Question

A lamina is made by removing a small disc of diameter $2R$ from a bigger disc of uniform mass density and radius $2R$, as shown in the figure. The moment of inertia of this lamina about axes passing through $O$ and $P$ is $I_O$ and $I_P$, respectively. Both these axes are perpendicular to the plane of the lamina. The ratio $\frac{I_P}{I_O}$ to the nearest integer is:

• A. $\frac{13}{37}$
• B. $\frac{37}{13}$
• C. $\frac{73}{31}$
• D. $\frac{8}{13}$

Answer 1

The correct option is A $\frac{13}{37}$

$I_p$ = $[\frac{4m{R}^2}{2}$+$4m{R}^2]$ -$[\frac{m{R}^2}{8}$ + $\frac{5m{R}^2}{4}]$ (by the metod of superposition of moment of inertia)
$I_p$ = $\frac{37m{R}^2}{8}$.........$\left(1\right)$
$I_0$ = $[\frac{4m{R}^2}{2}]$ - $[\frac{m{R}^2}{8}$ + $\frac{m{R}^2}{4}]$
$I_0$ = $\frac{13m{R}^2}{8}$...........$\left(2\right)$
So, $\frac{I_p}{I_0}$ = $\frac{37}{13}$
$Approximately3$