In forced oscillation of a particle the amplitude is maximum for a frequency `omega_(1)` of the force while the energy is maximum for a frequency `omega_(2)` of the force, then .

Options

A. `omega_(1)` = `omega_(2)`
B. `omega_(1)gtomega_(2)`
C. `omega_(1)ltomega_(2)`when damping is small and `omega_(1)gtomega_(2)` when damping is large
D. `omega_(1)ltomega_(2)`

Explanation

To solve the problem, we need to analyze the conditions for maximum amplitude and maximum energy in the context of forced oscillations.

### Step-by-Step Solution:

1. **Understanding Forced Oscillation**:
- In forced oscillation, a particle is subjected to an external periodic force. The system will respond at the frequency of this external force.

2. **Resonance Condition**:
- Resonance occurs when the frequency of the external force matches the natural frequency of the system. At resonance, the amplitude of oscillation reaches its maximum value.

3. **Amplitude Maximum Condition**:
- The amplitude of oscillation is maximum at a frequency \( \omega_1 \). This implies that \( \omega_1 \) is equal to the natural frequency \( \omega_0 \) of the system:
\[ \omega_1 = \omega_0 \]

4. **Energy Maximum Condition**:
- The energy of the oscillating system is also maximized when the system is in resonance. Therefore, at the frequency \( \omega_2 \), the energy is maximum, which also implies:
\[ \omega_2 = \omega_0 \]

5. **Equating Frequencies**:
- Since both conditions lead to the same natural frequency \( \omega_0 \), we can equate \( \omega_1 \) and \( \omega_2 \):
\[ \omega_1 = \omega_2 \]

6. **Conclusion**:
- Thus, the relationship between \( \omega_1 \) and \( \omega_2 \) is that they are equal. Therefore, the answer is:
\[ \omega_1 = \omega_2 \]

### Final Answer:
The correct relation is \( \omega_1 = \omega_2 \).