Discussion Forum

 The densities of two solids spheres A and B  of the same radii R vary with radial distance r as   $\rho_{A}(r)=k\left(\frac{r}{R}\right)$ and $\rho_{B}(r)=k\left(\frac{r}{R}\right)^{5}$ , respectively where k is a constant. The moment 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:

Explanation:

Consider a shell of radius r and thickness dr

  $dI= (dm)r^{2}$

$\Rightarrow dI=\frac{2}{3}(\rho 4 \pi r^{2}dr)r^{2}\Rightarrow I=\int_{}^{} dI$

  $\frac{I_{B}}{I_{A}}=\frac{\int_{0}^{R}\frac{2}{3}k\frac{r^{5}}{R^{5}.}.4\pi r^{2}dr r^{2} }{\int_{0}^{R}\frac{2}{3}k\frac{r^{}}{R^{}.}.4\pi r^{2}dr r^{2}}=\frac{6}{10}$