Two particles execute S.H.M. along the same line at the same frequency. They move in opposite direction at the mean position. The phase difference will be :

Options

A. `2 pi`
B. `2pi//3`
C. `pi`
D. `pi//2`

Answer

To solve the problem of determining the phase difference between two particles executing simple harmonic motion (S.H.M.) in opposite directions at the mean position, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Motion**:
- We have two particles executing S.H.M. along the same line with the same frequency.
- They are moving in opposite directions at the mean position.

2. **Write the Displacement Equations**:
- For the first particle, we can express its displacement as:
\[ x_1 = A_1 \sin(\omega t) \]
- For the second particle, we can express its displacement as:
\[ x_2 = A_2 \sin(\omega t + \phi) \]
- Here, \( \phi \) is the phase difference between the two particles.

3. **Determine the Mean Position**:
- The mean position corresponds to the point where the displacement is zero, i.e., \( x = 0 \).
- For the first particle:
\[ A_1 \sin(\omega t) = 0 \implies \sin(\omega t) = 0 \]
This occurs when \( \omega t = n\pi \) (where \( n \) is an integer).

4. **Evaluate the Second Particle's Displacement**:
- For the second particle at the same time \( t \):
\[ x_2 = A_2 \sin(\omega t + \phi) = A_2 \sin(n\pi + \phi) \]
- Since \( \sin(n\pi) = 0 \), we have:
\[ A_2 \sin(n\pi + \phi) = A_2 \sin(\phi) \]

5. **Condition for Opposite Directions**:
- The problem states that the particles are moving in opposite directions at the mean position. This means that when one particle is at the mean position moving in one direction, the other must be at the mean position moving in the opposite direction.
- Therefore, for the second particle to be moving in the opposite direction, we require:
\[ \sin(\phi) = -1 \]
- This occurs when:
\[ \phi = \frac{3\pi}{2} + 2n\pi \quad (n \text{ is an integer}) \]

6. **Conclusion**:
- The simplest case for the phase difference that satisfies the condition of opposite directions is:
\[ \phi = \pi \]
- Thus, the phase difference between the two particles is \( \pi \) radians.

### Final Answer:
The phase difference will be \( \pi \) radians.

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