Which of the following quantities is always negative is SHM ?
Here, x is displacement , from mean position.

Options

A. `F.a`
B. `v.s`
C. `a.s`
D. `F.v`

Explanation

To solve the question of which quantity is always negative in Simple Harmonic Motion (SHM), we need to analyze the options provided in the context of SHM.

### Step-by-Step Solution:

1. **Understanding SHM**: In SHM, the displacement \( x \) from the mean position can be expressed as:
\[ x = A \sin(\omega t) \]
where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( t \) is time.

2. **Identifying the Forces**: The restoring force \( F \) acting on the mass in SHM is given by Hooke's Law:
\[ F = -kx \]
This indicates that the force is always directed towards the mean position and is negative when \( x \) is positive (i.e., when the mass is displaced to the right).

3. **Analyzing Velocity and Displacement**:
- The velocity \( v \) can be derived from the displacement:
\[ v = \frac{dx}{dt} = \omega A \cos(\omega t) \]
The angle between velocity and displacement can vary, which means \( v \cdot x \) can be positive or negative.

4. **Analyzing Acceleration**:
- The acceleration \( a \) is given by:
\[ a = \frac{d^2x}{dt^2} = -\omega^2 A \sin(\omega t) \]
The acceleration is always directed towards the mean position, and it is negative when \( x \) is positive.

5. **Dot Products**:
- **Force and Acceleration**: Since both force and acceleration are directed towards the mean position, their dot product \( F \cdot a \) is positive.
- **Velocity and Displacement**: As discussed, \( v \cdot x \) can be either positive or negative.
- **Acceleration and Displacement**: The dot product \( a \cdot x \) will always be negative because they are always in opposite directions (acceleration is directed towards the mean position while displacement is away from it).

6. **Conclusion**: The only quantity that is always negative in SHM is the dot product of acceleration and displacement, \( a \cdot x \).

### Final Answer:
The quantity that is always negative in SHM is \( a \cdot x \).