In SHM which of the following statement is/are correct?

Options

A. Displacement and velocity may be in the same direction
B. Displacement and velocity can never be in the same direction
C. Velocity and acceleration may be in the same direction
D. Displacement and acceleration can never be in the same direction.

Answer

To determine which statements about Simple Harmonic Motion (SHM) are correct, we will analyze the properties of SHM step-by-step.

### Step 1: Understand the Definition of SHM
Simple Harmonic Motion is a type of periodic motion where an object oscillates back and forth around an equilibrium position. The motion is characterized by a restoring force that is proportional to the displacement from the equilibrium position and directed towards it.

**Hint:** Remember that SHM involves oscillation around a central point (equilibrium position).

### Step 2: Analyze the Displacement and Velocity Relationship
In SHM, when the particle is at the mean position (equilibrium), its velocity is at its maximum, while the displacement is zero. As the particle moves away from the mean position, the displacement increases, but the velocity decreases until it reaches zero at the extreme positions (A and B).

**Hint:** Consider the relationship between displacement and velocity at different points in the motion.

### Step 3: Analyze the Displacement and Acceleration Relationship
The restoring force in SHM is given by Hooke's law: \( F = -kx \), where \( k \) is the spring constant and \( x \) is the displacement from the mean position. The acceleration \( a \) can be expressed as \( a = \frac{F}{m} = -\frac{k}{m}x \). This shows that acceleration is always directed towards the mean position and is opposite to the displacement.

**Hint:** Focus on how acceleration behaves in relation to displacement in SHM.

### Step 4: Evaluate Each Statement
1. **Statement A:** Displacement and velocity can be in the same direction.
- This is correct when the particle is moving towards or away from the mean position.

2. **Statement B:** Displacement and velocity can never be in the same direction.
- This is incorrect, as explained above.

3. **Statement C:** Velocity and acceleration can be in the same direction.
- This is correct when the particle is moving towards the mean position.

4. **Statement D:** Displacement and acceleration can never be in the same direction.
- This is correct, as acceleration is always opposite to displacement.

### Conclusion
Based on the analysis:
- Correct statements are A, C, and D.
- Incorrect statement is B.

### Final Answer
The correct options are A, C, and D.