A particle executes S.H.M. along x-axis. The force acting on it is given by :

Options

A. A cos (kx)
B. `Ae^(-kx)`
C. Akx
D. `-Akx`

Explanation

To solve the problem of identifying the correct force acting on a particle executing Simple Harmonic Motion (S.H.M.) along the x-axis, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Force in S.H.M.**:
The force acting on a particle in simple harmonic motion is given by Hooke's Law, which states that the force (F) is proportional to the displacement (x) from the equilibrium position and is directed towards that position. Mathematically, this is expressed as:
\[ F = -kx \]
where \( k \) is a positive constant and the negative sign indicates that the force is directed opposite to the displacement.

**Hint**: Remember that in S.H.M., the force always acts towards the equilibrium position.

2. **Analyzing the Given Options**:
We need to evaluate the provided options for the force and determine which one matches the form of \( F = -kx \).

- **Option A: \( A \cos(kx) \)**
This expression does not have a linear relationship with \( x \) and does not satisfy the form of \( -kx \). It also does not change direction based on the sign of \( x \).

**Hint**: Check if the force is linear with respect to displacement.

- **Option B: \( -kx^2 \)**
This expression is quadratic in \( x \) and does not fit the linear form required for S.H.M. The force cannot be proportional to the square of the displacement.

**Hint**: The force should be linearly proportional to displacement, not quadratic.

- **Option C: \( A kx \)**
This expression is linear but lacks the negative sign. Therefore, it does not represent a restoring force as required in S.H.M.

**Hint**: Look for the negative sign that indicates the force is restoring.

- **Option D: \( -A kx \)**
This expression is in the correct form of \( F = -kx \) where \( A k \) can be considered as a constant (let's denote it as \( k' \)). This indicates that the force is proportional to the displacement and directed towards the equilibrium position.

**Hint**: Ensure that the force has the correct direction indicated by the negative sign.

3. **Conclusion**:
After analyzing all the options, we find that the only expression that correctly represents the force acting on a particle in S.H.M. is:
\[ F = -A kx \]
Therefore, the correct answer is **Option D**.

### Final Answer:
The correct force acting on the particle executing S.H.M. along the x-axis is given by:
\[ F = -A kx \]