Find the ratio of moments of inertia of the two disks of same mass and thickness but one of density 7.2 g/cm³ and the other of density 8.9 g/cm³.

We know that moment of inertia

$\mathrm{I}=\frac{{\mathrm{MR}}^2}{2}$

(of disk wrt principle axis)

Given ${\mathrm{M}}_1={\mathrm{M}}_2$

$\begin{array}{l}{\mathrm{V}}_1{\mathrm{d}}_1={\mathrm{V}}_2{\mathrm{d}}_2\\ \pi {\mathrm{R}}_1^2{\mathrm{t}}_1{\mathrm{d}}_1=\pi {\mathrm{R}}_2^2{\mathrm{t}}_2{\mathrm{d}}_2\dots \text{ (i) }\end{array}$

Given ${\mathrm{t}}_1={\mathrm{t}}_2$ (thickness)

So $\frac{{\mathrm{R}}_1^2}{{\mathrm{R}}_2^2}=\frac{{\mathrm{d}}_2}{{\mathrm{d}}_1}$ (from eq. (i))

So $\frac{{\mathrm{M}}_1{\mathrm{R}}_1^2}{{\mathrm{M}}_2{\mathrm{R}}_2^2}=\frac{{\mathrm{d}}_2}{{\mathrm{d}}_1}\left(\text{ as }{\mathrm{M}}_1={\mathrm{M}}_2\right)$

$\begin{array}{r}\Rightarrow \frac{{\mathrm{I}}_1}{{\mathrm{I}}_2}=\frac{{\mathrm{d}}_2}{{\mathrm{d}}_1}\\ \Rightarrow \frac{{\mathrm{I}}_1}{{\mathrm{I}}_2}=\frac{8.9}{7.2}\end{array}$