Which of the following quantities is/are always non-negative in a simple harmonic motion along a straight line?

Options

A. `vec(F).vec(a)`
B. `vec(v).vec(r)`
C. `vec(a).vec(r)`
D. `vec(F).vec(r)`

Explanation

To determine which quantities are always non-negative in simple harmonic motion (SHM) along a straight line, we can analyze each option provided in the question.

### Step-by-Step Solution:

1. **Understanding Simple Harmonic Motion (SHM)**:
- In SHM, a particle oscillates about a mean position. The maximum displacement from the mean position is called the amplitude.

2. **Analyzing Option A: \( F \cdot A \)**:
- Here, \( F \) is the force and \( A \) is the acceleration.
- By Newton's second law, \( F = m \cdot a \) (where \( m \) is mass and \( a \) is acceleration).
- The dot product \( F \cdot A \) can be expressed as \( (m \cdot a) \cdot a = m \cdot (a \cdot a) \).
- The dot product of a vector with itself is always non-negative (since it equals the square of the magnitude of the vector).
- Therefore, \( F \cdot A \) is always non-negative.

3. **Analyzing Option B: \( V \cdot R \)**:
- Here, \( V \) is the velocity and \( R \) is the position vector.
- The angle between \( V \) and \( R \) can vary. When the particle moves towards the mean position, the angle between \( V \) and \( R \) is 180 degrees, making \( V \cdot R < 0 \).
- Thus, \( V \cdot R \) is not always non-negative.

4. **Analyzing Option C: \( A \cdot R \)**:
- Here, \( A \) is the acceleration and \( R \) is the position vector.
- The acceleration in SHM is always directed towards the mean position, while the position vector points away from the mean position when the particle is displaced.
- This means the angle between \( A \) and \( R \) is also 180 degrees at maximum displacement, leading to \( A \cdot R < 0 \).
- Therefore, \( A \cdot R \) is not always non-negative.

5. **Analyzing Option D: \( F \cdot R \)**:
- Here, \( F \) is the force and \( R \) is the position vector.
- Since \( F = m \cdot A \), and we have already established that \( A \cdot R \) can be negative, it follows that \( F \cdot R \) can also be negative.
- Thus, \( F \cdot R \) is not always non-negative.

### Conclusion:
After analyzing all options, the only quantity that is always non-negative in simple harmonic motion along a straight line is:

**Answer: Option A: \( F \cdot A \)**