Two bodies with moment of inertia `I_1` and `I_2 (I_1 gt I_2)` have equal angular momenta. If their kinetic energy of rotation are `E_1` and `E_2` respectively, then.

Options

A. `E_1 = E_2`
B. `E_1 lt E_2`
C. `E_1 gt E_2`
D. `E_1 E_2`

Explanation

To solve the problem, we need to establish the relationship between the kinetic energies \( E_1 \) and \( E_2 \) of the two bodies with moments of inertia \( I_1 \) and \( I_2 \), given that their angular momenta are equal.

### 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:
\[ L = I \omega \]
where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.

2. **Express angular velocity in terms of angular momentum:**
Since both bodies have equal angular momentum, we can write:
\[ L_1 = L_2 \implies I_1 \omega_1 = I_2 \omega_2 \]
From this, we can express the angular velocities:
\[ \omega_1 = \frac{L}{I_1} \quad \text{and} \quad \omega_2 = \frac{L}{I_2} \]

3. **Write the expression for kinetic energy:**
The kinetic energy \( E \) of a rotating body is given by:
\[ E = \frac{1}{2} I \omega^2 \]
Therefore, for the two bodies, we have:
\[ E_1 = \frac{1}{2} I_1 \omega_1^2 \quad \text{and} \quad E_2 = \frac{1}{2} I_2 \omega_2^2 \]

4. **Substitute the expressions for angular velocities:**
Substituting \( \omega_1 \) and \( \omega_2 \) into the kinetic energy equations:
\[ E_1 = \frac{1}{2} I_1 \left(\frac{L}{I_1}\right)^2 = \frac{1}{2} \frac{L^2}{I_1} \]
\[ E_2 = \frac{1}{2} I_2 \left(\frac{L}{I_2}\right)^2 = \frac{1}{2} \frac{L^2}{I_2} \]

5. **Compare the kinetic energies:**
Now we can compare \( E_1 \) and \( E_2 \):
\[ \frac{E_1}{E_2} = \frac{\frac{1}{2} \frac{L^2}{I_1}}{\frac{1}{2} \frac{L^2}{I_2}} = \frac{I_2}{I_1} \]
Since \( I_1 > I_2 \), it follows that:
\[ E_1 < E_2 \]

### Conclusion:
Thus, we conclude that the kinetic energy of rotation of body 1 is less than that of body 2:
\[ E_1 < E_2 \]