Question

(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

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

Step-by-step solution

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/m²) 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.

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 \(1.0 \mathrm{W/m}^{2}\).

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 = \(\sqrt{2 \rho v \times I}\)

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 \(25.9 \mathrm{Pa}\).

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 × A where A is the eardrum area.

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 \(0.0142 \mathrm{N}\).