Which of the following statements is`//` are true for a simple harmonic oscillator ?

Options

A. Force acting is directly proportional to displacement from the mean position and opposite to it.
B. Motion is periodic
C. Acceleration of the oscillator is constant
D. The velocity is periodic

Explanation

To determine which statements are true for a simple harmonic oscillator (SHO), we will analyze each statement based on the principles of simple harmonic motion.

### Step-by-Step Solution:

1. **Understanding Simple Harmonic Motion (SHM)**:
- A simple harmonic oscillator is a system that oscillates about an equilibrium position. The motion is periodic, and the restoring force acting on the oscillator is proportional to the displacement from the equilibrium position and acts in the opposite direction.

2. **Analyzing the Statements**:
- **Statement 1**: "Force acting is directly proportional to displacement from the mean position and opposite to it."
- This statement is true. According to Hooke's Law, the force \( F \) acting on a mass in SHM is given by \( F = -kx \), where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position. The negative sign indicates that the force acts in the opposite direction of the displacement.

- **Statement 2**: "Motion is periodic."
- This statement is also true. Simple harmonic motion is characterized by its periodic nature, meaning that the oscillator returns to its mean position after a fixed interval of time, known as the time period \( T \).

- **Statement 3**: "Acceleration of the oscillator is constant."
- This statement is false. In SHM, the acceleration \( a \) is given by \( a = -\frac{k}{m}x \), where \( m \) is the mass of the oscillator. Since the displacement \( x \) changes as the oscillator moves, the acceleration is not constant; it varies with displacement.

- **Statement 4**: "Velocity is periodic."
- This statement is true. The velocity of the oscillator changes as it moves through its cycle, but it also exhibits periodic behavior, repeating its values over each cycle of motion.

3. **Conclusion**:
- The true statements regarding a simple harmonic oscillator are:
- Statement 1: True
- Statement 2: True
- Statement 3: False
- Statement 4: True

Thus, the correct answers are statements 1, 2, and 4.

### Summary of True Statements:
- **True Statements**: 1, 2, and 4.
- **False Statement**: 3.