Which one of the following equations of motion represents simple harmonic motion ?

Options

A. Acceleration `= -k_(0)x+k_(1)x^(2)`
B. Acceleration `= -k(x+a)`
C. Acceleration `= k(x+a)`
D. Acceleration `= kx`

Answer

To determine which equation of motion represents simple harmonic motion (SHM), we need to analyze the given equations based on the characteristics of SHM.

### Step-by-Step Solution:

1. **Understanding SHM**:
- Simple Harmonic Motion is defined as motion where the acceleration of the object is directly proportional to its displacement from the mean position and is always directed towards the mean position. Mathematically, this can be expressed as:
\[ a = -k \cdot x \]
where \( a \) is acceleration, \( k \) is a positive constant, and \( x \) is the displacement.

2. **Analyzing the Options**:
- We will analyze each option to see if it conforms to the SHM equation.

3. **Option A**:
- Given: \( a = -k \cdot x^2 \)
- Analysis: This equation has \( x^2 \), which does not conform to the SHM condition (should be proportional to \( x \) and not \( x^2 \)). Therefore, this does not represent SHM.

4. **Option B**:
- Given: \( a = -k \cdot (x + a) \)
- Analysis: This equation can be rewritten as \( a = -k \cdot x - k \cdot a \). The term \( -k \cdot x \) indicates that the acceleration is proportional to the displacement \( x \) and has a negative sign, which is characteristic of SHM. Thus, this represents SHM.

5. **Option C**:
- Given: \( a = k \cdot x + 1 \)
- Analysis: Here, the term \( k \cdot x \) is positive and does not have a negative sign, which means the acceleration is not directed towards the mean position. Therefore, this does not represent SHM.

6. **Option D**:
- Given: \( a = k \cdot x \)
- Analysis: Similar to option C, the absence of a negative sign means that the acceleration is not directed towards the mean position, hence this does not represent SHM.

7. **Conclusion**:
- After analyzing all options, we find that only **Option B** conforms to the characteristics of simple harmonic motion.

### Final Answer:
**The equation that represents simple harmonic motion is Option B.**

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