Which of the following relationships between the acceleration a and the displacement x of a particle executing simple harmonic motion?

Options

A. `a=2x^(2)`
B. `a=-2x^(2)`
C. `a=2x`
D. `a=-2x`

Explanation

To solve the problem, we need to analyze the relationships given for the acceleration \( a \) and the displacement \( x \) of a particle executing simple harmonic motion (SHM).

### Step-by-Step Solution:

1. **Understanding SHM**: In simple harmonic motion, the acceleration \( a \) of a particle is related to its displacement \( x \) from the mean position by the formula:
\[ a = -\omega^2 x \]
where \( \omega \) is the angular frequency of the motion.

2. **Analyzing the Options**: We need to evaluate the given options against the standard SHM equation:
- Option 1: \( a = 2x^2 \)
- Option 2: \( a = -2x^2 \)
- Option 3: \( a = 2x \)
- Option 4: \( a = -2x \)

3. **Evaluating Options 1 and 2**:
- Both options 1 and 2 involve \( x^2 \), which indicates that acceleration is proportional to the square of the displacement. This is not consistent with the SHM equation, where acceleration is directly proportional to \( x \) (not \( x^2 \)). Therefore, both options 1 and 2 are incorrect.

4. **Evaluating Options 3 and 4**:
- Both options 3 and 4 involve a linear relationship with \( x \). We can rewrite the SHM equation as:
\[ a = -\omega^2 x \]
- For option 3, \( a = 2x \): This suggests that \( \omega^2 = -2 \), which is not possible since \( \omega^2 \) must be a positive value.
- For option 4, \( a = -2x \): This suggests that \( \omega^2 = 2 \), which is valid since \( \omega^2 \) can indeed be a positive number.

5. **Conclusion**: Based on the analysis, the only correct relationship that matches the standard SHM equation is:
\[ a = -2x \]
Therefore, the correct answer is **Option 4**.