To find the amplitude of vibration of the particles in the medium when a sound wave travels through air, we can use the relationship between pressure variation, bulk modulus, and amplitude. Let's break down the steps involved in solving this problem.
Key Relationships
The amplitude of a sound wave is related to the pressure variation in the medium. The maximum pressure difference (ΔP) is given by the equation:
ΔP = B * (Δx / λ)
Where:
- ΔP is the difference between maximum and minimum pressures (1.0 x 10-3 N/m2 in this case),
- B is the bulk modulus of the medium (1.4 x 105 N/m2 for air),
- Δx is the amplitude of vibration we are trying to find, and
- λ is the wavelength of the sound wave (40 cm or 0.4 m).
Rearranging the Formula
We need to rearrange the equation to isolate Δx:
Δx = ΔP * λ / B
Substituting Values
Now we can substitute the known values into the rearranged equation:
Δx = (1.0 x 10-3 N/m2) * (0.4 m) / (1.4 x 105 N/m2)
Calculating Each Step
Let's perform the calculations step by step:
1. **Multiply ΔP and λ**:
ΔP * λ = (1.0 x 10-3) * (0.4) = 0.0004 N·m/m2
2. **Now divide by B**:
Δx = 0.0004 / (1.4 x 105) = 0.0004 / 140000 = 2.857 x 10-9 m
Final Result
To express this in more understandable units, convert meters to centimeters (1 m = 100 cm):
Δx ≈ 2.857 x 10-7 cm
Thus, the amplitude of vibration of the particles of air, when a sound wave of 40 cm wavelength produces a pressure difference of 1.0 x 10-3 N/m2, is approximately 2.857 x 10-7 cm. This value represents the maximum displacement of air molecules due to the sound wave.
