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Chapter 23: Problem 52

A whisper has an intensity level of about \(15 \mathrm{~dB}\). What is the corresponding intensity of the sound?

Short Answer

Expert verified
The intensity of the sound is approximately \( 3.1622 \times 10^{-11} \mathrm{~W/m}^2 \).

Step by step solution

01

Understand the Relationship

The relationship between sound intensity level in decibels (dB) and actual sound intensity in watts per square meter (W/m虏) is given by the formula: \[ L = 10 \log_{10} \left( \frac{I}{I_0} \right) \] where \( L \) is the sound intensity level in decibels, \( I \) is the sound intensity, and \( I_0 \) is the reference sound intensity, typically \( 10^{-12} \mathrm{~W/m}^2 \).
02

Setup the Formula

Given that the sound intensity level \( L = 15 \mathrm{~dB} \), we substitute into the formula: \[ 15 = 10 \log_{10} \left( \frac{I}{10^{-12}} \right) \] and solve for \( I \).
03

Solve for the Intensity

Divide both sides of the equation by 10 to isolate the logarithm: \[ 1.5 = \log_{10} \left( \frac{I}{10^{-12}} \right) \] To eliminate the logarithm, apply the inverse, which is exponentiation with base 10: \[ 10^{1.5} = \frac{I}{10^{-12}} \].
04

Calculate the Intensity

Solving \( 10^{1.5} = 31.622 \), we multiply both sides by \( 10^{-12} \) to find \( I \): \[ I = 31.622 \times 10^{-12} = 3.1622 \times 10^{-11} \mathrm{~W/m}^2 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Decibels
Decibels (dB) are used to measure the intensity level of sound. It is a logarithmic unit, which means it compares the intensity of a sound to a reference level in a way that reflects how humans perceive loudness. Since the human ear can detect a wide range of sound intensities, from the faintest whisper to the roar of a jet engine, the logarithmic scale compresses this large range of numbers into a more manageable form. The formula to convert sound intensity to decibels is given by:\[ L = 10 \log_{10} \left( \frac{I}{I_0} \right) \]Here:
  • \( L \) is the sound intensity level in decibels.
  • \( I \) is the actual sound intensity in watts per square meter (W/m虏).
  • \( I_0 \) is the reference intensity, typically set at \( 10^{-12} \mathrm{~W/m}^2 \), which is roughly the quietest sound that can be heard by the average human ear.
Using this formula, we can determine the sound intensity from its decibel level and vice versa.
Acoustics
Acoustics is the science of sound. It studies how sound is produced, transmitted, and received. Understanding acoustics involves knowledge of sound waves and their properties, such as frequency, wavelength, and speed. In the context of this exercise, acoustics helps explain how a whisper has a specific intensity level. This intensity is much lower than louder sounds like speaking or shouting. Some key components of acoustics include:
  • **Sound Waves**: These are the vibrations that travel through the air and reach our ears. Each wave has peaks and troughs, much like waves on the ocean.
  • **Frequency**: This determines the pitch of the sound and is measured in Hertz (Hz). It indicates how many waves pass a certain point per second.
  • **Intensity**: This is the power carried by the sound waves and is perceived by the ear as loudness, measured in watts per square meter.
A proper understanding of acoustics allows us to optimize environments for better sound clarity, such as in concert halls or recording studios.
Physics Problem Solving
Solving physics problems requires a strategic approach that involves understanding the problem, setting up equations, and performing calculations. This structured method enables students to tackle complex questions systematically and logically. For this exercise, we applied the physics problem-solving strategy to find the sound intensity from a given decibel level. Here鈥檚 how we approached it:
  • **Understand the Relationship**: Know how decibels relate to sound intensity using the given formula. This involves recognizing that the decibel level is a logarithmic representation of the intensity compared to a reference point.
  • **Set Up the Formula**: Based on the given decibel level, substitute the known values into the formula. This step transforms the problem into a solvable equation.
  • **Solve for Intensity**: Divide and manipulate the equation as needed (like handling logarithms) to isolate and calculate the unknown variable, completing the solution through precise mathematical operations.
Utilizing these steps with clarity and precision ensures effective problem solving and develops strong analytical skills in physics.

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Most popular questions from this chapter

Compute the speed of sound in helium gas at \(800^{\circ} \mathrm{C}\). [Hint: Take \(\gamma\) to be \(1.67 .]\)

A disk has 40 holes around its circumference and is rotating at 1200 rpm. Determine the frequency and wavelength of the tone produced by the disk when a jet of air is blown against it. The temperature is \(15^{\circ} \mathrm{C}\).

Helium is a monatomic gas that has a density of \(0.179 \mathrm{~kg} / \mathrm{m}^{3}\) at a pressure of \(76.0 \mathrm{~cm}\) of mercury and a temperature of precisely 0 \({ }^{\circ} \mathrm{C}\). Find the speed of compression waves (sound) in helium at this temperature and pressure.

Back in the days before computers, a single typist typing furiously could generate an average sound level nearby of \(60.0 \mathrm{~dB}\). What would be the decibel level in the vicinity if three equally noisy typists were working close to one another? If each typist emits the same amount of sound energy, then the final sound intensity \(I_{f}\) should be three times the initial intensity \(I_{i}\) We have $$ \begin{array}{l} \beta_{f}=10 \log _{10}\left(\frac{I_{f}}{I_{0}}\right)=10 \log _{10} I_{f}-10 \log _{10} I_{0} \\ \text { and } \beta_{i}=10 \log _{10} I_{i}-10 \log _{10} I_{0} \end{array} $$ Subtracting these yields the change in sound level in going from \(I_{i}\) $$ \begin{array}{l} \text { to } I_{f}=3 I_{i}, \\ \qquad \begin{aligned} \beta_{f}-\beta_{i}=10 \log _{10} I_{f}-10 \log _{10} I_{i} \\ \quad \beta_{f}=\beta_{i}+10 \log _{10}\left(\frac{I_{I}}{I_{i}}\right)=60.0 \mathrm{~dB}+10 \log _{10} 3=64.8 \mathrm{~dB} \end{aligned} \end{array} $$ The sound level, being a logarithmic measure, rises very slowly with the number of sources.

A rock band might easily produce a sound level of \(107 \mathrm{~dB}\) in a room. To two significant figures, what is the sound intensity at \(107 \mathrm{~dB} ?\)
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