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

Options

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

Answer

To solve the question of which quantities are always negative in simple harmonic motion (SHM), we need to analyze the given options based on the definitions and relationships of the vectors involved.

### Step-by-Step Solution

1. **Understanding the Vectors**:
- **F vector (Force)**: The force acting on the particle in SHM.
- **a vector (Acceleration)**: The acceleration of the particle in SHM.
- **r vector (Position Vector)**: The position of the particle relative to the mean position.
- **v vector (Velocity)**: The velocity of the particle in SHM.

2. **Analyzing the Relationships**:
- In SHM, the acceleration \( a \) is given by the formula:
\[ a = -\omega^2 x \]
This indicates that acceleration is always directed towards the mean position and is negative when the particle is on the opposite side of the mean position.

3. **Dot Product Analysis**:
- The dot product of two vectors \( A \) and \( B \) is given by:
\[ A \cdot B = |A||B| \cos \theta \]
where \( \theta \) is the angle between the two vectors. The dot product is negative when \( \theta > 90^\circ \).

4. **Evaluating Each Option**:
- **Option 1: \( F \cdot a \)**:
- Since \( F = ma \) and both force and acceleration are in the same direction, their dot product is always positive. Hence, this option is incorrect.

- **Option 2: \( V \cdot r \)**:
- The velocity \( v \) can be in the same or opposite direction to the position vector \( r \) depending on the position of the particle. Thus, this dot product is not always negative. Hence, this option is incorrect.

- **Option 3: \( a \cdot r \)**:
- The acceleration \( a \) is always directed towards the mean position while the position vector \( r \) points away from the mean position. Therefore, the angle between \( a \) and \( r \) is \( 180^\circ \) when \( r \) is positive, making \( a \cdot r \) negative. Hence, this option is correct.

- **Option 4: \( F \cdot r \)**:
- Similar to \( a \cdot r \), since \( F \) is directed towards the mean position and \( r \) points away from it, the angle between them is also \( 180^\circ \), making \( F \cdot r \) negative. Hence, this option is correct.

5. **Conclusion**:
- The quantities that are always negative in simple harmonic motion are:
- \( a \cdot r \)
- \( F \cdot r \)

### Final Answer:
The correct options are **Option 3: \( a \cdot r \)** and **Option 4: \( F \cdot r \)**.

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