Question

Two discs have same mass and thickness. Their materials are of densities ${\rho }_1$ and ${\rho }_{2.}$ The ratio of their moment of inertia about central axis will be

• A. ${\rho }_1:{\rho }_2$
• B. ${\rho }_1:{\rho }_2:1$
• C. $1:{\rho }_1{\rho }_2$
• D. ${\rho }_2:{\rho }_1$

Answer 1

Answer: (D)

Moment of inertia of disc
$I=\frac{1}{2}M{R}^2=\frac{1}{2}M\left(\frac{M}{\pi \rho t}\right)$
$\therefore I=\frac{1}{2}\frac{M^2}{\pi \rho t}$
$($ As $\rho =\frac{\text{ Mass }}{\text{ Volume }}=\frac{M}{\rho R^{2}t}$, therefore, $R^2=\frac{M}{\pi \rho t})$
$\therefore I\propto \frac{1}{\rho }$ (If $M$ and $t$ are constants)
$\Rightarrow \frac{I_1}{I_2}=\frac{{\rho }^2}{{\rho }_1}$