The equation of motion of a simple harmonic motion is

Options

A. `(d^(2)x)/(dt^(2))=-omega^(2)x`
B. `(d^(2)x)/(dt^(2))=-omega^(2)t`
C. `(d^(2)x)/(dt^(2))=-omegax`
D. `(d^(2)x)/(dt^(2))=-omegat`

Explanation

To determine which equation represents simple harmonic motion (SHM), we need to analyze the given options based on the standard form of the equation of motion for SHM.

### Step-by-Step Solution:

1. **Understanding the Equation of Motion for SHM**:
The standard equation of motion for simple harmonic motion is given by:
\[ \frac{d^2x}{dt^2} = -\omega^2 x \]
Here, \(x\) is the displacement, \(\omega\) is the angular frequency, and \(\frac{d^2x}{dt^2}\) represents the acceleration.

2. **Analyzing the Given Options**:
We have four options to evaluate:
- Option A: \(\frac{d^2x}{dt^2} = -\omega^2 x\)
- Option B: \(\frac{d^2x}{dt^2} = -\omega^2 t\)
- Option C: \(\frac{d^2x}{dt^2} = -\omega x\)
- Option D: \(\frac{d^2x}{dt^2} = -\omega t\)

3. **Evaluating Each Option**:
- **Option A**: This matches the standard form of SHM. Therefore, this is a valid equation for SHM.
- **Option B**: This equation has \(t\) on the right side, which means it does not depend on the displacement \(x\). Thus, it does not represent SHM.
- **Option C**: This equation has \(-\omega x\) instead of \(-\omega^2 x\). The absence of the square means it does not represent SHM.
- **Option D**: Similar to option B, this equation has \(t\) on the right side and does not depend on \(x\), so it does not represent SHM.

4. **Conclusion**:
The only equation that correctly represents simple harmonic motion is:
\[ \frac{d^2x}{dt^2} = -\omega^2 x \]
Therefore, the correct answer is **Option A**.