A particle in SHM starts its journey (at `t = 0`) from `x = - (A)/(2)` in negative direction. Write `x - t` equation corresponding to given condition. Angular frequency of oscillations is `omega`.

Explanation

To solve the problem of finding the \( x - t \) equation for a particle in Simple Harmonic Motion (SHM) that starts from \( x = -\frac{A}{2} \) and moves in the negative direction, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Initial Conditions**:
- The particle starts at \( x = -\frac{A}{2} \) at \( t = 0 \) and moves in the negative direction.

2. **Determine the General Equation of SHM**:
- The general equations for SHM can be expressed as:
\[ x(t) = A \sin(\omega t + \phi) \quad \text{or} \quad x(t) = A \cos(\omega t + \phi) \]
- Here, \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase constant.

3. **Choose the Appropriate Form**:
- Since the particle starts at \( x = -\frac{A}{2} \), we can use the sine function for this scenario. The sine function will allow us to incorporate the phase shift easily.

4. **Determine the Phase Constant \( \phi \)**:
- At \( t = 0 \), we have:
\[ x(0) = A \sin(\phi) = -\frac{A}{2} \]
- This implies:
\[ \sin(\phi) = -\frac{1}{2} \]
- The angle \( \phi \) that satisfies this equation is \( \phi = -\frac{\pi}{6} \) (or \( 210^\circ \) in degrees).

5. **Write the Equation**:
- Substituting \( \phi \) into the sine equation gives:
\[ x(t) = A \sin(\omega t - \frac{\pi}{6}) \]

6. **Adjust for Direction of Motion**:
- Since the particle is moving in the negative direction at \( t = 0 \), we need to adjust the sine function to reflect this. The equation can be written as:
\[ x(t) = -\frac{A}{2} + A \sin(\omega t - \frac{\pi}{6}) \]

### Final Equation:
The final equation for the position of the particle as a function of time is:
\[ x(t) = -\frac{A}{2} + A \sin(\omega t - \frac{\pi}{6}) \]