In a string wave , all particles of the medium cross the mean position with

Options

A. Different speeds at different instants
B. Different speeds at same instant
C. Same speed at different instants
D. Same speed at same instant

Answer

To solve the question regarding the behavior of particles in a string wave as they cross the mean position, we can break down the explanation into clear steps.

### Step-by-Step Solution:

1. **Understanding the Wave Motion**:
- In a string wave, particles of the medium oscillate around a mean position. This oscillation can be described mathematically using the sine function, which represents simple harmonic motion (SHM).

2. **Equation of the Wave**:
- The displacement of particles can be expressed as:
\[ y = 2a \sin(kx) \sin(\omega t) \]
- Here, \(2a\) represents the amplitude, \(k\) is the wave number, \(x\) is the position along the string, \(\omega\) is the angular frequency, and \(t\) is time.

3. **Crossing the Mean Position**:
- The mean position is where the displacement \(y = 0\). At this point, all particles of the medium cross the mean position simultaneously.

4. **Velocity of Particles**:
- The velocity of a particle in SHM is given by:
\[ v = A \omega \cos(\omega t) \]
- At the mean position, the particles have maximum velocity, which occurs when \(\cos(\omega t) = 1\). Thus, the maximum velocity is:
\[ v_{max} = A \omega \]
- However, at the extremes of motion (maximum displacement), the velocity is zero.

5. **Conclusion on Particle Behavior**:
- All particles cross the mean position at the same time, but their velocities are different depending on their position in the wave. The particles at the mean position have maximum velocity, while those at the nodes (extremes) have zero velocity.

### Final Answer:
All particles of the medium cross the mean position simultaneously, but their speeds are different.

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