Question

A system of solid discs, each of mass $M$ and radius $R$ is shown in figure.


Find the MOI about an axis passing through point $\text{O}$ and perpendicular to the plane.

• A. $\frac{9M{R}^2}{2}$
• B. $9M{R}^2$
• C. $18M{R}^2$
• D. $27M{R}^2$

Answer 1

The correct option is D $27M{R}^2$

This is a regular hexagon made by joining all the disc centers whose every edge is equal to $2R$.


The moment of inertia of solid disc about its center and perpendicular to its plane is
$I=\frac{M{R}^2}{2}$
Here, distance of each of the points from point $\text{O}$ is $2R$ each. Hence MOI will be same for all the discs about point $\text{O}$.
Therefore,
$I=I_1+I_2+I_3+I_4+I_5+I_6$
$I=6\times I_1$
$I=6\times [\frac{M{R}^2}{2}+M(2R{)}^2]$
$I=6\times \left[\frac{9M{R}^2}{2}\right]$
$I=27M{R}^2$