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. Amplitude and initial phase
B. Amplitude and total energy of oscillation
C. Initial phase and total energy of oscillation
D. Initial position and initial velocity

Explanation

To determine the complete motion of a simple harmonic oscillator (SHM) given the angular frequency \( \omega \), we need two arbitrary initial conditions. Let's analyze what these initial conditions can be.

### Step-by-Step Solution:

1. **Understanding SHM Equation**:
The equation for simple harmonic motion can be expressed as:
\[ x(t) = A \sin(\omega t + \phi) \]
where:
- \( x(t) \) is the displacement at time \( t \),
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( \phi \) is the phase constant.

2. **Identifying Initial Conditions**:
To fully determine the motion, we need to know two initial conditions. The initial conditions can be:
- Initial position \( x(0) \) at \( t = 0 \).
- Initial velocity \( v(0) \) at \( t = 0 \).

3. **Using Initial Position**:
At \( t = 0 \):
\[ x(0) = A \sin(\phi) \]
Knowing the initial position \( x(0) \) allows us to solve for the phase constant \( \phi \) if the amplitude \( A \) is known.

4. **Using Initial Velocity**:
The velocity in SHM is given by:
\[ v(t) = \frac{dx}{dt} = A \omega \cos(\omega t + \phi) \]
At \( t = 0 \):
\[ v(0) = A \omega \cos(\phi) \]
Knowing the initial velocity \( v(0) \) allows us to find the phase constant \( \phi \) or the amplitude \( A \).

5. **Conclusion**:
Therefore, the two arbitrary initial conditions that are necessary and sufficient to determine the motion completely are:
- Initial position \( x(0) \) and initial velocity \( v(0) \).
- Alternatively, one could also use the amplitude \( A \) and the initial phase \( \phi \).

### Final Answer:
The two arbitrary initial conditions necessary and sufficient to determine the motion completely in simple harmonic motion are:
1. Initial position \( x(0) \) and initial velocity \( v(0) \).
2. Amplitude \( A \) and initial phase \( \phi \).

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