Two circular loops P and Q of radii ' r 'and ' nr ' are made respectively from a uniform wire. Moment of inertia of loop Q about its axis is four times that of loop P about its axis. The value of ' \(n\) ' is

The two loops are formed from the same wire. Hence, the linear density will remain constant.
\(\therefore \lambda=\frac{\mathrm{M}_{\mathrm{P}}}{\mathrm{~L}_{\mathrm{p}}}=\frac{\mathrm{M}_{\mathrm{Q}}}{\mathrm{~L}_{\mathrm{Q}}}\)
\(\therefore \mathrm{M}_{\mathrm{P}}=\lambda \times \mathrm{L}_{\mathrm{P}}=\lambda \times\left(2 \pi \mathrm{R}_{\mathrm{P}}\right)=2 \pi \mathrm{r} \lambda \)
\( \text { Similarly, } \)
\( \mathrm{M}_{\mathrm{Q}}=\lambda \times \mathrm{L}_{\mathrm{Q}}=\lambda \times\left(2 \pi \mathrm{R}_{\mathrm{Q}}\right)=2 \pi(\mathrm{nr}) \lambda \)
\( \text { Also, M.I. of circular loop } \mathrm{P} \text { is, } \)
\( \mathrm{I}_{\mathrm{P}}=\mathrm{M}_{\mathrm{P}} \mathrm{r}^2=2 \pi \mathrm{r} \lambda(\mathrm{r})^2=2 \pi \mathrm{r}^3 \lambda \)
\( \text { and that of loop } \mathrm{Q}, \)
\( \mathrm{I}_{\mathrm{Q}}=\mathrm{M}_{\mathrm{Q}}(\mathrm{nr})^2=2 \pi(\mathrm{nr}) \lambda \times(\mathrm{nr})^2=2 \pi \mathrm{n}^3 \mathrm{r}^3 \lambda \)
\( \mathrm{Given}^2 \text { that, } \mathrm{I}_{\mathrm{Q}}=4 \mathrm{I}_{\mathrm{P}} \)
\( \therefore 2 \pi \mathrm{n}^3 \mathrm{r}^3 \lambda=4 \times 2 \pi \mathrm{r}^3 \lambda \)
\( \therefore \mathrm{n}^3=4 \)
\( \therefore \mathrm{n}=(4)^{\frac{1}{3}}=(2)^{\frac{2}{3}}\)