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(D) The moment of inertia $I$ of a solid sphere is given by $I = \frac{2}{5} mr^{2}$.Since the spheres are made of the same material (iron), their den

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$1 : 32$

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(D) The moment of inertia $I$ of a solid sphere is given by $I = \frac{2}{5} mr^{2}$.Since the spheres are made of the same material (iron), their density $d$ is the same.The mass $m$ of a sphere is given by $m = 	ext{Volume} 	imes 	ext{Density} = \frac{4}{3} \pi r^{3} d$.Substituting the expression for mass into the moment of inertia formula:$I = \frac{2}{5} (\frac{4}{3} \pi r^{3} d) r^{2} = \frac{8}{15} \pi d r^{5}$.This shows that $I \propto r^{5}$.Given the ratio of radii $r_{1} : r_{2} = 1 : 2$, the ratio of their moments of inertia is:$\frac{I_{1}}{I_{2}} = (\frac{r_{1}}{r_{2}})^{5} = (\frac{1}{2})^{5} = \frac{1}{32}$.Therefore, the ratio is $1 : 32$.

Two solid spheres of iron have radii in the ratio $1 : 2$. Their moments of inertia will be in the ratio

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