Two bodies A and B having same angular momentum and `I_(A) gt I_(B)`, then the relation between `(K.E.)_(A) and (K.E.)_(B) ` will be :

Options

A. `(K.E.)_(A) gt (K.E.)_(B)`
B. `(K.E.)_(A)=(K.E.)_(B)`
C. `(K.E.)_(A) lt (K.E.)_(B)`
D. `(K.E.)_(A)""^(1)(K.E.)_(B)`

Answer

To solve the problem, we need to analyze the relationship between the angular momentum and kinetic energy of two rotating bodies A and B, given that they have the same angular momentum and that the moment of inertia of A is greater than that of B.

### Step-by-Step Solution:

1. **Understand 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 Conditions**:
We know that:
- \( L_A = L_B \) (same angular momentum for both bodies)
- \( I_A > I_B \) (moment of inertia of A is greater than that of B)

3. **Express Angular Velocity**:
From the angular momentum formula, we can express the angular velocities of A and B:
\[ \omega_A = \frac{L_A}{I_A} \quad \text{and} \quad \omega_B = \frac{L_B}{I_B} \]
Since \( L_A = L_B \), we can write:
\[ \omega_A = \frac{L}{I_A} \quad \text{and} \quad \omega_B = \frac{L}{I_B} \]

4. **Kinetic Energy of Rotating Bodies**:
The kinetic energy \( K.E. \) of a rotating body is given by:
\[ K.E. = \frac{1}{2} I \omega^2 \]
Substituting the expressions for angular velocity, we get:
\[ K.E._A = \frac{1}{2} I_A \left(\frac{L}{I_A}\right)^2 = \frac{L^2}{2 I_A} \]
\[ K.E._B = \frac{1}{2} I_B \left(\frac{L}{I_B}\right)^2 = \frac{L^2}{2 I_B} \]

5. **Compare Kinetic Energies**:
Now we can compare the kinetic energies:
\[ K.E._A = \frac{L^2}{2 I_A} \quad \text{and} \quad K.E._B = \frac{L^2}{2 I_B} \]
Since \( I_A > I_B \), it follows that:
\[ \frac{1}{I_A} < \frac{1}{I_B} \]
Therefore:
\[ K.E._A < K.E._B \]

6. **Conclusion**:
The relation between the kinetic energies of the two bodies is:
\[ K.E._A < K.E._B \]

### Final Answer:
The kinetic energy of body A is less than the kinetic energy of body B, i.e., \( K.E._A < K.E._B \).