The densities of two solid spheres $A$ and $B$ of the same radii $R$ vary with radial distance $r$ as $\rho_{A}(r) = k\frac{r}{R}$ and $\rho_{B}(r) = k\left(\frac{r}{R}\right)^{5}$, respectively, where $k$ is a constant. The moments of inertia of the individual spheres about axes passing through…

Question

The densities of two solid spheres $A$ and $B$ of the same radii $R$ vary with radial distance $r$ as $\rho_{A}(r) = k\frac{r}{R}$ and $\rho_{B}(r) = k\left(\frac{r}{R}\right)^{5}$, 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

Aicharya · Accepted Answer

Consider a shell of radius r and thickness dr

d I = \frac{2}{3} (d m) r^{2}

∴ d I = \frac{2}{3} [(\frac{k r}{R}) 4 π r^{2} d r] r^{2}

\int d I = (\frac{8 π}{3} \frac{k}{R}) \int r^{5} d r

I_{A} = (\frac{8 π k}{3 R}) (\frac{R^{6}}{6})

I_{A} = (\frac{8 π k}{18}) R^{5}

...(i)
Similarly

I_{B} = \frac{8 k π}{3 R^{5}} \int_{0}^{R} r^{9} d r

∴ I_{B} = \frac{8 π k}{30} R^{5}

...(ii)

∴ \frac{I_{B}}{I_{A}} = \frac{6}{10}

so

n = 6

The densities of two solid spheres $A$ and $B$ of the same radii $R$ vary with radial distance $r$ as…

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