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  4. A uniform thin bar of mass 6m and length 12L is bent to make a regular hexagon. Its moment of inertia about an axis passing through the centre of mass and perpendicular to the plane of hexagon is

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Q. A uniform thin bar of mass $6m$ and length $12L$ is bent to make a regular hexagon. Its moment of inertia about an axis passing through the centre of mass and perpendicular to the plane of hexagon is

JIPMERJIPMER 2012System of Particles and Rotational Motion

Solution:

Length of each side of hexagon = $2L$ and mass of each side = $m$.
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Let $O$ be the centre of mass of hexagon. Therefore, perpendicular distance of $O$ from each side, $r = L \tan 60° = L \sqrt{3}$ .
The desired moment of inertia of hexagon about $O$ is
$I = 6\left[I_{one side}\right] =6\left[\frac{m\left(2L\right)^{2}}{12}+mr^{2}\right] = 6\left[\frac{mL^{2}}{3} +m\left(L\sqrt{3}\right)^{2}\right] =20mL^2$