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Chapter 12: Problem 11

A sound wave with an intensity level of \(80.0 \mathrm{dB}\) is incident on an eardrum of area \(0.600 \times 10^{-4} \mathrm{m}^{2} .\) How much energy is absorbed by the eardrum in 3.0 min?

Short Answer

Expert verified
Answer: The eardrum absorbs 108 x 10鈦烩伕 Joules of energy in 3 minutes.

Step by step solution

01

Convert the intensity level to intensity

Given the intensity level (L) of the sound wave is \(80.0 \mathrm{dB}\). To convert this value to intensity (I) in watts per square meter, we use the decibel formula: $$L = 10 \cdot \log_{10}\left(\frac{I}{I_0}\right)$$ where \(I_0=10^{-12} \mathrm{W/m^2}\) is the reference intensity. We need to solve this equation for I: $$I = I_0 \cdot 10 ^{\frac{L}{10}}$$ Now, substituting the given values: $$I = 10^{-12} \cdot 10 ^{\frac{80}{10}}$$
02

calculate the intensity

Performing the calculation of intensity: $$I = 10^{-12} \cdot 10^{8} = 10^{-4} \mathrm{W/m^2}$$
03

Find the power absorbed by the eardrum

The power absorbed by the eardrum (P) can be calculated by multiplying the intensity (I) by the area of the eardrum (A): $$P = I \cdot A$$ Given the area of the eardrum, A is \(0.600 \times 10^{-4} \mathrm{m^2}\). Now substituting the intensity and area of the eardrum to find the power: $$P = 10^{-4} \cdot 0.600 \times 10^{-4}$$
04

Calculate the power absorbed by the eardrum

Performing the calculation of power absorbed: $$P = 10^{-4} \cdot 0.600 \times 10^{-4} = 0.6 \times 10^{-8} \mathrm{W}$$
05

Calculate the energy absorbed by the eardrum

To find the energy absorbed by the eardrum in 3 minutes, we can use the formula: $$E = P \cdot t$$ where t is the time in seconds. Given the time t is 3 minutes, convert it to seconds: $$t = 3 \cdot 60 = 180 \,\text{seconds}$$ Now substituting the power and time to find the energy absorbed: $$E = 0.6 \times 10^{-8} \cdot 180$$
06

Calculate the energy absorbed

Performing the calculation of energy absorbed: $$E = 0.6 \times 10^{-8} \cdot 180 = 108 \times 10^{-8} \mathrm{Joules}$$ So, the eardrum absorbs \(108 \times 10^{-8} \mathrm{J}\) of energy in 3 minutes.

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Most popular questions from this chapter

At a rock concert, the engineer decides that the music isn't loud enough. He turns up the amplifiers so that the amplitude of the sound, where you're sitting, increases by \(50.0 \%\). (a) By what percentage does the intensity increase? (b) How does the intensity level (in \(\mathrm{dB}\) ) change?
A source and an observer are each traveling at 0.50 times the speed of sound. The source emits sound waves at \(1.0 \mathrm{kHz}\). Find the observed frequency if (a) the source and observer are moving towand each other, (b) the source and observer are moving away from each other; (c) the source and observer are moving in the same direction.
Doppler ultrasound is used to measure the speed of blood flow (see Problem 42). The reflected sound interferes with the emitted sound, producing beats. If the speed of red blood cells is \(0.10 \mathrm{m} / \mathrm{s},\) the ultrasound frequency used is \(5.0 \mathrm{MHz},\) and the speed of sound in blood is \(1570 \mathrm{m} / \mathrm{s},\) what is the beat frequency?
Humans can hear sounds with frequencies up to about \(20.0 \mathrm{kHz},\) but dogs can hear frequencies up to about \(40.0 \mathrm{kHz} .\) Dog whistles are made to emit sounds that dogs can hear but humans cannot. If the part of a dog whistle that actually produces the high frequency is made of a tube open at both ends, what is the longest possible length for the tube?
Some bats determine their distance to an object by detecting the difference in intensity between cchoes.(a) If intensity falls off at a rate that is inversely proportional to the distance squared, show that the echo intensity is inversely proportional to the fourth power of distance. (b) The bat was originally \(0.60 \mathrm{m}\) from one object and \(1.10 \mathrm{m}\) from another. After flying closer, it is now \(0.50 \mathrm{m}\) from the first and at \(1.00 \mathrm{m}\) from the second object. What is the percentage increase in the intensity of the ccho from each object?
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