The rotational kinetic energy of two bodies of moment of inertia `9 kg m^(2)` and `1kg m^(2)` are same . The ratio of their angular momenta is

Options

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

Explanation

To solve the problem, we need to find the ratio of the angular momenta of two bodies with the same rotational kinetic energy but different moments of inertia. Let's denote the moments of inertia as \( I_1 = 9 \, \text{kg m}^2 \) and \( I_2 = 1 \, \text{kg m}^2 \).

### Step-by-Step Solution:

1. **Understanding Rotational Kinetic Energy**:
The rotational kinetic energy \( K \) of a rotating body is given by the formula:
\[ K = \frac{L^2}{2I} \]
where \( L \) is the angular momentum and \( I \) is the moment of inertia.

2. **Setting Up the Equation**:
Since the rotational kinetic energies of both bodies are the same, we can write:
\[ K_1 = K_2 \]
Thus, we have:
\[ \frac{L_1^2}{2I_1} = \frac{L_2^2}{2I_2} \]

3. **Canceling Common Factors**:
We can simplify this equation by multiplying both sides by \( 2 \):
\[ \frac{L_1^2}{I_1} = \frac{L_2^2}{I_2} \]

4. **Cross-Multiplying**:
Cross-multiplying gives us:
\[ L_1^2 \cdot I_2 = L_2^2 \cdot I_1 \]

5. **Finding the Ratio of Angular Momenta**:
Rearranging the equation to find the ratio \( \frac{L_1^2}{L_2^2} \):
\[ \frac{L_1^2}{L_2^2} = \frac{I_1}{I_2} \]
Taking the square root of both sides, we get:
\[ \frac{L_1}{L_2} = \sqrt{\frac{I_1}{I_2}} \]

6. **Substituting the Values**:
Now, substituting the given values of moments of inertia:
\[ \frac{L_1}{L_2} = \sqrt{\frac{9 \, \text{kg m}^2}{1 \, \text{kg m}^2}} = \sqrt{9} = 3 \]

7. **Final Ratio**:
Thus, the ratio of their angular momenta is:
\[ \frac{L_1}{L_2} = 3:1 \]

### Conclusion:
The ratio of the angular momenta of the two bodies is \( 3:1 \).