Angular momentum L and rotational kinetic energy `K_R` of a body are related to each other by the relation. (I = moment of inertia)

Options

A. 1. `K_R` = 2IL
B. 2. `K_R = L^(2)/(2I)`
C. 3. `K_R = (2I)/L`
D. 4. `K_R = L^(2)/I`

Explanation

To solve the problem of finding the relationship between angular momentum \( L \) and rotational kinetic energy \( K_R \) of a body in terms of moment of inertia \( I \), we can follow these steps:

### Step-by-Step Solution

1. **Understand the formulas**:
- The formula for rotational kinetic energy \( K_R \) is given by:
\[ K_R = \frac{1}{2} I \omega^2 \]
- The formula for angular momentum \( L \) is given by:
\[ L = I \omega \]

2. **Express \( \omega \) in terms of \( L \)**:
- From the angular momentum formula, we can express \( \omega \) as:
\[ \omega = \frac{L}{I} \]

3. **Substitute \( \omega \) into the kinetic energy formula**:
- Now, substitute \( \omega \) into the formula for \( K_R \):
\[ K_R = \frac{1}{2} I \left(\frac{L}{I}\right)^2 \]

4. **Simplify the equation**:
- This simplifies to:
\[ K_R = \frac{1}{2} I \cdot \frac{L^2}{I^2} \]
- Further simplification gives:
\[ K_R = \frac{L^2}{2I} \]

5. **Final relation**:
- Thus, the relationship between angular momentum \( L \) and rotational kinetic energy \( K_R \) in terms of moment of inertia \( I \) is:
\[ K_R = \frac{L^2}{2I} \]

### Conclusion
The final relation we derived is:
\[ K_R = \frac{L^2}{2I} \]