The equation of stationary wave along a stretched string is given by `y=5 sin (pix)/(3) cos 40 pi t `, where x and y are in cm and t in second. The separation between two adjacent nodes is

Options

A. `1.5 cm`
B. 3 cm
C. 6 cm
D. 4 cm

Answer

To find the separation between two adjacent nodes in the stationary wave described by the equation \( y = 5 \sin\left(\frac{\pi x}{3}\right) \cos(40 \pi t) \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the wave equation**: The given equation is \( y = 5 \sin\left(\frac{\pi x}{3}\right) \cos(40 \pi t) \).

2. **Understand the structure of the wave equation**: The general form of a stationary wave can be expressed as:
\[ y = 2A \sin(kx) \cos(\omega t) \]
where \( A \) is the amplitude, \( k \) is the wave number, and \( \omega \) is the angular frequency.

3. **Extract the wave number \( k \)**: From the equation, we can see that:
\[ k = \frac{\pi}{3} \]

4. **Relate wave number to wavelength**: The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Therefore, we can set up the equation:
\[ \frac{\pi}{3} = \frac{2\pi}{\lambda} \]

5. **Solve for the wavelength \( \lambda \)**: Rearranging the equation gives:
\[ \lambda = \frac{2\pi}{\frac{\pi}{3}} = 2 \times 3 = 6 \text{ cm} \]

6. **Calculate the distance between adjacent nodes**: The distance between two adjacent nodes is half the wavelength:
\[ \text{Distance between nodes} = \frac{\lambda}{2} = \frac{6}{2} = 3 \text{ cm} \]

### Final Answer:
The separation between two adjacent nodes is **3 cm**.

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