A body is rotating with angular momentum L. If I is its moment of inertia about the axis of rotation is I, its kinetic energy of rotation is

Options

A. `(1)/(2)IL^(2)`
B. `(1)/(2)IL`
C. `(1)/(2)(I^(2)//L)`
D. `(1)/(2).(L^(2))/(I)`

Answer

To find the kinetic energy of a body rotating with angular momentum \( L \) and moment of inertia \( I \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship**:
We know that the kinetic energy \( K \) of a rotating body is given by the formula:
\[ K = \frac{1}{2} I \omega^2 \]
where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.

2. **Relate Angular Momentum and Angular Velocity**:
Angular momentum \( L \) is related to moment of inertia \( I \) and angular velocity \( \omega \) by the equation:
\[ L = I \omega \]
From this equation, we can express angular velocity \( \omega \) in terms of angular momentum \( L \) and moment of inertia \( I \):
\[ \omega = \frac{L}{I} \]

3. **Substitute \( \omega \) in the Kinetic Energy Formula**:
Now, we can substitute \( \omega \) back into the kinetic energy formula:
\[ K = \frac{1}{2} I \left( \frac{L}{I} \right)^2 \]

4. **Simplify the Expression**:
Simplifying the expression gives:
\[ K = \frac{1}{2} I \cdot \frac{L^2}{I^2} \]
\[ K = \frac{L^2}{2I} \]

5. **Final Result**:
Therefore, the kinetic energy of rotation is:
\[ K = \frac{L^2}{2I} \]

### Conclusion:
The kinetic energy of the rotating body is given by \( K = \frac{L^2}{2I} \).

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