Two particles P and Q describe S.H.M. of same amplitude a, same frequency f along the same straight line. The maximum distance between the two particles is `a sqrt2` . The initial phase difference between the particle is –

Options

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

Answer

To solve the problem of finding the initial phase difference between two particles P and Q that describe Simple Harmonic Motion (SHM) with the same amplitude and frequency, we can follow these steps:

### Step 1: Understand the Problem
We know that both particles P and Q have the same amplitude \( a \) and frequency \( f \). The maximum distance between the two particles is given as \( a\sqrt{2} \).

### Step 2: Set Up the Equations for SHM
The displacement of the two particles can be expressed as:
- For particle P:
\[ x_P = a \sin(\omega t + \phi_P) \]
- For particle Q:
\[ x_Q = a \sin(\omega t + \phi_Q) \]
Where \( \phi_P \) and \( \phi_Q \) are the initial phases of particles P and Q, respectively.

### Step 3: Calculate the Maximum Distance
The maximum distance \( D \) between the two particles can be given by:
\[ D = |x_P - x_Q| = |a \sin(\omega t + \phi_P) - a \sin(\omega t + \phi_Q)| \]
Using the sine subtraction formula, we can express this as:
\[ D = a |\sin(\omega t + \phi_P) - \sin(\omega t + \phi_Q)| \]

### Step 4: Use the Given Maximum Distance
Given that the maximum distance is \( a\sqrt{2} \), we can set up the equation:
\[ D_{max} = a\sqrt{2} \]
This implies:
\[ |\sin(\omega t + \phi_P) - \sin(\omega t + \phi_Q| = \sqrt{2} \]

### Step 5: Relate the Phase Difference
The maximum value of \( |\sin A - \sin B| \) occurs when \( A \) and \( B \) are \( 90^\circ \) apart. Therefore, we can express the phase difference \( \Delta \phi = \phi_Q - \phi_P \) as:
\[ \Delta \phi = 90^\circ \text{ or } \frac{\pi}{2} \text{ radians} \]

### Step 6: Determine the Initial Phase Difference
Since the initial phase difference is defined as:
\[ \text{Initial Phase Difference} = \phi_Q - \phi_P \]
If we assume \( \phi_P = 0 \), then \( \phi_Q \) would be \( \frac{\pi}{2} \). Thus, the initial phase difference is:
\[ \Delta \phi = \frac{\pi}{2} \text{ radians} \]

### Final Answer
The initial phase difference between the two particles is:
\[ \Delta \phi = \frac{\pi}{2} \text{ radians} \]