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

Answer: 11

Let $I_0=Km{l}^2$
$I_{DEF}=K\left(\frac{m}{4}\right){\left(\frac{l}{2}\right)}^2=\left(\frac{I_0}{16}\right)$
and $I_{ADE}=I_{BDE}=I_{EFC}=I$
$3I=I_0-\frac{I_0}{16}=\frac{15I_0}{16}$
$\Rightarrow I=\frac{5I_0}{16}$
$I_{\text{remaining}}=2I+\frac{I_0}{16}=\frac{11I_0}{16}$