Question

From a given sample of uniform wire, two circular loops P and Q are made, P of radius r and Q of radius nr. If the M.I. of Q about its axis is four times that of P about its axis. If $n$ is $x^{1/3}$, then $x$ is:

Answer 1

Given that $I_Q=4I_p$
Here $I=\text{mass}\times (\text{radius}{)}^2$. If $\sigma$ is the linear mass density (mass/length) of the wire, then
$\sigma \times 2\pi \left(nr\right)\times n^2{r}^2=4(\sigma \times 2\pi r\times r^2)$
$\Rightarrow n^3=4\Rightarrow n=(4{)}^{\frac{1}{3}}$