The paritcle displacement (in cm) in a stationary wave is given by `y(x,t)=2sin(0.1pix)cos(100pit)`. The distance between a node and the next antinode is

Options

A. `2.5cm`
B. `7.5cm`
C. `5 cm`
D. `10cm`

Explanation

To solve the problem, we need to find the distance between a node and the next antinode in the given stationary wave described by the equation:

\[ y(x,t) = 2 \sin(0.1 \pi x) \cos(100 \pi t) \]

### Step-by-Step Solution:

1. **Identify the Wave Equation**:
The given equation is in the form of a stationary wave, which can be expressed as:
\[ y(x,t) = A \sin(kx) \cos(\omega t) \]
Here, \( A = 2 \), \( k = 0.1 \pi \), and \( \omega = 100 \pi \).

2. **Determine the Wavelength (\( \lambda \))**:
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
We can rearrange this to find \( \lambda \):
\[ \lambda = \frac{2\pi}{k} \]
Substituting \( k = 0.1 \pi \):
\[ \lambda = \frac{2\pi}{0.1 \pi} = \frac{2}{0.1} = 20 \text{ cm} \]

3. **Calculate the Distance Between a Node and the Next Antinode**:
In a stationary wave, the distance between one node and the next antinode is given by:
\[ \text{Distance} = \frac{\lambda}{4} \]
Substituting \( \lambda = 20 \text{ cm} \):
\[ \text{Distance} = \frac{20 \text{ cm}}{4} = 5 \text{ cm} \]

### Final Answer:
The distance between a node and the next antinode is **5 cm**.

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