Which of the following is a necessary and sufficient condition for S.H.M.

Options

A. Constant period
B. Constant acceleration
C. Proportionality between acceleration and velocity
D. Proportionality between restoring force and displacement.

Explanation

To determine the necessary and sufficient condition for Simple Harmonic Motion (S.H.M.), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of S.H.M.**:
Simple Harmonic Motion is defined as a type of periodic motion where the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction.

2. **Identify the Mathematical Condition**:
The necessary and sufficient condition for S.H.M. can be expressed mathematically by the second-order differential equation:
\[ \frac{d^2x}{dt^2} + \omega^2 x = 0 \]
Here, \( \frac{d^2x}{dt^2} \) is the acceleration, \( x \) is the displacement, and \( \omega \) is the angular frequency.

3. **Rearranging the Equation**:
Rearranging the equation gives:
\[ \frac{d^2x}{dt^2} = -\omega^2 x \]
This indicates that the acceleration is proportional to the negative of the displacement.

4. **Relate to Newton's Second Law**:
By multiplying both sides of the equation by mass \( m \), we can relate this to force:
\[ m \frac{d^2x}{dt^2} = -m \omega^2 x \]
According to Newton's second law, \( F = ma \), we can rewrite this as:
\[ F = -m \omega^2 x \]
This equation shows that the force is proportional to the displacement and acts in the opposite direction, which is characteristic of S.H.M.

5. **Conclusion**:
Thus, the necessary and sufficient condition for S.H.M. is that the restoring force \( F \) is proportional to the displacement \( x \) and acts in the opposite direction. Therefore, the correct option is that there is a proportionality between the restoring force and displacement.