Two rotating bodies have same angular momentum but their moments of inertia are I1 and I2 respectively `(I_1 gt I_2)`. Which body will have higher kinetic energy of rotation :-

Options

A. first
B. second
C. both will have same kinetic energy
D. not possible to predict

Answer

To solve the problem, we need to determine which body has a higher kinetic energy of rotation given that both bodies have the same angular momentum but different moments of inertia.

### Step-by-Step Solution:

1. **Understanding Angular Momentum**:
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. **Given Information**:
We have two bodies with moments of inertia \( I_1 \) and \( I_2 \) such that \( I_1 > I_2 \). We know that both bodies have the same angular momentum \( L \).

3. **Relating Angular Momentum to Angular Velocity**:
Since both bodies have the same angular momentum, we can express the angular velocities in terms of their moments of inertia:
\[ \omega_1 = \frac{L}{I_1} \quad \text{and} \quad \omega_2 = \frac{L}{I_2} \]

4. **Calculating Kinetic Energy**:
The rotational kinetic energy \( K \) of a body is given by:
\[ K = \frac{1}{2} I \omega^2 \]
For the two bodies, we can write:
\[ K_1 = \frac{1}{2} I_1 \omega_1^2 \quad \text{and} \quad K_2 = \frac{1}{2} I_2 \omega_2^2 \]

5. **Substituting Angular Velocity**:
We substitute \( \omega_1 \) and \( \omega_2 \) from step 3 into the kinetic energy equations:
\[ K_1 = \frac{1}{2} I_1 \left(\frac{L}{I_1}\right)^2 = \frac{1}{2} \frac{L^2}{I_1} \]
\[ K_2 = \frac{1}{2} I_2 \left(\frac{L}{I_2}\right)^2 = \frac{1}{2} \frac{L^2}{I_2} \]

6. **Comparing Kinetic Energies**:
Now we can compare \( K_1 \) and \( K_2 \):
\[ K_1 = \frac{1}{2} \frac{L^2}{I_1} \quad \text{and} \quad K_2 = \frac{1}{2} \frac{L^2}{I_2} \]
Since \( I_1 > I_2 \), it follows that \( \frac{1}{I_1} < \frac{1}{I_2} \). Therefore:
\[ K_1 < K_2 \]

7. **Conclusion**:
Hence, the body with moment of inertia \( I_2 \) has a higher kinetic energy of rotation than the body with moment of inertia \( I_1 \).

### Final Answer:
The body with moment of inertia \( I_2 \) will have a higher kinetic energy of rotation.