The equation of a stationary wave in a medium is given as `y=sinomegatcoskx`. The length of a loop in fundamental mode is

Options

A. `(pi)/(2K)`
B. `(pi)/(K)`
C. `(2pi)/(K)`
D. `(K)/(pi)`

Explanation

To find the length of a loop in the fundamental mode of a stationary wave given by the equation \( y = \sin(\omega t) \cos(kx) \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Wave Equation**: The given equation \( y = \sin(\omega t) \cos(kx) \) represents a stationary wave. In this equation, \( \sin(\omega t) \) indicates the time-dependent part, while \( \cos(kx) \) indicates the spatial part of the wave.

2. **Identify the Fundamental Mode**: In the fundamental mode of vibration, the stationary wave pattern consists of one antinode in the center and two nodes at the ends. This means that the wave oscillates between maximum displacement (antinode) and zero displacement (nodes).

3. **Determine the Length of the Loop**: The length of the loop in the fundamental mode corresponds to the distance between two consecutive nodes. In the fundamental mode, this length is equal to half of the wavelength (\( \lambda \)). Therefore, we can express the length of the loop (\( L \)) as:
\[ L = \frac{\lambda}{2} \]

4. **Relate Wavelength to Wave Number**: The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
From this, we can solve for \( \lambda \):
\[ \lambda = \frac{2\pi}{k} \]

5. **Substitute Wavelength into Length of Loop**: Now, substituting the expression for \( \lambda \) back into the equation for \( L \):
\[ L = \frac{1}{2} \left( \frac{2\pi}{k} \right) = \frac{\pi}{k} \]

6. **Final Answer**: Thus, the length of the loop in the fundamental mode is:
\[ L = \frac{\pi}{k} \]

### Summary:
The length of a loop in the fundamental mode of the stationary wave described by the equation \( y = \sin(\omega t) \cos(kx) \) is given by:
\[ L = \frac{\pi}{k} \]