Which of the following quantities are always positive in a simple harmonic motion?

Options

A. `vecF.veca.`
B. `vecv.vecr`
C. `veca.vecr`
D. `vecF.vecr`

Answer

To determine which quantities are always positive in simple harmonic motion (SHM), we will analyze each option provided in the question.

### Step-by-Step Solution:

1. **Understanding the quantities involved**:
- We have four options to evaluate:
1. \( \mathbf{f} \cdot \mathbf{a} \)
2. \( \mathbf{v} \cdot \mathbf{r} \)
3. \( \mathbf{a} \cdot \mathbf{r} \)
4. \( \mathbf{f} \cdot \mathbf{r} \)

2. **Analyzing Option 1: \( \mathbf{f} \cdot \mathbf{a} \)**:
- The force \( \mathbf{f} \) acting on a particle in SHM is given by \( \mathbf{f} = m \mathbf{a} \), where \( m \) is the mass (always positive) and \( \mathbf{a} \) is the acceleration.
- Since both \( \mathbf{f} \) and \( \mathbf{a} \) are in the same direction (towards the mean position), the dot product \( \mathbf{f} \cdot \mathbf{a} \) is always positive.
- **Conclusion**: This option is correct.

3. **Analyzing Option 2: \( \mathbf{v} \cdot \mathbf{r} \)**:
- The velocity \( \mathbf{v} \) can change direction as the particle moves back and forth in SHM.
- When the particle is moving towards the mean position, \( \mathbf{v} \) and \( \mathbf{r} \) (position vector) are in opposite directions, leading to a negative dot product.
- **Conclusion**: This option is not always positive.

4. **Analyzing Option 3: \( \mathbf{a} \cdot \mathbf{r} \)**:
- The acceleration \( \mathbf{a} \) in SHM is given by \( \mathbf{a} = -\omega^2 \mathbf{x} \), which means it always points towards the mean position (opposite to the position vector \( \mathbf{r} \)).
- Therefore, the dot product \( \mathbf{a} \cdot \mathbf{r} \) is always negative.
- **Conclusion**: This option is not correct.

5. **Analyzing Option 4: \( \mathbf{f} \cdot \mathbf{r} \)**:
- Since \( \mathbf{f} \) is proportional to \( \mathbf{a} \) (both point towards the mean position), and \( \mathbf{a} \cdot \mathbf{r} \) is negative, it follows that \( \mathbf{f} \cdot \mathbf{r} \) will also be negative.
- **Conclusion**: This option is not correct.

### Final Answer:
The only quantity that is always positive in simple harmonic motion is \( \mathbf{f} \cdot \mathbf{a} \).