Physics - Rotation Question with Solution | TestHub

Mass of a disc is given by $M=\text{volume}\times \text{density}=\pi R^{2}t\rho$, where $t$ is the thickness and $\rho$ is the density.

Moment of inertia of a disc about its axis is $I=\frac{1}{2}M{R}^2$.

Substituting the expression for mass, we get $I=\frac{1}{2}(\pi R^{2}t\rho )R^2=\frac{1}{2}\pi t\rho R^4$.

Since both discs have the same material (same $\rho$) and same thickness (same $t$), $I\propto R^4$.

Therefore, the ratio of their moments of inertia is $\frac{I_A}{I_B}={\left(\frac{R_A}{R_B}\right)}^4={\left(\frac{R}{3R}\right)}^4=\frac{1}{81}$.

The ratio is $1:81$.

Answer: 1:81