A string is rigidly tied at two ends and its equation of vibration is given by `y=cos 2pix.` Then minimum length of string is

Options

A. 1m
B. `1/2 M`
C. 5M
D. `2 pi M`

Answer

To find the minimum length of a string that is rigidly tied at both ends and has a vibration equation given by \( y = \cos(2\pi x) \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Wave Equation**: The given equation of vibration is \( y = \cos(2\pi x) \). This is in the form of a wave equation \( y = \cos(kx) \), where \( k \) is the wave number.

2. **Determine the Wave Number \( k \)**: From the equation \( y = \cos(2\pi x) \), we can see that \( k = 2\pi \).

3. **Relate Wave Number to Wavelength**: The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Substituting \( k = 2\pi \) into the equation gives:
\[ 2\pi = \frac{2\pi}{\lambda} \]

4. **Solve for Wavelength \( \lambda \)**: Rearranging the equation to solve for \( \lambda \):
\[ \lambda = 1 \text{ meter} \]

5. **Determine Minimum Length of the String**: The minimum length of the string that can vibrate in a fundamental mode (first harmonic) is half of the wavelength:
\[ L = \frac{\lambda}{2} = \frac{1}{2} \text{ meter} = 0.5 \text{ meter} \]

### Final Answer:
The minimum length of the string is \( 0.5 \) meters.

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