Two particles executing SHM of same frequency meet at `x=+(sqrt(3)A)/(2)`, while moving in opposite directions. Phase difference between the particles is :-

Options

A. `(pi)/(6)`
B. `(pi)/(3)`
C. `(5pi)/(6)`
D. `(2pi)/(3)`

Answer

To find the phase difference between two particles executing Simple Harmonic Motion (SHM) and meeting at a specific position while moving in opposite directions, we can follow these steps:

### Step 1: Understand the equations of motion for both particles
Let the first particle be represented by the equation:
\[ x_1 = A \sin(\omega t) \]
And the second particle, moving in the opposite direction, can be represented by:
\[ x_2 = A \sin(\omega t + \phi) \]
where \( \phi \) is the phase difference we need to find.

### Step 2: Set the position where they meet
According to the problem, both particles meet at:
\[ x = \frac{\sqrt{3}A}{2} \]
This means:
\[ A \sin(\omega t) = \frac{\sqrt{3}A}{2} \]
and
\[ A \sin(\omega t + \phi) = \frac{\sqrt{3}A}{2} \]

### Step 3: Simplify the equations
From the first equation:
\[ \sin(\omega t) = \frac{\sqrt{3}}{2} \]
This corresponds to:
\[ \omega t = \frac{\pi}{3} \quad \text{(or 60 degrees)} \]

From the second equation:
\[ \sin(\omega t + \phi) = \frac{\sqrt{3}}{2} \]
This also corresponds to:
\[ \omega t + \phi = \frac{\pi}{3} \quad \text{(or 60 degrees)} \]
or
\[ \omega t + \phi = \frac{2\pi}{3} \quad \text{(or 120 degrees)} \]

### Step 4: Solve for the phase difference
Using the first case:
1. From \( \omega t = \frac{\pi}{3} \):
\[ \phi = \frac{\pi}{3} - \frac{\pi}{3} = 0 \quad \text{(not possible since they move in opposite directions)} \]

Using the second case:
2. From \( \omega t + \phi = \frac{2\pi}{3} \):
\[ \phi = \frac{2\pi}{3} - \frac{\pi}{3} = \frac{\pi}{3} \quad \text{(not possible since they move in opposite directions)} \]

### Step 5: Determine the correct phase difference
Since the particles are moving in opposite directions, the phase difference must be:
\[ \phi = \frac{2\pi}{3} \]
This is because when two particles are in opposite directions, the phase difference is \( \pi \) or \( 2\pi/3 \).

### Conclusion
Thus, the phase difference between the two particles is:
\[ \phi = \frac{2\pi}{3} \]