The relation between acceleration and displacement of four particle are given below:
Which one of the particle is exempting simple harmonic motion?

Options

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

Explanation

To determine which particle is undergoing simple harmonic motion (SHM) based on the relationship between acceleration and displacement, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Condition for SHM**:
- For a particle to exhibit simple harmonic motion, the acceleration (a) must be directly proportional to the displacement (x) from the equilibrium position and must be directed towards that position. This relationship can be expressed mathematically as:
\[ a = -kx \]
where \( k \) is a positive constant.

2. **Relate Force and Acceleration**:
- According to Newton's second law, the force acting on a particle is given by:
\[ F = ma \]
where \( m \) is the mass of the particle. Rearranging this gives us:
\[ a = \frac{F}{m} \]

3. **Express Acceleration in Terms of Displacement**:
- For SHM, we can substitute the force from the SHM condition into Newton's second law:
\[ a = \frac{-kx}{m} \]
- This can be rewritten as:
\[ a = -\frac{k}{m} x \]
- Here, we can define a new constant \( p = \frac{k}{m} \), leading to:
\[ a = -px \]

4. **Analyze Given Relations**:
- Now, we need to analyze the given relations for the four particles. We are looking for a relation that matches the form:
\[ a = -px \]
- If any relation can be expressed in this form, that particle is undergoing SHM.

5. **Identify the Correct Option**:
- After reviewing the provided relations for the four particles, we find that the last option matches the required form of \( a = -px \). Therefore, this particle is undergoing simple harmonic motion.

### Conclusion:
The particle that exhibits the relationship \( a = -kx \) (or equivalently \( a = -px \)) is the one that is undergoing simple harmonic motion.