} $$
From Eq. (i),
$$\eqalign{ & I = \frac{2}{3}m{r^2} + 2 \times \frac{5}{3}m{r^2} = m{r^2}\left( {\frac{2}{3} + \frac{{10}}{3}} \right) \cr & I = 4m{r^2} \cr} $$
