Two solid spheres (A and B) are made of metals having densities ρ A and ρ B respectively. If there masses are equal then ratio of their moments of inertia ( I A I B ) about their respective

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Two solid spheres (A and B) are made of metals having densities ρ A ​ and ρ B ​ respectively. If there masses are equal then ratio of their moments of inertia ( I A ​ I B ​ ​ ) about their respective diameter is

TestHub · Accepted Answer

Moment of inertia of a solid sphere about its diameter is $I = \frac{2}{5} M R^{2}$ .

Given that the masses of both spheres are equal, $M_{A} = M_{B} = M$ .

Mass is related to density and radius by $M = \frac{4}{3} π R^{3} ρ$ .

Since $M$ is constant, $R^{3} \propto \frac{1}{ρ} \Rightarrow R \propto ρ^{- 1/3}$ .

The moment of inertia $I \propto M R^{2}$ . Since $M$ is constant, $I \propto R^{2} \propto ρ^{- 2/3}$ .

Therefore, the ratio of their moments of inertia is $\frac{I _{B}}{I _{A}} = (\frac{ρ _{A}}{ρ _{B}})^{2/3}$ .

Answer: $(\frac{ρ _{A}}{ρ _{B}})^{2/3}$ .

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