The relation between acceleration and displacement of four particles are given below:

Options

A. `a_(x)=+2x`
B. `a_(x)=2x^(2)`
C. `a_(x)=02x^(2)`
D. `a_(x)=-2x`

Explanation

To determine which of the four particles (A, B, C, and D) is moving in Simple Harmonic Motion (SHM), we need to analyze the relationship between acceleration (A) and displacement (X) for each particle.

### Step-by-Step Solution:

1. **Understanding SHM**:
- In SHM, the acceleration of a particle is directly proportional to its displacement from the equilibrium position and is directed towards that position. Mathematically, this is expressed as:
\[ A = -kX \]
- Here, \( k \) is a positive constant, and the negative sign indicates that the acceleration is directed opposite to the displacement.

2. **Rewriting the Equation**:
- We can express the relationship in terms of mass (M) and acceleration (A):
\[ F = ma \implies A = \frac{F}{M} \]
- Thus, we can rewrite the equation for SHM as:
\[ A = -\frac{k}{M}X \]
- This shows that \( A \) is linearly proportional to \( -X \).

3. **Analyzing Each Particle**:
- We need to check the relationships given for each particle (A, B, C, and D) to see if they match the form \( A \propto -X \).

- **Particle A**: If the relationship is \( A = kX \) (positive proportionality), it does not satisfy SHM.
- **Particle B**: If the relationship is \( A = kX^2 \) (quadratic dependence), it does not satisfy SHM.
- **Particle C**: If the relationship is \( A = kX^2 \) (also quadratic), it does not satisfy SHM.
- **Particle D**: If the relationship is \( A = -kX \) (negative linear dependence), it satisfies the condition for SHM.

4. **Conclusion**:
- The only particle that exhibits the required relationship for SHM is **Particle D**.

### Final Answer:
The particle moving in Simple Harmonic Motion is **Particle D**.

---