Which of the following is correct about a `SHM`, along a straight line?

Options

A. Ratio of acceleration to velocity is constant.
B. Ratio of acceleration to potential energy is constant.
C. Ratio of acceleration to displacement from the mean position is constant.
D. Ratio of accleration to kinetic energy is constant.

Answer

To solve the question regarding the characteristics of Simple Harmonic Motion (SHM) along a straight line, we will analyze the provided options step by step.

### Step-by-Step Solution:

1. **Understanding SHM**:
- Simple Harmonic Motion is defined as the motion of a particle where the acceleration is always directed towards a fixed point (the mean position) and is proportional to the displacement from that point.

2. **Key Equations in SHM**:
- The velocity \( v \) of a particle in SHM is given by:
\[ v = \pm \omega \sqrt{A^2 - x^2} \]
where \( A \) is the amplitude, \( x \) is the displacement, and \( \omega \) is the angular frequency.
- The acceleration \( a \) is given by:
\[ a = -\omega^2 x \]
- The potential energy \( PE \) is:
\[ PE = \frac{1}{2} k x^2 \]
where \( k \) is the spring constant.
- The kinetic energy \( KE \) is:
\[ KE = \frac{1}{2} k (A^2 - x^2) \]

3. **Analyzing Each Option**:
- **Option A**: Ratio of acceleration to velocity \( \frac{a}{v} \)
\[ \frac{a}{v} = \frac{-\omega^2 x}{\pm \omega \sqrt{A^2 - x^2}} = \frac{-\omega x}{\sqrt{A^2 - x^2}} \]
This ratio depends on \( x \), hence it is not constant. **(Incorrect)**

- **Option B**: Ratio of acceleration to potential energy \( \frac{a}{PE} \)
\[ \frac{a}{PE} = \frac{-\omega^2 x}{\frac{1}{2} k x^2} = \frac{-2\omega^2}{k} \cdot \frac{1}{x} \]
This ratio also depends on \( x \), hence it is not constant. **(Incorrect)**

- **Option C**: Ratio of acceleration to displacement \( \frac{a}{x} \)
\[ \frac{a}{x} = \frac{-\omega^2 x}{x} = -\omega^2 \]
This ratio is constant as it only depends on \( \omega \). **(Correct)**

- **Option D**: Ratio of acceleration to kinetic energy \( \frac{a}{KE} \)
\[ \frac{a}{KE} = \frac{-\omega^2 x}{\frac{1}{2} k (A^2 - x^2)} = \frac{-2\omega^2 x}{k (A^2 - x^2)} \]
This ratio depends on \( x \), hence it is not constant. **(Incorrect)**

4. **Conclusion**:
- The only option that is correct regarding SHM along a straight line is **Option C**: The ratio of acceleration to displacement from the mean position is constant.