The densities of two solid spheres $\mathrm{A}$ and $\mathrm{B}$ of the same radii $\mathrm{R}$ vary with radial distance $\mathrm{r}$ as ${\mathrm{\rho }}_{\mathrm{A}}\left(\mathrm{r}\right)=\mathrm{k}\left(\frac{\mathrm{r}}{\mathrm{R}}\right)$ and ${\mathrm{\rho }}_{\mathrm{B}}\left(\mathrm{r}\right)=\mathrm{k}{\left(\frac{\mathrm{r}}{\mathrm{R}}\right)}^5$ , respectively, where $\mathrm{k}$ is a constant. The moments of inertia of the individual spheres about axes passing through their centres are ${\mathrm{I}}_{\mathrm{A}}$ and ${\mathrm{I}}_{\mathrm{B}}$ , respectively. If $\frac{{\mathrm{I}}_{\mathrm{B}}}{{\mathrm{I}}_{\mathrm{A}}}=\frac{\mathrm{n}}{10}$ , the value of $\mathrm{n}$ is

Solution

Consider a shell of radius r and thickness dr
$\mathrm{d}\mathrm{I}=\frac{2}{3}\left(\mathrm{d}\mathrm{m}\right){\mathrm{r}}^2$
$\therefore \mathrm{d}\mathrm{I}=\frac{2}{3}\left[\left(\frac{\mathrm{k}\mathrm{r}}{\mathrm{R}}\right)4\mathrm{\pi }{\mathrm{r}}^2\mathrm{d}\mathrm{r}\right]{\mathrm{r}}^2\mathrm{ }$
$\int \mathrm{d}\mathrm{I}=\left(\frac{8\mathrm{\pi }}{3}\frac{\mathrm{k}}{\mathrm{R}}\right)\int {\mathrm{r}}^5\mathrm{d}\mathrm{r}\mathrm{ }$
${\mathrm{I}}_{\mathrm{A}}=\left(\frac{8\pi k}{3R}\right)\left(\frac{R^6}{6}\right)$
${\mathrm{I}}_{\mathrm{A}}=\left(\frac{8\mathrm{\pi }\mathrm{k}}{18}\right){\mathrm{R}}^5$ ...(i)
Similarly
${\mathrm{I}}_{\mathrm{B}}=\frac{8\mathrm{k}\mathrm{\pi }}{3{\mathrm{R}}^5}{\int }_0^{\mathrm{R}}{\mathrm{r}}^9\mathrm{d}\mathrm{r}$
$\therefore {\mathrm{I}}_{\mathrm{B}}=\frac{8\mathrm{\pi }\mathrm{k}}{30}{\mathrm{R}}^5$ ...(ii)
$\therefore \frac{{\mathrm{I}}_{\mathrm{B}}}{{\mathrm{I}}_{\mathrm{A}}}=\frac{6}{10}$ so $\mathrm{n}=6$

Hence, the correct answer is 6.