Two particles undergo SHM along parallel lines with the same time period (T) and equal amplitudes. At particular instant, one particle is at its extreme position while the other is at its mean position. The move in the same direction. They will cross each other after a further time.

Options

A. `T//2`
B. `3T//8`
C. `T//6`
D. `3T//4`

Answer

To solve the problem step by step, we will analyze the motion of the two particles undergoing Simple Harmonic Motion (SHM) and determine the time it takes for them to cross each other.

### Step-by-Step Solution:

1. **Understanding the Initial Positions**:
- Let Particle A be at its mean position (x = 0) and moving towards the positive amplitude.
- Let Particle B be at its extreme position (x = A) and also moving towards the positive amplitude.

2. **Identifying the Motion**:
- Both particles have the same time period \( T \) and equal amplitudes \( A \).
- Since they are moving in the same direction, we need to find out when they will cross each other.

3. **Using SHM Properties**:
- In SHM, the displacement of a particle can be described by the equation:
\[ x(t) = A \sin\left(\frac{2\pi}{T} t + \phi\right) \]
- For Particle A at the mean position, we can assume its phase \( \phi_A = 0 \):
\[ x_A(t) = A \sin\left(\frac{2\pi}{T} t\right) \]
- For Particle B at the extreme position, we can assume its phase \( \phi_B = \frac{\pi}{2} \):
\[ x_B(t) = A \sin\left(\frac{2\pi}{T} t + \frac{\pi}{2}\right) = A \cos\left(\frac{2\pi}{T} t\right) \]

4. **Setting Up the Crossing Condition**:
- The particles will cross each other when \( x_A(t) = x_B(t) \):
\[ A \sin\left(\frac{2\pi}{T} t\right) = A \cos\left(\frac{2\pi}{T} t\right) \]
- Dividing both sides by \( A \) (assuming \( A \neq 0 \)):
\[ \sin\left(\frac{2\pi}{T} t\right) = \cos\left(\frac{2\pi}{T} t\right) \]

5. **Using Trigonometric Identity**:
- The equation \( \sin(x) = \cos(x) \) can be rewritten as:
\[ \tan\left(\frac{2\pi}{T} t\right) = 1 \]
- This implies:
\[ \frac{2\pi}{T} t = \frac{\pi}{4} + n\pi \quad (n \in \mathbb{Z}) \]

6. **Finding the Time for the First Crossing**:
- For the first crossing (taking \( n = 0 \)):
\[ \frac{2\pi}{T} t = \frac{\pi}{4} \]
- Solving for \( t \):
\[ t = \frac{T}{8} \]

7. **Conclusion**:
- The two particles will cross each other after a time of \( \frac{T}{8} \).

### Final Answer:
The two particles will cross each other after a time of \( \frac{T}{8} \).

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