Question

$ABC$ is a plane lamina of the shape of an equilateral triangle. $D,E$ are mid points of $AB,AC$ and $G$ is the centroid of the lamina. Moment of inertia of the lamina about an axis passing through $G$ and perpendicular to the plane $ABC$ is $I_{0}.$ If part $ADE$ is removed, the moment of inertia of the remaining part about the same axis is $\frac{N I_{0}}{16}$ where $N$ is an integer. Value of $N$ is________.

Answer 1

Let $I_{0}=Kml^{2}$
$I_{D E F}=K\left(\frac{m}{4}\right)\left(\frac{l}{2}\right)^{2}=\left(\frac{I_{0}}{16}\right)$
and $I_{A D E}=I_{B D E}=I_{E F C}=I$
$3I=I_{0}-\frac{I_{0}}{16}=\frac{15 I_{0}}{16}$
$\Rightarrow I=\frac{5 I_{0}}{16}$
$I_{\text{remaining}}=2I+\frac{I_{0}}{16}=\frac{11 I_{0}}{16}$