The equation y=4cos`((2pix)/(50))sin(100pit)` represents a stationary wave, where x and y are in cm, and 't' is in s. Then a node occurs at a distance of origin

Options

A. 12.5 cm
B. 50 cm
C. 20 cm
D. `100//2pi` cm

Explanation

To solve the problem, we need to analyze the given stationary wave equation and determine where the nodes occur.

### Step-by-Step Solution:

1. **Identify the Wave Equation**:
The given stationary wave equation is:
\[ y = 4 \cos\left(\frac{2\pi x}{50}\right) \sin(100\pi t) \]

2. **Understanding Nodes and Antinodes**:
- A **node** is a point where the displacement is always zero.
- An **antinode** is a point where the displacement is maximum.

3. **Finding Nodes**:
To find the nodes, we need to set the cosine term to zero, since the sine term will oscillate between -1 and 1. The cosine function is zero at:
\[ \cos(kx) = 0 \quad \text{where } k = \frac{2\pi}{\lambda} \]
For our equation, we have:
\[ k = \frac{2\pi}{50} \quad \Rightarrow \quad \lambda = 50 \text{ cm} \]

4. **Setting the Cosine Term to Zero**:
The cosine term is zero when:
\[ \frac{2\pi x}{50} = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]
Simplifying this gives:
\[ x = \frac{50}{2} \left(\frac{1}{2} + n\right) = 25(1 + 2n) \text{ cm} \]

5. **Finding the First Node**:
The first node occurs at \( n = 0 \):
\[ x = 25(1 + 0) = 25 \text{ cm} \]

6. **Conclusion**:
Therefore, a node occurs at a distance of 25 cm from the origin.

### Summary:
The distance from the origin where the first node occurs is **25 cm**.

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