A steel wire of length `1m`, mass `0.1kg` and uniform cross-sectional area `10^(-6)m^(2)` is rigidly fixed at both ends. The temperature of the wire is lowered by `20^(@)C`. If transverse waves are set up by plucking the string in the middle.Calculate the frequency of the fundamental mode of vibration.
Given for steel `Y = 2 xx 10^(11)N//m^(2)`
`alpha = 1.21 xx 10^(-5) per ^(@)C`

Explanation

To calculate the frequency of the fundamental mode of vibration of a steel wire that is fixed at both ends and has undergone a temperature change, we can follow these steps:

### Step 1: Calculate the mass per unit length (μ) of the wire
The mass per unit length (μ) is given by the formula:
\[ \mu = \frac{m}{L} \]
where:
- \( m = 0.1 \, \text{kg} \) (mass of the wire)
- \( L = 1 \, \text{m} \) (length of the wire)

Substituting the values:
\[ \mu = \frac{0.1 \, \text{kg}}{1 \, \text{m}} = 0.1 \, \text{kg/m} \]

### Step 2: Calculate the change in tension (T) in the wire due to temperature change
The tension in the wire can be calculated using Young's modulus (Y) and the change in length due to temperature change. The formula for tension is:
\[ T = Y \cdot \text{strain} \]
The strain can be calculated as:
\[ \text{strain} = \alpha \cdot \Delta T \]
where:
- \( \alpha = 1.21 \times 10^{-5} \, \text{per } ^\circ C \) (coefficient of linear expansion)
- \( \Delta T = -20 \, ^\circ C \) (change in temperature)

Calculating strain:
\[ \text{strain} = 1.21 \times 10^{-5} \cdot (-20) = -2.42 \times 10^{-4} \]

Now substituting into the tension formula:
\[ T = Y \cdot \text{strain} = (2 \times 10^{11} \, \text{N/m}^2) \cdot (-2.42 \times 10^{-4}) = -4.84 \times 10^{7} \, \text{N} \]
(Note: The negative sign indicates a decrease in length, but tension is taken as a positive quantity.)

### Step 3: Calculate the velocity (v) of the wave in the wire
The velocity of the wave is given by:
\[ v = \sqrt{\frac{T}{\mu}} \]
Substituting the values of T and μ:
\[ v = \sqrt{\frac{4.84 \times 10^{7} \, \text{N}}{0.1 \, \text{kg/m}}} = \sqrt{4.84 \times 10^{8}} \approx 22040.3 \, \text{m/s} \]

### Step 4: Calculate the frequency (f) of the fundamental mode of vibration
The frequency of the fundamental mode of vibration is given by:
\[ f = \frac{v}{2L} \]
Substituting the values:
\[ f = \frac{22040.3 \, \text{m/s}}{2 \cdot 1 \, \text{m}} = \frac{22040.3}{2} \approx 11020.15 \, \text{Hz} \]

### Final Result
The frequency of the fundamental mode of vibration is approximately:
\[ f \approx 11020.15 \, \text{Hz} \]