If the ratio of moment of inertia of two rings of radius. r₁ and r₂ is 0.25, then find the value of ratio $\frac{r_2}{r_1}.$

Ratio of moments of inertia is given as

$\frac{{\mathrm{I}}_2}{{\mathrm{I}}_1}=4$

and we know that $I=m{r}^2$

So $\frac{{\mathrm{I}}_2}{{\mathrm{I}}_1}=\frac{{\mathrm{m}}_2{\mathrm{r}}_2^2}{{\mathrm{m}}_1{\mathrm{r}}_1^2}\dots \left(i\right)$

mass = volume $\times$ density,

density and thickness of both loops are same here,

So $\mathrm{m}\propto 2\pi \mathrm{r}\Rightarrow \mathrm{m}\propto \mathrm{r}$

So from eq. (i)

$\begin{array}{l}\frac{I_2}{I_1}=\frac{r_2^3}{r_1^3}\\ \frac{4}{I}={\left(\frac{r_2}{r_1}\right)}^3\Rightarrow \frac{r_2}{r_1}=(4{)}^{\frac{1}{3}}\end{array}$