A particle, with restoring force proportional to displacement and resulting force proportional to velocity is subjected to a force `F sin omega t`. If the amplitude of the particle is maximum for `omega = omega_(1)`, and the energy of the particle is maximum for `omega=omega_(2)`, then

Options

A. `omega_(1) = omega_(0)` and `omega_(2) != omega_(0)`
B. `omega_(1) = omega_(0)` and `omega_(2) = omega_(0)`
C. `omega_(1) != omega_(0)` and `omega_(2) = omega_(0)`
D. `omega_(1) != omega_(0)` and `omega_(2) != omega_(0)`

Answer

To solve the problem, we need to analyze the behavior of a damped harmonic oscillator subjected to an external periodic force. The key points to consider are the relationships between the frequencies, amplitude, and energy of the oscillating particle.

### Step-by-Step Solution:

1. **Understanding the System**:
- The particle experiences a restoring force proportional to its displacement, which indicates simple harmonic motion (SHM).
- It also experiences a damping force proportional to its velocity, which means the system is a damped oscillator.
- The external force acting on the particle is given by \( F \sin(\omega t) \).

2. **Resonance and Amplitude**:
- In a damped oscillator, the amplitude of oscillation is maximized at a specific frequency known as the resonance frequency.
- This frequency, denoted as \( \omega_1 \), is where the driving frequency matches the natural frequency of the system, adjusted for damping effects.

3. **Energy Considerations**:
- The energy of the oscillating particle is also maximized at a specific frequency. For a damped oscillator, this frequency is typically the natural frequency of the system, denoted as \( \omega_2 \).
- The energy is maximum when the system oscillates at its natural frequency, which is less affected by damping.

4. **Relationship Between Frequencies**:
- For a damped oscillator, it is known that the frequency at which the amplitude is maximum (\( \omega_1 \)) is different from the frequency at which the energy is maximum (\( \omega_2 \)).
- Generally, we have \( \omega_1 < \omega_2 \) because the amplitude is affected by damping, while energy is maximized at the natural frequency.

5. **Conclusion**:
- Therefore, we conclude that \( \omega_1 \) (frequency for maximum amplitude) is not equal to \( \omega_2 \) (frequency for maximum energy).
- The correct relationships are:
- \( \omega_1 \neq \omega_2 \)
- \( \omega_1 < \omega_2 \)

### Final Answer:
The correct conclusion based on the analysis is that \( \omega_1 \) is not equal to \( \omega_2 \) and \( \omega_1 < \omega_2 \).