The densities of two solid spheres A and B of the same radii R vary with radial distance r as ρ _ A (r) = k((r)/(R)) and ρ _ B (r) = k ((r)/(R)) ^ 5 , respectively, where k is a constant. The moments of inertia of the

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The densities of two solid spheres A and B of the same radii R vary with radial distance r as ρ _ A (r) = k((r)/(R))

Consider a shell of radius r and thickness dr

dI = \dfrac{2}{3}\left(dm\right){r}^{2}

∴ dI = \dfrac{2}{3}\left[\left(\dfrac{kr}{R}\right)4\pi{r}^{2}dr\right]{r}^{2}

∫ dI = \left(\dfrac{8\pi}{3}\dfrac{k}{R}\right)∫ {r}^{5}dr

{I}_{A} = \left(\dfrac{8\pik}{3R}\right)\left(\dfrac{{R}^{6}}{6}\right)

{I}_{A} = \left(\dfrac{8\pik}{18}\right){R}^{5}

...(i)
Similarly

{I}_{B} = \dfrac{8k\pi}{3{R}^{5}}{∫}_{0}^{R}{r}^{9}dr

∴ {I}_{B} = \dfrac{8\pik}{30}{R}^{5}

...(ii)

∴ \dfrac{{I}_{B}}{{I}_{A}} = \dfrac{6}{10}

so

n = 6

The densities of two solid spheres A and B of the same radii R vary...

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