Question

The ratio of radii of gyration of a ring to a disc (both circular of same radii and mass), about a tangential axis perpendicular to the plane is

• A. \( \frac{2}{\sqrt{3}} \)
• B. \( \sqrt{2} \)
• C. \( \frac{\sqrt{3}}{\sqrt{2}} \)
• D. \( \frac{2}{\sqrt{5}} \)

Hint:

In rotational motion, the moment of inertia plays a crucial role in calculating the radius of gyration. For a ring, the moment of inertia is greater than that for a disc, leading to a different radius of gyration.

Answer 1

The Correct Option is A

Step 1: Understanding radii of gyration.
The radius of gyration \( k \) is given by the formula \( k = \sqrt{\frac{I}{m}} \), where \( I \) is the moment of inertia and \( m \) is the mass. We need to find the ratio of the radii of gyration of a ring and a disc about a tangential axis perpendicular to the plane.
Step 2: Moment of inertia.
For the ring, the moment of inertia about the tangential axis is \( I_{\text{ring}} = 2mr^2 \), where \( r \) is the radius. For the disc, the moment of inertia is \( I_{\text{disc}} = \frac{1}{2}mr^2 \).
Step 3: Calculating the ratio.
The ratio of the radii of gyration is: \[ \frac{k_{\text{ring}}}{k_{\text{disc}}} = \frac{\sqrt{\frac{I_{\text{ring}}}{m}}}{\sqrt{\frac{I_{\text{disc}}}{m}}} = \frac{\sqrt{\frac{2mr^2}{m}}}{\sqrt{\frac{\frac{1}{2}mr^2}{m}}} = \frac{\sqrt{2}r}{\sqrt{\frac{1}{2}}r} = \frac{2}{\sqrt{3}} \]
Step 4: Conclusion.
The ratio of the radii of gyration is \( \frac{2}{\sqrt{3}} \), so the correct answer is (A) \( \frac{2}{\sqrt{3}} \).