Question

Two solid spheres A and B each of radius ${}^{\prime }{R}^{\prime }$ are made of materials of densities ${\rho }_A$ and ${\rho }_B$ respectively. Their moments of inertia about a diameter are $I_A$ and $I_B$ respectively. The value of $\frac{I_A}{I_B}$ is:

• A. $\sqrt{\frac{{\rho }_A}{{\rho }_B}}$
• B. $\sqrt{\frac{{\rho }_B}{{\rho }_A}}$
• C. $\frac{{\rho }_A}{{\rho }_B}$
• D. $\frac{{\rho }_B}{{\rho }_A}$

Answer 1

Answer: (C)

$\frac{{\rho }_A}{{\rho }_B}$

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

Thus $I_A=\frac{2}{5}{M}_A{R}^2$

$=\frac{2}{5}{\rho }_A\frac{4}{3}\pi R^3.R^2=\frac{8}{15}{\rho }_A\pi R^5$

and $I_B=\frac{8}{15}{\rho }_B\pi R^5$

$⟹\frac{I_A}{I_B}=\frac{{\rho }_A}{{\rho }_B}$