Two particle `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 `asqrt(2)` The phase difference between the two particle is

Options

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

Explanation

To solve the problem, we need to find the phase difference between two particles \( P \) and \( Q \) that are performing Simple Harmonic Motion (SHM) with the same amplitude \( a \) and frequency \( f \). The maximum distance between the two particles is given as \( a\sqrt{2} \).

### Step-by-Step Solution:

1. **Define the Position of the Particles**:
The position of particle \( P \) can be expressed as:
\[ x_1 = a \sin(\omega t) \]
The position of particle \( Q \) can be expressed as:
\[ x_2 = a \sin(\omega t + \phi) \]
where \( \phi \) is the phase difference between the two particles.

2. **Calculate the Difference in Position**:
The difference in position between the two particles is:
\[ \Delta x = x_2 - x_1 = a \sin(\omega t + \phi) - a \sin(\omega t) \]

3. **Use the Sine Difference Formula**:
We can use the sine difference identity:
\[ \Delta x = a \left( \sin(\omega t + \phi) - \sin(\omega t) \right) = 2a \sin\left(\frac{\phi}{2}\right) \cos\left(\omega t + \frac{\phi}{2}\right) \]

4. **Determine Maximum Distance**:
The maximum value of \( \Delta x \) occurs when \( \cos\left(\omega t + \frac{\phi}{2}\right) = 1 \):
\[ \Delta x_{\text{max}} = 2a \sin\left(\frac{\phi}{2}\right) \]
We are given that \( \Delta x_{\text{max}} = a\sqrt{2} \).

5. **Set Up the Equation**:
Setting the maximum distance equal to \( a\sqrt{2} \):
\[ 2a \sin\left(\frac{\phi}{2}\right) = a\sqrt{2} \]
Dividing both sides by \( a \) (assuming \( a \neq 0 \)):
\[ 2 \sin\left(\frac{\phi}{2}\right) = \sqrt{2} \]

6. **Solve for \( \sin\left(\frac{\phi}{2}\right) \)**:
Dividing both sides by 2:
\[ \sin\left(\frac{\phi}{2}\right) = \frac{\sqrt{2}}{2} \]

7. **Find \( \frac{\phi}{2} \)**:
The angle whose sine is \( \frac{\sqrt{2}}{2} \) is \( \frac{\pi}{4} \):
\[ \frac{\phi}{2} = \frac{\pi}{4} \]

8. **Calculate \( \phi \)**:
Multiplying both sides by 2 gives:
\[ \phi = \frac{\pi}{2} \]

### Final Answer:
The phase difference \( \phi \) between the two particles is:
\[ \phi = \frac{\pi}{2} \text{ radians} \quad \text{or} \quad 90^\circ \]