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Q.
Two rings of radius $R$ and $n R$ made of same material have the ratio of moment of inertia about an axis passing through centre in $1: 8$. The value of $n$ is
Ratio of moment of inertia of the rings
$\frac{ I _{1}}{ I _{2}}=\left(\frac{ M _{1}}{ M _{2}}\right)\left(\frac{ R _{1}}{ R _{2}}\right)^{2}=\left(\frac{\lambda L _{1}}{\lambda L _{2}}\right)^{2}\left(\frac{ R _{1}}{ R _{2}}\right)^{2} $
$=\left(\frac{2 \pi R }{2 \pi n R }\right)\left(\frac{ R }{ nR }\right)^{2} $
$[\lambda=$ linear density of wire $=$ constant ]
$\Rightarrow \frac{ L _{1}}{ L _{2}}+\frac{1}{ n _{3}}+\frac{1}{8} $ (given)
$\therefore n ^{3}=8$
$\Rightarrow n =2$