Question

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

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

Hint:

The radius of gyration is simply the square root of the numerical coefficient in the moment of inertia formula multiplied by $R$. For a ring (coefficient 1), $k = R$. For a disc (coefficient 1/2), $k = R/\sqrt{2}$.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to find the ratio of the radius of gyration ($k$) of a circular ring to that of a circular disc, given they have identical masses and radii.

Step 2: Key Formula or Approach:
The radius of gyration $k$ is defined by the equation $I = Mk^2$, where $I$ is the moment of inertia and $M$ is the mass.
Therefore, $k = \sqrt{\frac{I}{M}}$.

Step 3: Detailed Explanation:
Let the mass be $M$ and the radius be $R$ for both objects.

For the circular ring:
The moment of inertia about its central perpendicular axis is $I_{\text{ring}} = MR^2$.
$$M k_{\text{ring}}^2 = MR^2 \implies k_{\text{ring}} = R$$

For the circular disc:
The moment of inertia about its central perpendicular axis is $I_{\text{disc}} = \frac{1}{2}MR^2$.
$$M k_{\text{disc}}^2 = \frac{1}{2}MR^2 \implies k_{\text{disc}} = \frac{R}{\sqrt{2}}$$

Ratio:
$$\frac{k_{\text{ring}}}{k_{\text{disc}}} = \frac{R}{\frac{R}{\sqrt{2}}} = \frac{\sqrt{2}}{1}$$

Step 4: Final Answer:
The ratio is $\sqrt{2} : 1$, matching option (B).