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Chapter 16: Problem 96

The average sound intensity inside a busy neighborhood restaurant is \(3.2 \times 10^{-5} \mathrm{W} / \mathrm{m}^{2}\) . How much energy goes into each ear (area \(=\) \(2.1 \times 10^{-3} \mathrm{m}^{2} )\) during a one-hour meal?

Short Answer

Expert verified
Each ear absorbs approximately \(2.42 \times 10^{-7}\) Joules.

Step by step solution

01

Understand Energy Transfer Formula

The energy transferred to a surface via sound is given by the formula \(E = I \cdot A \cdot t\) where \(E\) is the energy, \(I\) is the intensity, \(A\) is the area, and \(t\) is the time.
02

Identify Given Values

From the problem statement, we know the intensity \(I = 3.2 \times 10^{-5} \, \mathrm{W/m^2}\), the area \(A = 2.1 \times 10^{-3} \, \mathrm{m^2}\), and the time \(t = 1\) hour. To use the formula, convert the time to seconds: \(1\) hour = 3600 seconds.
03

Substitute Values Into Formula

Plug the values into the energy formula: \(E = (3.2 \times 10^{-5}) \times (2.1 \times 10^{-3}) \times (3600)\).
04

Calculate the Energy

Perform the calculation: \[E = 3.2 \times 2.1 \times 3600 \times 10^{-8} = 24.192 \times 10^{-8} \, \mathrm{J}\] Simplifying this gives us \(E = 2.4192 \times 10^{-7} \, \mathrm{J}\).
05

Interpretation

Each ear absorbs an energy of approximately \(2.42 \times 10^{-7}\) Joules during the hour-long meal.

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Key Concepts

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

Energy Transfer
Energy transfer in sound involves the movement of acoustic energy from one point to another. In the context of sound waves, this means transferring energy from the sound source to a listener's ear. To understand how much energy is transferred, we use a specific formula:
  • \( E = I \cdot A \cdot t \)
Where:
  • \( E \) is the energy transferred in Joules.
  • \( I \) is the sound intensity, measured in \( \, \mathrm{W/m^2} \) (watts per square meter).
  • \( A \) is the area through which the sound passes, measured in square meters.
  • \( t \) is the time duration in seconds.
In our example, when eating at a busy restaurant, sound waves constantly hit your ear. Understanding energy transfer helps quantify how much sound energy is absorbed over a specific period. By knowing the sound intensity and the area of the ear, we can calculate the total energy received during a duration, like an hour-long meal.
Acoustic Energy
Acoustic energy is the energy carried by sound waves. These waves are vibrations that move through air, water, or other mediums. The strength of these waves, or how much energy they carry, is described by their intensity.Intensity (\( I \)), in the case of a busy restaurant, tells us how concentrated the acoustic energy is in a particular area. It's like measuring how loud or powerful a sound is in a given space. This is crucial because it affects how much energy reaches our ears.Some key points about acoustic energy:
  • Higher intensity means more energy is being transferred.
  • The amount of acoustic energy received depends on the listener's distance from the sound source.
During an hour-long restaurant meal, the acoustic energy continuously accumulates in the ear, dependent on both sound intensity and duration.
Physical Acoustics
Physical acoustics is the study of sound and its interactions with the environment. In simpler terms, it is how sound behaves physically, affecting and being affected by materials and the environment it moves through. In the restaurant scenario, physical acoustics helps us understand:
  • How sound waves originate from various sources like conversations or clinking cutlery.
  • How these waves travel through the space and bounce off surfaces like walls and tables.
  • How our ears perceive these waves as they enter and resonate inside.
Sound intensity and acoustic energy are therefore crucial components of physical acoustics, helping us predict how sound will behave in varied environments. This understanding lets us better design spaces to control sound, like reducing echo or minimizing noise for acoustic comfort.

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Most popular questions from this chapter

When an earthquake occurs, two types of sound waves are gen- erated and travel through the earth. The primary, or P, wave has a speed of about 8.0 km/s and the secondary, or S, wave has a speed of about 4.5 km/s. A seismograph, located some distance away, records the arrival of the P wave and then, 78 s later, records the arrival of the S wave. Assuming that the waves travel in a straight line, how far is the seismograph from the earthquake?

You are flying in an ultralight aircraft at a speed of 39 m/s. An eagle, whose speed is 18 m/s, is flying directly toward you. Each of the given speeds is relative to the ground. The eagle emits a shrill cry whose frequency is 3400 Hz. The speed of sound is 330 m/s. What frequency do you hear?

From a vantage point very close to the track at a stock car race, you hear the sound emitted by a moving car. You detect a frequency that is 0.86 times as small as the frequency emitted by the car when it is stationary. The speed of sound is 343 m/s. What is the speed of the car?

Sound is passing perpendicularly through an open window whose dimensions are 1.1 \(\mathrm{m} \times 0.75 \mathrm{m}\) . The sound intensity level is 95 \(\mathrm{dB}\) above the threshold of hearing. How much sound energy comes through the window in one hour?

A source emits sound uniformly in all directions. A radial line is drawn from this source. On this line, determine the positions of two points, 1.00 m apart, such that the intensity level at one point is 2.00 dB greater than the intensity level at the other.
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