Question

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

Answer 1

Moment of inertia of a hollow sphere is $\frac{2}{3}m{r}^2$
Consider a thin shell of width dr at radius r from the center of the sphere.
Mass of the thin shell is ${\rho }_{(}r)4\pi r^{2}dr$
$I_A={\int }_0^R\frac{2}{3}{\rho }_A\left(r\right)4\pi r^{2}dr)r^2=\frac{2}{3}(\left(k\frac{r}{R}\right)4\pi r^{2}dr)r^2==\frac{2}{3}\frac{4\pi k}{R}|\frac{r^6}{6}{|}_0^R=\frac{2}{3}\frac{4\pi k}{R}|\frac{R^6}{6}$
$I_B={\int }_0^R\frac{2}{3}{\rho }_B\left(r\right)4\pi r^{2}dr)r^2=\frac{2}{3}(k\left(\frac{r^5}{R^5}\right)4\pi r^{2}dr)r^2=\frac{2}{3}\frac{4\pi k}{R^5}|\frac{r^{10}}{10}{|}_0^R=\frac{2}{3}\frac{4\pi k}{R^5}|\frac{R^{10}}{10}$
$\therefore \frac{I_B}{I_A}=\frac{6}{10}$
$\Rightarrow n=6$