A particle moving along the X-axis executes simple harmonic motion, then the force acting on it is given by
where, A and K are positive constants.

Options

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

Answer

To solve the problem, we need to analyze the motion of a particle executing simple harmonic motion (SHM) along the X-axis and derive the expression for the force acting on it.

### Step-by-Step Solution:

1. **Understanding Simple Harmonic Motion (SHM)**:
- A particle in SHM experiences a restoring force that is proportional to its displacement from the equilibrium position. The general form of the equation of motion for SHM can be expressed as:
\[ \frac{d^2x}{dt^2} + \omega^2 x = 0 \]
where \( \omega \) is the angular frequency of the motion.

2. **Relating Acceleration to Displacement**:
- From the equation of motion, we can identify that the acceleration \( a \) of the particle is given by:
\[ a = \frac{d^2x}{dt^2} = -\omega^2 x \]
- This indicates that the acceleration is directly proportional to the negative of the displacement \( x \).

3. **Using Newton's Second Law**:
- According to Newton's second law, the force \( F \) acting on the particle is related to its mass \( m \) and acceleration \( a \) by:
\[ F = m \cdot a \]
- Substituting the expression for acceleration, we have:
\[ F = m \cdot (-\omega^2 x) \]
- Rearranging this gives:
\[ F = -m \omega^2 x \]

4. **Identifying the Force Expression**:
- The force can also be expressed in the form \( F = -kx \), where \( k \) is a positive constant (the force constant).
- By comparing the two expressions for force, we can identify that:
\[ k = m \omega^2 \]
- Here, \( k \) is a constant that relates to the mass of the particle and the angular frequency of the motion.

5. **Conclusion**:
- Since the problem states that the force acting on the particle is given in terms of positive constants \( A \) and \( K \), we can conclude that:
\[ F = -Kx \]
- Therefore, the correct expression for the force acting on the particle in SHM is:
\[ F = -A \cdot K \cdot x \]
- This leads us to conclude that the force acting on the particle is proportional to its displacement, confirming the nature of SHM.

### Final Answer:
The force acting on the particle executing simple harmonic motion is given by \( F = -Kx \), where \( K \) is a positive constant.

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