Question

What is the ratio of gyration of ring and disc?

• A. 1:2
• B. 1:4
• C. 2:1
• D. 1:1

Hint:

The moment of inertia and radius of gyration of an object depend on its mass distribution relative to the axis of rotation. A ring has a larger radius of gyration compared to a disc of the same mass and radius.

Answer 1

The Correct Option is A

The radius of gyration \( k \) is defined as \( k = \sqrt{\frac{I}{m}} \), where \( I \) is the moment of inertia and \( m \) is the mass of the object. For a ring and a disc: - Moment of inertia of a ring (about its center of mass) is \( I_{\text{ring}} = m r^2 \), where \( r \) is the radius of the ring. - Moment of inertia of a disc (about its center of mass) is \( I_{\text{disc}} = \frac{1}{2} m r^2 \), where \( r \) is the radius of the disc. The radius of gyration for each is: - For the ring: \( k_{\text{ring}} = \sqrt{\frac{I_{\text{ring}}}{m}} = \sqrt{r^2} = r \) - For the disc: \( k_{\text{disc}} = \sqrt{\frac{I_{\text{disc}}}{m}} = \sqrt{\frac{1}{2} r^2} = \frac{r}{\sqrt{2}} \) Thus, the ratio of the radii of gyration is: \[ \frac{k_{\text{ring}}}{k_{\text{disc}}} = \frac{r}{\frac{r}{\sqrt{2}}} = \sqrt{2} \approx 1.414:1 \] Therefore, the ratio of the gyration radii is approximately 1:2.