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 solve the problem of finding the kinetic energy of rotation for a body with angular momentum \( L \) and moment of inertia \( I \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between angular momentum and angular velocity**:
The angular momentum \( L \) of a rotating body is given by the formula:
\[ L = I \omega \]
where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.

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

3. **Write the formula for kinetic energy of rotation**:
The kinetic energy \( K \) of a rotating body is given by:
\[ K = \frac{1}{2} I \omega^2 \]

4. **Substitute the expression for angular velocity**:
Now, substitute \( \omega = \frac{L}{I} \) into the kinetic energy formula:
\[ K = \frac{1}{2} I \left(\frac{L}{I}\right)^2 \]

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

6. **Final result**:
Thus, the kinetic energy of rotation is:
\[ K = \frac{L^2}{2I} \]

### Final Answer:
The kinetic energy of rotation is given by:
\[ K = \frac{L^2}{2I} \]