What is the ratio of the radius of gyration of a uniform circular disc about its diameter to the rad

Question

What is the ratio of the radius of gyration of a uniform circular disc about its diameter to the radius of gyration of a thin circular ring about its diameter, assuming both have the same radius $R$ and mass $M$?

Professor PC Thomas Classes · Answer

The moment of inertia of a circular disc about its diameter is $I_{disc} = \frac{1}{4}MR^2$. Equating this to $Mk^2$ gives the radius of gyration $k_{disc} = \frac{R}{2}$. The moment of inertia of a thin circular ring about its diameter is $I_{ring} = \frac{1}{2}MR^2$, giving $k_{ring} = \frac{R}{\sqrt{2}}$. The ratio is $\frac{R/2}{R/\sqrt{2}} = \frac{1}{\sqrt{2}}$.