Two particles are executing simple harmonic of the same amplitude (A) and frequency `omega` along the x-axis . Their mean position is separated by distance `Xo. (Xo>A). If the maximum separation between them is (Xo+A), the phase difference between their motion is:

Options

A. `pi//2`
B. `pi//3`
C. `pi//4`
D. `pi//6`

Explanation

To solve the problem, we need to determine the phase difference between two particles executing simple harmonic motion (SHM) under the given conditions. Let's break down the solution step by step.

### Step 1: Understand the Problem
We have two particles executing SHM with the same amplitude \( A \) and frequency \( \omega \). Their mean positions are separated by a distance \( X_0 \) where \( X_0 > A \). The maximum separation between the two particles is given as \( X_0 + A \).

### Step 2: Analyze the Maximum Separation
The maximum separation between the two particles occurs when they are at their extreme positions. When one particle is at its maximum positive displacement \( +A \), the other must be at its maximum negative displacement \( -A \).

### Step 3: Set Up the Equation
Given that the mean positions of the two particles are separated by \( X_0 \), we can express the positions of the two particles as follows:
- Particle 1: \( x_1 = A \cos(\omega t) \)
- Particle 2: \( x_2 = X_0 + A \cos(\omega t + \phi) \)

Where \( \phi \) is the phase difference we need to find.

### Step 4: Determine the Maximum Separation Condition
The maximum separation between the two particles can be expressed as:
\[ \text{Maximum Separation} = |x_2 - x_1| = |(X_0 + A \cos(\omega t + \phi)) - A \cos(\omega t)| \]
This simplifies to:
\[ |X_0 + A \cos(\omega t + \phi) - A \cos(\omega t)| \]
At maximum separation, we have:
\[ X_0 + A = |X_0 + A \cos(\omega t + \phi) - A \cos(\omega t)| \]

### Step 5: Analyze the Condition
For maximum separation \( X_0 + A \), the two particles must be in opposite phases when they reach their extreme positions:
\[ X_0 + A = X_0 + A + A \Rightarrow 0 = 2A \cos(\phi) \]
This implies that \( \cos(\phi) = -\frac{1}{2} \).

### Step 6: Find the Phase Difference
The value of \( \phi \) for which \( \cos(\phi) = -\frac{1}{2} \) corresponds to:
\[ \phi = \frac{2\pi}{3} \text{ or } \phi = \frac{4\pi}{3} \]
However, since we are interested in the phase difference, we can take:
\[ \phi = \frac{2\pi}{3} \text{ (or equivalently } 120^\circ\text{)} \]

### Conclusion
The phase difference between the two particles is:
\[ \phi = \frac{2\pi}{3} \text{ radians} \text{ or } 120^\circ \]