Question

Two solid spheres $(A$ and $B)$ are made of metals of different densities ${\rho }_A$ and ${\rho }_B$ respectively. If their masses are equal, the ratio of their moments of inertia $\left(I_B/I_A\right)$ about their respective diameter is

• A. ${\left(\frac{{\rho }_B}{{\rho }_A}\right)}^{2/3}$
• B. ${\left(\frac{{\rho }_A}{{\rho }_B}\right)}^{2/3}$
• C. $\frac{{\rho }_A}{{\rho }_B}$
• D. $\frac{{\rho }_B}{{\rho }_A}$

Answer 1

Answer: (A)

As two solid spheres are equal in masses, so
$m_A=m_B$
$\Rightarrow \frac{4}{3}\pi R_A^3{\rho }_A=\frac{4}{3}\pi R_B^3{\rho }_B$
$\Rightarrow \frac{R_A}{R_B}={\left(\frac{{\rho }_B}{{\rho }_A}\right)}^{1/3}$
The moment of inertia of sphere about diameter
$I=\frac{2}{5}m{R}^2$
$\Rightarrow \frac{I_A}{I_B}={\left(\frac{R_A}{R_B}\right)}^2\left(\text{as }{m}_A=m_B\right)$
$\Rightarrow \frac{I_A}{I_B}={\left(\frac{{\rho }_B}{{\rho }_A}\right)}^{2/3}$