In a standing transerse wave on a string :

Options

A. In one time period all the particles are simultaneously at rest twice
B. All the particles must be at their positive extremes simultaneously once in a time period
C. All the particles may be at their positive extremes simultaneously twice in a time period
D. All the particles are never at rest simultaneously

Answer

To solve the question regarding standing transverse waves on a string, we will analyze the behavior of particles in a standing wave over one complete time period.

### Step-by-Step Solution:

1. **Understanding Standing Waves**:
- A standing wave is formed by the superposition of two waves traveling in opposite directions. It consists of nodes (points of no displacement) and antinodes (points of maximum displacement).

2. **Wave Equation**:
- The general equation for a standing transverse wave can be expressed as:
\[ y(x, t) = A \sin(kx) \sin(\omega t) \]
- Here, \(A\) is the amplitude, \(k\) is the wave number, and \(\omega\) is the angular frequency.

3. **Velocity of Particles**:
- The velocity of the particles in the wave can be derived by taking the partial derivative of \(y\) with respect to time \(t\):
\[ v(x, t) = \frac{\partial y}{\partial t} = A \sin(kx) \omega \cos(\omega t) \]
- This simplifies to:
\[ v(x, t) = A \omega \sin(kx) \cos(\omega t) \]

4. **Condition for Particles at Rest**:
- For the particles to be at rest, the velocity \(v\) must equal zero:
\[ A \omega \sin(kx) \cos(\omega t) = 0 \]
- This can occur if either \(\sin(kx) = 0\) (at nodes) or \(\cos(\omega t) = 0\).

5. **Finding Times When Particles are at Rest**:
- The condition \(\cos(\omega t) = 0\) gives:
\[ \omega t = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]
- This corresponds to:
\[ t = \frac{T}{4}, \frac{3T}{4} \quad \text{where } T = \frac{2\pi}{\omega} \]
- Thus, within one complete time period \(T\), the particles are at rest at two distinct times: \(t = \frac{T}{4}\) and \(t = \frac{3T}{4}\).

6. **Analyzing the Options**:
- **Option 1**: All particles are simultaneously at rest twice in one time period. **True**.
- **Option 2**: All particles must be at the positive extreme simultaneously once in a time period. **Not necessarily true**.
- **Option 3**: All particles may be at the positive extreme simultaneously twice in a time period. **True**.
- **Option 4**: All particles are never at rest simultaneously. **False**.

### Conclusion:
The correct answer is **Option 1**: In one time period, all the particles are simultaneously at rest twice.