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 axis passing through their centres are $I_A$ and $I_B$ respectively. If $\frac{I_B}{I_A}=\frac{n}{10}$, then value of $n$ is-

Answer 1

For solid sphere$A,{\rho }_A\left(r\right)=k\left(\frac{r}{R}\right)$
Consider a spherical shell of radius $x$ and thickness $dx$.
Mass of the shell, $dm=$ density $\times$ volume
$=\left(k\frac{x}{R}\right)\left(4\pi x^{2}dx\right)$
So, moment of inertia of shell about its diameter,
$dI=\frac{2}{3}\left(dm\right)x^2=\frac{2}{3}\left(k\frac{x}{R}\right)\left(4\pi x^{2}dx\right)x^2=\left(\frac{8\pi }{3}\frac{k}{R}\right)x^{5}dx$
$\therefore$ Moment of inertia of the sphere A,
$I_A={\int }_0^{R}dI=\frac{8\pi k}{3R}{\int }_0^R{x}^{5}dx=\frac{8\pi k}{3R}{\left[\frac{x^6}{6}\right]}_0^R=\left(\frac{8\pi k}{18}\right)R^5....\left(i\right)$
Similarly, for sphere B
$I_B=\frac{8\pi k}{3R^5}{\int }_0^R{x}^{9}dx=\left(\frac{8\pi k}{3R^5}\right){\left[\frac{x^{10}}{10}\right]}_0^R$
$\therefore I_B=\frac{8\pi k}{30}{R}^5.....\left(ii\right)$
From eqns. (i) and (ii), we get
$\frac{I_B}{I_A}=\frac{18}{30}=\frac{6}{10}=\frac{n}{10}\therefore n=6$