A solid spherical ball of density rho_{1} and | vedclass

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(D) Let $M$ be the mass of both spheres. For the solid sphere: $M = \rho_{1} \frac{4}{3} \pi R^{3}$.For the hollow sphere with inner radius $r$: $M =

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$\frac{\rho_{2}}{\rho_{1}}\left[1-\left(1-\frac{\rho_{1}}{\rho_{2}}\right)^{\frac{5}{3}}\right]$

Vedclass · Answer

(D) Let $M$ be the mass of both spheres. For the solid sphere: $M = \rho_{1} \frac{4}{3} \pi R^{3}$.For the hollow sphere with inner radius $r$: $M = \rho_{2} \frac{4}{3} \pi (R^{3} - r^{3})$.Equating the masses: $\rho_{1} R^{3} = \rho_{2} (R^{3} - r^{3})$.This gives $\frac{\rho_{1}}{\rho_{2}} = 1 - \frac{r^{3}}{R^{3}}$, so $\frac{r^{3}}{R^{3}} = 1 - \frac{\rho_{1}}{\rho_{2}}$, which implies $\frac{r}{R} = (1 - \frac{\rho_{1}}{\rho_{2}})^{1/3}$.The moment of inertia of the solid sphere is $I_{S} = \frac{2}{5} M R^{2}$.The moment of inertia of the hollow sphere is $I_{H} = \frac{2}{5} M \frac{R^{5} - r^{5}}{R^{3} - r^{3}}$.Using $M = \rho_{2} \frac{4}{3} \pi (R^{3} - r^{3})$, we get $I_{H} = \frac{2}{5} (\rho_{2} \frac{4}{3} \pi (R^{3} - r^{3})) \frac{R^{5} - r^{5}}{R^{3} - r^{3}} = \frac{2}{5} \rho_{2} \frac{4}{3} \pi (R^{5} - r^{5})$.Since $M = \rho_{1} \frac{4}{3} \pi R^{3}$, we have $I_{S} = \frac{2}{5} (\rho_{1} \frac{4}{3} \pi R^{3}) R^{2} = \frac{2}{5} \rho_{1} \frac{4}{3} \pi R^{5}$.Therefore, $\frac{I_{H}}{I_{S}} = \frac{\rho_{2} (R^{5} - r^{5})}{\rho_{1} R^{5}} = \frac{\rho_{2}}{\rho_{1}} [1 - (\frac{r}{R})^{5}]$.Substituting $\frac{r}{R} = (1 - \frac{\rho_{1}}{\rho_{2}})^{1/3}$, we get $\frac{I_{H}}{I_{S}} = \frac{\rho_{2}}{\rho_{1}} [1 - (1 - \frac{\rho_{1}}{\rho_{2}})^{5/3}]$.

$A$ solid spherical ball of density $\rho_{1}$ and a hollow spherical ball of density $\rho_{2}$ have the same outer radius $R$ and the same mass $M$. What is the ratio of the moment of inertia of the hollow sphere to that of the solid sphere about an axis passing through their centers?

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