Which among the following is the necessary condition for simple harmonic motion ?

Options

A. Constant period
B. Constant acceleration
C. Displacement and acceleration are proportional
D. Displacement and torque are proportional

Explanation

To determine the necessary condition for simple harmonic motion (SHM), we need to analyze the fundamental characteristics that define SHM. Here’s a step-by-step solution:

### Step 1: Understand the Definition of SHM
Simple Harmonic Motion is 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.

**Hint:** Remember that SHM is characterized by a restoring force that tries to bring the system back to equilibrium.

### Step 2: Identify the Relationship Between Displacement and Acceleration
In SHM, the acceleration (a) of the object is always directed towards the equilibrium position and is proportional to the negative of the displacement (y) from that position. This can be mathematically expressed as:
\[ a \propto -y \]
This means:
\[ a = -k \cdot y \]
where \( k \) is a positive constant.

**Hint:** The negative sign indicates that the acceleration is directed opposite to the displacement.

### Step 3: Analyze the Given Options
Now, let’s analyze the options provided in the question:
1. Constant period
2. Constant acceleration
3. Displacement and acceleration are proportional
4. Displacement and torque are proportional

From our understanding of SHM:
- **Option 1 (Constant period)**: While SHM has a constant period, this is not a necessary condition for SHM.
- **Option 2 (Constant acceleration)**: This is incorrect as the acceleration changes with displacement.
- **Option 3 (Displacement and acceleration are proportional)**: This is the correct option, but it must be noted that they are proportional in opposite directions.
- **Option 4 (Displacement and torque are proportional)**: This is not relevant to the definition of SHM.

**Hint:** Focus on the relationship between displacement and acceleration to identify the correct option.

### Step 4: Conclusion
The necessary condition for simple harmonic motion is that the acceleration is directly proportional to the displacement from the equilibrium position and acts in the opposite direction. Therefore, the correct answer is:
**Displacement and acceleration are proportional (in opposite directions).**

**Final Answer:** The necessary condition for simple harmonic motion is that the acceleration is proportional to the displacement and directed opposite to it.