At STP sound waves of wavelength 2 m are produced in air by a tuning fork of frequency 200 Hz. What will be increase of wavelength at a temperature of `27^(@)C` ?

Answer

To solve the problem, we need to find the increase in wavelength of sound waves when the temperature changes from 0°C (STP) to 27°C. Here’s a step-by-step solution:

### Step 1: Understand the given data
- Wavelength at STP (λ₀) = 2 m
- Frequency (ν) = 200 Hz
- Temperature at STP = 0°C = 273 K
- Temperature at 27°C = 27°C = 300 K

### Step 2: Calculate the speed of sound at STP
The speed of sound (V) can be calculated using the formula:
\[ V = ν \cdot λ \]
Substituting the values:
\[ V = 200 \, \text{Hz} \times 2 \, \text{m} = 400 \, \text{m/s} \]

### Step 3: Use the relationship of speed of sound with temperature
The speed of sound in air is related to temperature by the formula:
\[ V \propto \sqrt{T} \]
Thus, we can write:
\[ \frac{V_{27}}{V_{0}} = \sqrt{\frac{T_{27}}{T_{0}}} \]
Where:
- \( T_{27} = 300 \, \text{K} \)
- \( T_{0} = 273 \, \text{K} \)

### Step 4: Calculate the speed of sound at 27°C
Using the relationship:
\[ V_{27} = V_{0} \cdot \sqrt{\frac{T_{27}}{T_{0}}} \]
Substituting the values:
\[ V_{27} = 400 \, \text{m/s} \cdot \sqrt{\frac{300}{273}} \]

Calculating the square root:
\[ \sqrt{\frac{300}{273}} \approx 1.0607 \]
Thus,
\[ V_{27} \approx 400 \, \text{m/s} \cdot 1.0607 \approx 424.28 \, \text{m/s} \]

### Step 5: Calculate the new wavelength at 27°C
Using the formula for wavelength:
\[ λ_{27} = \frac{V_{27}}{ν} \]
Substituting the values:
\[ λ_{27} = \frac{424.28 \, \text{m/s}}{200 \, \text{Hz}} \approx 2.1214 \, \text{m} \]

### Step 6: Calculate the increase in wavelength
The increase in wavelength (Δλ) is given by:
\[ Δλ = λ_{27} - λ_{0} \]
Substituting the values:
\[ Δλ = 2.1214 \, \text{m} - 2 \, \text{m} \approx 0.1214 \, \text{m} \]

### Final Answer
The increase in wavelength at a temperature of 27°C is approximately **0.1214 meters**.

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