A particle performs SHM along a straight line and its position is `bar("R")` , acceleration is `bar(a)` , velocity is `bar(v)` is and force on particle is `bar(f)`. Then which of the following statements are true?
`hat(v)*hat(a)` is always +ve
`hat(v)*hat("R")` may be -ve
`hat(f)*hat("R")` is always -ve
`hat(v)` is parrallel to `bar(f)` sometimes

Options

A. ii, iii & iv
B. I, ii & iv
C. I, iii , & iv
D. ii & iii

Answer

To solve the problem, we need to analyze the statements regarding a particle performing Simple Harmonic Motion (SHM). We will examine each statement one by one.

### Step 1: Understand the Definitions
In SHM, the following relationships hold:
- The position vector \( \vec{R} \) points from the equilibrium position to the particle's position.
- The velocity vector \( \vec{v} \) is the rate of change of position.
- The acceleration vector \( \vec{a} \) is the rate of change of velocity.
- The force \( \vec{F} \) acting on the particle is proportional to the displacement from the equilibrium position and is directed towards the equilibrium position.

### Step 2: Analyze Each Statement

1. **Statement 1: \( \hat{v} \cdot \hat{a} \) is always positive.**
- In SHM, when the particle is moving towards the mean position, both velocity and acceleration are in the same direction, making the dot product positive. However, when the particle is moving away from the mean position, the velocity is in one direction and the acceleration (which is directed towards the mean position) is in the opposite direction, making the dot product negative.
- **Conclusion:** This statement is **false**.

2. **Statement 2: \( \hat{v} \cdot \hat{R} \) may be negative.**
- The position vector \( \hat{R} \) points away from the mean position. If the particle is moving towards the mean position, the velocity vector \( \hat{v} \) will be directed towards the mean position, which means they are in opposite directions, resulting in a negative dot product.
- **Conclusion:** This statement is **true**.

3. **Statement 3: \( \hat{F} \cdot \hat{R} \) is always negative.**
- The force \( \hat{F} \) in SHM is directed towards the mean position, while the position vector \( \hat{R} \) points away from the mean position. Therefore, the dot product will always be negative as they are in opposite directions.
- **Conclusion:** This statement is **true**.

4. **Statement 4: \( \hat{v} \) is parallel to \( \hat{F} \) sometimes.**
- The force \( \hat{F} \) is directed towards the mean position, while the velocity \( \hat{v} \) can be directed either towards or away from the mean position. Thus, there are instances when the velocity and force vectors can be parallel (when the particle is moving towards the mean position).
- **Conclusion:** This statement is **true**.

### Final Conclusion
- The true statements are: **2, 3, and 4**.
- The false statement is: **1**.