Question

The densities of two spheres A and B of the same radii R vary with radial distance r as $p_A\left(r\right)=k\left(\frac{r}{R}\right)$ and $p_B\left(r\right)=k(\frac{r}{R}{)}^5,$ 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 $I_0=\frac{2}{3}m{R}^2$
Here, for sphere A:
$I_A={\int }_0^R\frac{2}{3}{r}^2.4\pi r^{2}drK\frac{r}{R}$
$=\frac{8\pi K}{3R}{\int }_0^R{r}^{5}dr=\frac{8\pi K}{3R}\frac{R^6}{6}$
Also, for sphere B:
$I_B={\int }_0^R\frac{2}{3}{r}^2.4\pi r^{2}dr\frac{K}{R^5}.r^5=\frac{8\pi K}{3R^5}.\frac{R^{10}}{10}$
Ratio, $\frac{I_B}{I_A}=\frac{6}{10}$