Question

Find the ratio of radius of gyration about natural axis of a circular disk to that of a circular ring each having same mass and same radius.

• A. $1:1$
• B. $2:1$
• C. $1:\sqrt{2}$
• D. $\sqrt{2}:1$

Answer 1

The correct option is C $1:\sqrt{2}$

Let the mass and radius of the circular disk and circular ring are $M$ and $R$.
We know
Radius of gyration is given by $\left(K\right)$
$K=\sqrt{\frac{I}{M}}$
$\therefore$ Radius of gyration of disk $K_{disc}=\sqrt{\frac{I_{disc}}{M}}...\left(i\right)$
Radius of gyration of ring, $K_{ring}=\sqrt{\frac{I_{ring}}{M}}....\left(ii\right)$

Dividing equation $\left(i\right)$ and $\left(ii\right)$
$\frac{K_{disc}}{K_{ring}}=\sqrt{\frac{I_{disc}}{I_{ring}}}=\sqrt{\frac{\frac{M{R}^2}{2}}{M{R}^2}}=\frac{1}{\sqrt{2}}$
where $I_{disc}$ and $I_{ring}$ are the moment of inertia of the circular disk and circular ring about their natural axis.