Question

If two discs have same mass and thickness but their densities are ${\rho }_1$ and ${\rho }_2$, then the ratio of their moment of inertia about central axis will be

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

Answer 1

The correct option is D ${\rho }_2:{\rho }_1$

Moment of inertia of disc,
$I=\frac{1}{2}M{R}^2=\frac{1}{2}M\left(\frac{M}{\pi \rho t}\right)$
$[\because \rho =\frac{M}{V}=\frac{M}{\pi R^{2}t};\therefore R^2=\frac{M}{\pi \rho t}]$
$I=\frac{1}{2}\frac{M^2}{\pi \rho t}$
$\therefore I\propto \frac{1}{\rho }$ [As $M$ and $t$ are constants]
$\frac{I_1}{I_2}=\frac{{\rho }_2}{{\rho }_1}$.