Question

Two uniform circular discs having the same mass and the same thickness, but different radii, are made from different materials. The disc with the smaller rotational inertia is:

• A. the one made from the more dense material
• B. the one made from the less dense material
• C. the disc with the larger angular velocity
• D. the disc with the larger torque

Answer 1

The correct option is A the one made from the more dense material

Given : $M_1=M_2$ $t_1=t_2$

Let the densities of the two discs be ${\rho }_1$ and ${\rho }_2$ respectively.

Using $M_1=M_2$ we get ${\rho }_1\times \pi R_1^2{t}_1={\rho }_2\times \pi R_2^2{t}_2$

$⟹$ ${\rho }_1{R}_1^2={\rho }_2{R}_2^2$ or, $\frac{R_1^2}{R_2^2}=\frac{{\rho }_2}{{\rho }_1}$ .............(1)

Moment of inertia of circular disc $I=\frac{1}{2}M{R}^2$

$⟹$ $\frac{I_1}{I_2}=\frac{R_1^2}{R_2^2}=\frac{{\rho }_2}{{\rho }_1}$

Thus if ${\rho }_1>{\rho }_2$ $⟹$ $I_1<I_2$

Hence the disc made up of more dense material has smaller rotational inertia.