Question

Find the moment of inertia of the system about the axis passing through the $COM$ and perpendicular to its plane. All eight particles have the same mass $M$ and located at distances $R$ and $2R$ from the centre $\text{O}$, as shown in figure.

• A. $10M{R}^2$
• B. $20M{R}^2$
• C. $15M{R}^2$
• D. $30M{R}^2$

Answer 1

The correct option is B $20M{R}^2$


As we can see, from symmetry, center of mass of the system will be at point $\text{O}$.
Particle $1,2,3$ and $4$ are at a distance $R$ and particle $5,6,7$ and $8$ are at a distance of $2R$ from the $COM$.
Hence, $I_{COM}=\sum {mr}^2$
$I=4\times \left(M{R}^2\right)+4\times M(2R{)}^2$
$I=4M{R}^2+16M{R}^2$
$I=20M{R}^2$