The angular speed of a body changes from `omega_1` to `omega_2` without applying a torque but due to change in its moment of inertia. The ratio of radii of gyration in the two cases is :-

Options

A. (a)`sqrt(omega_(2)):sqrt(omega_(1))`
B. (b)`sqrt(omega_(1)):sqrt(omega_(2))`
C. (c)`omega_(2):omega_(1)`
D. (d)`omega_(1):omega_(2)`

Answer

To solve the problem, we need to find the ratio of the radii of gyration in two cases when the angular speed of a body changes from \(\omega_1\) to \(\omega_2\) without applying any torque, but due to a change in its moment of inertia.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We know that the angular momentum of a system is conserved when no external torque is applied.
- The angular momentum \(L\) is given by the product of the moment of inertia \(I\) and the angular velocity \(\omega\):
\[ L = I \omega \]

2. **Setting Up the Conservation of Angular Momentum**:
- Since no torque is applied, the angular momentum before the change must equal the angular momentum after the change:
\[ I_1 \omega_1 = I_2 \omega_2 \]

3. **Expressing Moment of Inertia in Terms of Radius of Gyration**:
- The moment of inertia \(I\) can be expressed in terms of the mass \(m\) and the radius of gyration \(k\):
\[ I = m k^2 \]
- Therefore, we can write:
\[ I_1 = m k_1^2 \quad \text{and} \quad I_2 = m k_2^2 \]

4. **Substituting into the Angular Momentum Equation**:
- Substitute \(I_1\) and \(I_2\) into the conservation of angular momentum equation:
\[ (m k_1^2) \omega_1 = (m k_2^2) \omega_2 \]
- The mass \(m\) cancels out from both sides:
\[ k_1^2 \omega_1 = k_2^2 \omega_2 \]

5. **Rearranging the Equation**:
- Rearranging gives:
\[ \frac{k_1^2}{k_2^2} = \frac{\omega_2}{\omega_1} \]

6. **Taking the Square Root**:
- Taking the square root of both sides to find the ratio of the radii of gyration:
\[ \frac{k_1}{k_2} = \sqrt{\frac{\omega_2}{\omega_1}} \]

### Final Result:
The ratio of the radii of gyration in the two cases is:
\[ \frac{k_1}{k_2} = \sqrt{\frac{\omega_2}{\omega_1}} \]