Two particles execute SHM of the same time period along the same straight lines. They cross each other at the mean position while going in opposite directions. Their phase difference is

Options

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

Explanation

To solve the problem of finding the phase difference between two particles executing simple harmonic motion (SHM) that cross each other at the mean position while moving in opposite directions, we can follow these steps:

### Step 1: Understand the SHM Characteristics
Both particles have the same time period and are moving along the same straight line. In SHM, the position of a particle can be described by the equation:
\[ x(t) = A \sin(\omega t + \phi) \]
where \( A \) is the amplitude, \( \omega \) is the angular frequency, \( t \) is time, and \( \phi \) is the phase constant.

### Step 2: Analyze the Crossing at the Mean Position
The mean position in SHM corresponds to the position \( x = 0 \). When the particles cross each other at this position, they are moving in opposite directions. This indicates that one particle is moving towards the mean position while the other is moving away from it.

### Step 3: Determine the Phase of Each Particle
1. **Particle 1**: Let’s assume Particle 1 is at the mean position and moving upwards (towards the positive amplitude). The phase angle for this position can be represented as:
\[ \phi_1 = \frac{\pi}{2} \text{ (90 degrees)} \]
This is because at \( \phi = \frac{\pi}{2} \), the sine function reaches its maximum value.

2. **Particle 2**: Since Particle 2 is also at the mean position but moving downwards (towards the negative amplitude), its phase angle can be represented as:
\[ \phi_2 = \frac{3\pi}{2} \text{ (270 degrees)} \]
At this phase, the sine function is at zero but moving in the negative direction.

### Step 4: Calculate the Phase Difference
The phase difference \( \Delta \phi \) between the two particles can be calculated as:
\[ \Delta \phi = \phi_2 - \phi_1 = \frac{3\pi}{2} - \frac{\pi}{2} = \pi \]
This indicates that the phase difference is \( \pi \) radians, or 180 degrees.

### Conclusion
The phase difference between the two particles executing SHM and crossing each other at the mean position while moving in opposite directions is:
\[ \Delta \phi = \pi \text{ radians (or 180 degrees)} \]