For a simple harmonic motion with given angular frequency `omega`, two arbitrary initial conditions are necessary and sufficient to determine the motion completely. These initial conditions may be

Options

A. initial position and initial velocity
B. amplitude and initial phase
C. total energy of oscillation and amplitude
D. total energy of oscillation and initial phase.

Explanation

To determine the complete motion of a simple harmonic oscillator with a given angular frequency \( \omega \), we need two arbitrary initial conditions. Here’s a step-by-step solution to identify the necessary conditions:

### Step 1: Understanding Simple Harmonic Motion (SHM)
The general equation for simple harmonic motion can be expressed as:
\[ x(t) = A \cos(\omega t + \delta) \]
where:
- \( x(t) \) is the displacement at time \( t \),
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( \delta \) is the phase constant.

### Step 2: Identifying Initial Conditions
To fully describe the motion, we need to determine both the amplitude \( A \) and the phase constant \( \delta \). This can be achieved through various combinations of initial conditions.

### Step 3: Analyzing Possible Initial Conditions
1. **Initial Position and Initial Velocity**:
- Knowing the initial position \( x(0) \) and initial velocity \( v(0) \) allows us to solve for both \( A \) and \( \delta \).

2. **Amplitude and Initial Phase**:
- If we know the amplitude \( A \) and the phase \( \delta \), we can describe the motion completely.

3. **Total Energy of Oscillation and Amplitude**:
- The total energy \( E \) in SHM is given by:
\[ E = \frac{1}{2} k A^2 \]
Knowing \( E \) and \( A \) does not provide enough information to determine \( \delta \).

4. **Total Energy of Oscillation and Initial Phase**:
- Knowing \( E \) and \( \delta \) allows us to find \( A \) using the energy formula, thus giving us a complete description of the motion.

### Step 4: Conclusion
From the analysis, the combinations of initial conditions that are sufficient to determine the motion completely are:
- Initial Position and Initial Velocity (Option A)
- Amplitude and Initial Phase (Option B)
- Total Energy of Oscillation and Initial Phase (Option D)

The combination of Total Energy of Oscillation and Amplitude (Option C) does not provide sufficient information to determine the phase constant \( \delta \).

### Final Answer
The initial conditions that are necessary and sufficient to determine the motion completely are:
- **Option A**: Initial Position and Initial Velocity
- **Option B**: Amplitude and Initial Phase
- **Option D**: Total Energy of Oscillation and Initial Phase