Which one of the following equations of motion represent 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`

Explanation

To determine which of the given equations of motion represent simple harmonic motion (SHM), we need to analyze the characteristics of SHM and the provided options.

### Step-by-Step Solution:

1. **Understanding Simple Harmonic Motion (SHM)**:
- In SHM, the restoring force is proportional to the displacement from the mean position and acts in the opposite direction. Mathematically, this is represented as:
\[ F = -kx \]
- According to Newton's second law, force can also be expressed as:
\[ F = ma \]
- By equating these two expressions, we have:
\[ ma = -kx \]
- Rearranging gives:
\[ a = -\frac{k}{m}x \]
- This shows that acceleration (a) is directly proportional to the displacement (x) but in the opposite direction, which is a defining characteristic of SHM.

2. **Identifying the Form of the Equation**:
- We can express the relationship as:
\[ a = -px \]
- Here, \( p = \frac{k}{m} \) is a positive constant.

3. **Analyzing the Given Options**:
- We need to check each option to see if it fits the form \( a = -px \).

**Option 1**: \( a = kx \)
- This indicates that acceleration and displacement are in the same direction. This does not represent SHM.

**Option 2**: \( a = -kx + a \)
- This can be rewritten as \( a = -kx + a \), which indicates a shift in the mean position. The term \( -kx \) still represents a restoring force, which is characteristic of SHM. Thus, this option represents SHM.

**Option 3**: \( a = -k_0x + k_1x^2 \)
- The presence of the \( k_1x^2 \) term indicates that acceleration depends on the square of the displacement, which is not characteristic of SHM. Hence, this does not represent SHM.

**Option 4**: \( a = kx \)
- Similar to Option 1, this indicates that acceleration is in the same direction as displacement, which does not represent SHM.

4. **Conclusion**:
- The only equation that represents simple harmonic motion is **Option 2**: \( a = -kx + a \).

### Final Answer:
The equation that represents simple harmonic motion is **Option 2**: \( a = -kx + a \).