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

Explanation

To derive the \( x-t \) equation for a particle in simple harmonic motion (SHM) that starts its journey from \( x = -\frac{A}{2} \) in the negative direction at \( t = 0 \), we can follow these steps:

### Step 1: Understand the Initial Conditions
The particle starts at position \( x = -\frac{A}{2} \) at \( t = 0 \) and moves in the negative direction. This indicates that the velocity \( v \) at \( t = 0 \) is negative.

### Step 2: General Equation of SHM
The general equation for SHM can be expressed in terms of sine or cosine functions:
- Using cosine:
\[ x(t) = A \cos(\omega t + \phi) \]
- Using sine:
\[ x(t) = A \sin(\omega t + \phi) \]
where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase constant.

### Step 3: Determine the Phase Constant
Since the particle starts at \( x = -\frac{A}{2} \), we need to find the appropriate phase constant \( \phi \).

1. **Using Cosine Function**:
\[ x(0) = A \cos(\phi) = -\frac{A}{2} \]
This gives:
\[ \cos(\phi) = -\frac{1}{2} \]
The angles that satisfy this equation are \( \phi = \frac{2\pi}{3} \) (or 120°) and \( \phi = \frac{4\pi}{3} \) (or 240°).

2. **Direction of Motion**: Since the particle is moving in the negative direction, we choose \( \phi = \frac{2\pi}{3} \) (or 120°) because it corresponds to a negative velocity at \( t = 0 \).

### Step 4: Write the Equation
Using the determined phase constant in the cosine function, the equation becomes:
\[ x(t) = A \cos(\omega t + \frac{2\pi}{3}) \]

### Step 5: Alternative Representation
Alternatively, we can express the equation using the sine function:
\[ x(t) = A \sin(\omega t + \phi) \]
Using the phase constant \( \phi = \frac{2\pi}{3} \):
\[ x(t) = A \sin(\omega t + \frac{2\pi}{3}) \]

### Final Equation
Thus, the final equation for the position of the particle in SHM starting from \( x = -\frac{A}{2} \) in the negative direction is:
\[ x(t) = A \cos(\omega t + \frac{2\pi}{3}) \quad \text{or} \quad x(t) = A \sin(\omega t + \frac{2\pi}{3}) \]

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