Two circular loops $\mathrm{P}$ and $\mathrm{Q}$ of radii $r$ and $n r$ respectively are made from a uniform…
Two circular loops $\mathrm{P}$ and $\mathrm{Q}$ of radii $r$ and $n r$ respectively are made from a uniform wire. Moment of inertia of $Q$ about its axis is four times that of $\mathrm{P}$ about its axis. The value of $n$ is
$(2)^{\frac{2}{3}}$
$(2)^{\frac{1}{3}}$
$2$
$\sqrt{2}$
Solution
Given, $I_Q=4 I_P$.
Let $m$ be the mass per unit length of wire.
$\therefore m \times 2 \pi n r(n r)^2=4\left[(m 2 \pi r) r^2\right]$
Implies $n^3=4$
$\Rightarrow n=4^{1 / 3}$