The angular velocity of a body changes from one revolution per 9 second to 1 revolution per second without applying any torque. The ratio of its radius of gyration in the two cases is

Options

A. `1 : 9 `
B. ` 3 : 1 `
C. `9 : 1 `
D. `1 : 3 `

Answer

To solve the problem, we need to find the ratio of the radius of gyration of a body when its angular velocity changes from one revolution per 9 seconds to one revolution per second, without applying any torque.

### Step-by-Step Solution:

1. **Convert Angular Velocities to Radians per Second:**
- The initial angular velocity (ω₁) is given as 1 revolution per 9 seconds.
- To convert this to radians per second:
\[ \omega_1 = \frac{1 \text{ revolution}}{9 \text{ seconds}} \times 2\pi \text{ radians/revolution} = \frac{2\pi}{9} \text{ radians/second} \]
- The final angular velocity (ω₂) is given as 1 revolution per second:
\[ \omega_2 = 1 \text{ revolution/second} \times 2\pi \text{ radians/revolution} = 2\pi \text{ radians/second} \]

2. **Apply the Conservation of Angular Momentum:**
- Since no torque is applied, angular momentum is conserved:
\[ I_1 \omega_1 = I_2 \omega_2 \]
- Here, \(I_1\) and \(I_2\) are the moments of inertia at the two states.

3. **Express the Ratio of Moments of Inertia:**
- Rearranging the equation gives:
\[ \frac{I_1}{I_2} = \frac{\omega_2}{\omega_1} \]
- Substitute the values of ω₁ and ω₂:
\[ \frac{I_1}{I_2} = \frac{2\pi}{\frac{2\pi}{9}} = 9 \]

4. **Relate the Radius of Gyration to the Moment of Inertia:**
- The radius of gyration \(k\) is defined as:
\[ k = \sqrt{\frac{I}{m}} \]
- Since mass (m) remains constant, the ratio of the radii of gyration can be expressed as:
\[ \frac{k_1}{k_2} = \sqrt{\frac{I_1}{I_2}} \]

5. **Calculate the Ratio of the Radii of Gyration:**
- Substitute the ratio of moments of inertia:
\[ \frac{k_1}{k_2} = \sqrt{9} = 3 \]

### Final Answer:
The ratio of the radius of gyration in the two cases is:
\[ \frac{k_1}{k_2} = 3 \]

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