Question

Four hollow spheres, each of mass $m=1\text{k}g$ and radius $0.5\text{m}$ are located with their centres on four corners of a square of side $2\text{m}$, find the moment of inertia of the system of four spheres about the diagonal of square.

• A. $5.72{\text{kgm}}^2$
• B. $4.67{\text{kgm}}^2$
• C. $3.48{\text{kgm}}^2$
• D. $4.81{\text{kgm}}^2$

Answer 1

The correct option is B $4.67{\text{kgm}}^2$


Due to symmetry and equal dimensions, the moment of inertia of spheres $\text{A}$ and $\text{C}$ are equal about $\text{AC}$.

$I_A=I_C=\frac{2}{3}m{R}^2=\frac{2}{3}\times 1\times (0.5{)}^2=0.166{\text{kgm}}^2$

Due to symmetry and equal dimensions, the moment of inertia of spheres $\text{B}$ and $\text{D}$ about $\text{AC}$ are equal.

$I_B=I_D=I_0+m{\left(\frac{a}{\sqrt{2}}\right)}^2=\frac{2}{3}m{R}^2+\frac{m{a}^2}{2}$

$=\frac{2}{3}\times 1\times (0.5{)}^2+\frac{1\times 2^2}{2}=2.166{\text{kgm}}^2$

$\therefore$ Total moment of inertia about the diagonal,

$I=I_A+I_B+I_C+I_D$

$=0.166+2.166+0.166+2.166$

$=4.664{\text{kgm}}^2$

Hence, $\left(B\right)$ is the correct answer.