If `overset(vec)(F)` is force vector, `overset(vec)(v)` is velocity , `overset(vec)(a)` is acceleration vector and `overset(vec)(r)` vector is displacement vector `w.r.t.` mean position than which of the following quantities are always non-negative in a simple harmonic motion along a straight line ?

Options

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

Explanation

To determine which quantities are always non-negative in simple harmonic motion (SHM) along a straight line, we need to analyze the given quantities: force vector (\(\overset{\vec}{F}\)), velocity vector (\(\overset{\vec}{v}\)), acceleration vector (\(\overset{\vec}{a}\)), and displacement vector (\(\overset{\vec}{r}\)) with respect to the mean position.

### Step-by-Step Solution:

1. **Understanding the Relationship in SHM**:
- In SHM, the acceleration (\(\overset{\vec}{a}\)) is proportional to the negative of the displacement (\(\overset{\vec}{r}\)):
\[ \overset{\vec}{a} = -\omega^2 \overset{\vec}{r} \]
- Here, \(\omega\) is the angular frequency.

2. **Analyzing the Acceleration**:
- Since \(\overset{\vec}{a} = -\omega^2 \overset{\vec}{r}\), the acceleration vector is always directed towards the mean position, and its magnitude is proportional to the distance from the mean position.
- The magnitude of acceleration (\(a\)) can be expressed as:
\[ a = \omega^2 r \]
- Thus, \(a^2\) is always non-negative since both \(\omega^2\) and \(r^2\) are non-negative.

3. **Analyzing the Force**:
- The force in SHM is given by:
\[ \overset{\vec}{F} = m \overset{\vec}{a} = -m \omega^2 \overset{\vec}{r} \]
- The force vector is also directed towards the mean position, and its magnitude is:
\[ F = m a = m \omega^2 r \]
- The force is negative when considering the direction towards the mean position, thus it is not always non-negative.

4. **Analyzing the Velocity**:
- The velocity (\(\overset{\vec}{v}\)) can be either positive or negative depending on the direction of motion. Therefore, it is not always non-negative.

5. **Analyzing the Dot Products**:
- The dot product of the acceleration and itself:
\[ \overset{\vec}{a} \cdot \overset{\vec}{a} = a^2 \]
is always non-negative.
- The dot product of the force and acceleration:
\[ \overset{\vec}{F} \cdot \overset{\vec}{a} = m \overset{\vec}{a} \cdot \overset{\vec}{a} = m a^2 \]
is also always non-negative since \(m\) and \(a^2\) are non-negative.

6. **Final Conclusion**:
- The quantities that are always non-negative in simple harmonic motion are:
- \( \overset{\vec}{a} \cdot \overset{\vec}{a} \) (acceleration squared)
- \( \overset{\vec}{F} \cdot \overset{\vec}{a} \) (force dot acceleration)

### Summary of Non-Negative Quantities:
- The quantities that are always non-negative in SHM are:
- \( a^2 \) (acceleration squared)
- \( \overset{\vec}{F} \cdot \overset{\vec}{a} \) (force dot acceleration)