Calculate the bulk modulus of air from the following data about a sound wave of wavelength 35 cm travelling in air. The pressure at a point varies between `(1.0xx 10^5+-14)` Pa and the particles of the air vibrate in simple harmonic motion of amplitude` 5.5 x 10^-5m.

Explanation

To calculate the bulk modulus of air using the provided data about a sound wave, we can follow these steps:

### Step 1: Understand the given data
- Wavelength (λ) = 35 cm = 0.35 m
- Pressure variation (ΔP) = ±14 Pa (the amplitude of pressure variation)
- Amplitude of particle vibration (s₀) = 5.5 x 10⁻⁵ m
- Average pressure (P₀) = 1.0 x 10⁵ Pa

### Step 2: Use the formula for bulk modulus
The bulk modulus (B) can be calculated using the formula:
\[ B = \frac{\Delta P \cdot \lambda}{2 \pi s₀} \]

Where:
- ΔP is the pressure amplitude
- λ is the wavelength
- s₀ is the displacement amplitude

### Step 3: Substitute the known values into the formula
Substituting the values we have:
- ΔP = 14 Pa
- λ = 0.35 m
- s₀ = 5.5 x 10⁻⁵ m

So,
\[ B = \frac{14 \, \text{Pa} \cdot 0.35 \, \text{m}}{2 \pi (5.5 \times 10^{-5} \, \text{m})} \]

### Step 4: Calculate the numerator
Calculate the numerator:
\[ 14 \cdot 0.35 = 4.9 \, \text{Pa m} \]

### Step 5: Calculate the denominator
Calculate the denominator:
\[ 2 \pi (5.5 \times 10^{-5}) \approx 2 \cdot 3.14 \cdot 5.5 \times 10^{-5} \approx 3.46 \times 10^{-4} \, \text{m} \]

### Step 6: Divide the numerator by the denominator
Now, divide the numerator by the denominator to find B:
\[ B = \frac{4.9}{3.46 \times 10^{-4}} \]

### Step 7: Perform the final calculation
Calculating this gives:
\[ B \approx 1.42 \times 10^{5} \, \text{N/m}^2 \]

### Conclusion
Thus, the bulk modulus of air is approximately:
\[ B \approx 1.4 \times 10^{5} \, \text{N/m}^2 \]

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