Question

The ratio of radii of gyration of a circular ring and a circular disc, of the same mass and radius, about an axis passing through their centres and perpendicular to their planes are :

• A. $ 1: \sqrt 2 $
• B. 3 : 2
• C. 2 : 1
• D. $ \sqrt 2 :1 $

Answer 1

The Correct Option is D

To find the ratio of the radii of gyration of a circular ring and a circular disc, of the same mass and radius, about an axis passing through their centers and perpendicular to their planes, we will use the following concepts:

1. Moment of Inertia: The moment of inertia (I) of an object about an axis is a measure of its resistance to rotational motion about that axis.
   • For a circular ring of mass M and radius R, the moment of inertia about an axis through its center and perpendicular to its plane is given by: I_{\text{ring}} = MR^2.
   • For a circular disc of mass M and radius R, the moment of inertia about an axis through its center and perpendicular to its plane is: I_{\text{disc}} = \frac{1}{2}MR^2.
2. Radius of Gyration (k): The radius of gyration is defined as the distance from the axis of rotation at which the entire mass of the body can be assumed to be concentrated to give the same moment of inertia.
   • The formula is k = \sqrt{\frac{I}{M}}.
   • For the ring, the radius of gyration is: k_{\text{ring}} = \sqrt{\frac{MR^2}{M}} = R.
   • For the disc, the radius of gyration is: k_{\text{disc}} = \sqrt{\frac{\frac{1}{2}MR^2}{M}} = \sqrt{\frac{R^2}{2}} = \frac{R}{\sqrt{2}}.
3. Calculation of the Ratio: The ratio of the radii of gyration of the ring to the disc is:
   • \text{Ratio} = \frac{k_{\text{ring}}}{k_{\text{disc}}} = \frac{R}{\frac{R}{\sqrt{2}}} = \sqrt{2}: 1.

Thus, the correct ratio of the radii of gyration of a circular ring and a circular disc is \sqrt{2}:1.