A piano string `1.5 m` long is made of steel of density `7.7 xx 10^(3) kg//m^(3) and gamma = 2 xx 10^(11) N//m^(2)`. It is maintained at a tension which produces an elastic strain of `1 %` in the string . What is the fundamental frequency of transverse vibration of the string ?

Options

A. 256 Hz
B. 178Hz
C. 170Hz
D. 200Hz.

Explanation

To find the fundamental frequency of transverse vibration of the piano string, we can follow these steps:

### Step 1: Identify the given data
- Length of the string, \( L = 1.5 \, \text{m} \)
- Density of steel, \( \rho = 7.7 \times 10^{3} \, \text{kg/m}^{3} \)
- Young's modulus, \( Y = 2 \times 10^{11} \, \text{N/m}^{2} \)
- Elastic strain, \( \epsilon = 1\% = 0.01 \)

### Step 2: Calculate the stress in the string
Stress (\( \sigma \)) can be calculated using the formula:
\[ \sigma = Y \cdot \epsilon \]
Substituting the values:
\[ \sigma = (2 \times 10^{11} \, \text{N/m}^{2}) \cdot (0.01) = 2 \times 10^{9} \, \text{N/m}^{2} \]

### Step 3: Calculate the velocity of the wave in the string
The velocity (\( v \)) of the wave in the string can be calculated using the formula:
\[ v = \sqrt{\frac{\sigma}{\rho}} \]
Substituting the values:
\[ v = \sqrt{\frac{2 \times 10^{9} \, \text{N/m}^{2}}{7.7 \times 10^{3} \, \text{kg/m}^{3}}} \]
Calculating the above expression:
\[ v = \sqrt{259.74 \times 10^{3}} \approx 509.7 \, \text{m/s} \]

### Step 4: Calculate the wavelength for the fundamental frequency
For the fundamental frequency of a string fixed at both ends, the wavelength (\( \lambda \)) is given by:
\[ \lambda = 2L \]
Substituting the length of the string:
\[ \lambda = 2 \times 1.5 \, \text{m} = 3 \, \text{m} \]

### Step 5: Calculate the fundamental frequency
The fundamental frequency (\( f \)) can be calculated using the formula:
\[ f = \frac{v}{\lambda} \]
Substituting the values:
\[ f = \frac{509.7 \, \text{m/s}}{3 \, \text{m}} \approx 169.9 \, \text{Hz} \]

### Final Answer
The fundamental frequency of transverse vibration of the string is approximately \( 170 \, \text{Hz} \).

---