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Chapter 12: Problem 15

(a) What is the pressure amplitude of a sound wave with an intensity level of \(120.0 \mathrm{dB}\) in air? (b) What force does this exert on an eardrum of area \(0.550 \times 10^{-4} \mathrm{m}^{2} ?\)

Short Answer

Expert verified
Answer: The force exerted on the eardrum is \(0.0142 \mathrm{N}\).

Step by step solution

01

Convert intensity level from dB to watts per square meter

The formula to convert intensity level in decibels (dB) to intensity (I) in watts per square meter (W/m虏) is: I = \(I_{0} \times 10^{\frac{\beta}{10}}\) where \(I_{0}\) is the threshold of hearing (\(10^{-12} \mathrm{W/m}^{2}\)) and \(\beta\) is the intensity level in decibels. We are given that the intensity level of the sound wave is \(120.0 \mathrm{dB}\). Using this information, we can find the intensity (I) of the sound wave.
02

Calculate the intensity (I) of the sound wave

Using the formula mentioned in step 1: I = \((10^{-12} \mathrm{W/m}^{2}) \times 10^{\frac{120}{10}}\) I = \((10^{-12} \mathrm{W/m}^{2}) \times 10^{12}\) I = \(1.0 \mathrm{W/m}^{2}\) So, the intensity of the sound wave is \(1.0 \mathrm{W/m}^{2}\).
03

Calculate the pressure amplitude (P) of the sound wave

The relationship between intensity (I), pressure amplitude (P), and the speed of sound (v) in the medium can be given as: I = \(\frac{P^2}{2 \rho v}\) In air, the density (\(\rho\)) is approximately \(1.20 \mathrm{kg/m}^3\) and the speed of sound (v) is approximately \(340 \mathrm{m/s}\). We can use these values and the intensity (I) found in step 2 to find the pressure amplitude (P). Rearranging the formula for P: P = \(\sqrt{2 \rho v \times I}\)
04

Calculate the pressure amplitude (P) of the sound wave using the given intensity and air properties

Using the values for density, speed of sound, and intensity: P = \(\sqrt{2 \times 1.20 \mathrm{kg/m}^3 \times 340 \mathrm{m/s} \times 1.0 \mathrm{W/m}^2}\) P = \(25.9 \mathrm{Pa}\) So, the pressure amplitude of the sound wave is \(25.9 \mathrm{Pa}\).
05

Calculate the force exerted on the eardrum

We are given that the area of the eardrum is \(0.550 \times 10^{-4} \mathrm{m}^{2}\). Using the pressure amplitude (P) found in step 4, we can find the force exerted on the eardrum (F) using the formula: F = P 脳 A where A is the eardrum area.
06

Calculate the force exerted on the eardrum using the pressure amplitude and eardrum area

Using the values for pressure and eardrum area: F = \(25.9 \mathrm{Pa} \times 0.550 \times 10^{-4} \mathrm{m}^{2}\) F = \(0.0142 \mathrm{N}\) So, the force exerted on the eardrum is \(0.0142 \mathrm{N}\).

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