Problem 65 - College Physics

Question

A very noisy chain saw operated by a tree surgeon emits a total acoustic power of 20.0 \(\mathrm{W}\) uniformly in all directions. At what distance from the source is the sound level equal to (a) 100 \(\mathrm{dB}\) . (b) 60 \(\mathrm{dB} ?\)

Step-by-Step Solution

Answer

The distance for 100 dB is approximately 3.99 meters, and for 60 dB, it's approximately 398.11 meters.

Step 1: Understanding the Problem

We need to find the distance from a sound source where the sound level is given in decibels (dB). We are provided with two sound levels to solve: 100 dB and 60 dB. The acoustic power emitted by the source is 20 W, and it radiates sound uniformly in all directions.

Step 2: Use the Sound Level Formula

The sound level in decibels can be calculated using the formula:\[ L = 10 \log_{10} \left( \frac{I}{I_0} \right) \]where:- \( L \) is the sound level in dB,- \( I \) is the intensity in W/m²,- \( I_0 = 1 \times 10^{-12} \) W/m² is the reference intensity.

Step 3: Relate Intensity to Acoustic Power and Distance

Since power is emitted uniformly in all directions, the intensity \( I \) at a distance \( r \) can be given by:\[ I = \frac{P}{4\pi r^2} \]where:- \( P = 20 \) W is the acoustic power,- \( r \) is the distance from the source in meters.

Step 4: Solve for Distance for 100 dB

Substitute \( L = 100 \) dB into the sound level formula:\[ 100 = 10 \log_{10} \left( \frac{P}{4\pi r^2 I_0} \right) \]Solving for \( r \), we work through the algebra:\[ 10 = \log_{10} \left( \frac{20}{4\pi r^2 \times 10^{-12}} \right) \]Take antilog on both sides:\[ 10^{10} = \frac{20}{4\pi r^2 \times 10^{-12}} \]Rearrange and solve for \( r \):\[ r^2 = \frac{20}{4\pi \times 10^{-2}} \]\[ r = \sqrt{\frac{20}{4\pi \times 10^{-2}}} \approx 3.99 \text{ meters} \]

Step 5: Solve for Distance for 60 dB

For 60 dB, the steps are similar:\[ 60 = 10 \log_{10} \left( \frac{20}{4\pi r^2 \times 10^{-12}} \right) \]Solve for \( r \) similarly:\[ 6 = \log_{10} \left( \frac{20}{4\pi r^2 \times 10^{-12}} \right) \]Take antilog:\[ 10^6 = \frac{20}{4\pi r^2 \times 10^{-12}} \] \\[ r^2 = \frac{20}{4\pi \times 10^{6-12}} \] \\[ r = \sqrt{\frac{20}{4\pi \times 10^{-6}}} \approx 398.11 \text{ meters} \]

Key Concepts

Sound Level

Sound level is a measure of how loud a sound is perceived and is expressed in decibels (dB). This concept is crucial for understanding how sound intensity changes based on distance from the source.
Sound level is logarithmic, meaning each increase of 10 dB represents a tenfold increase in intensity. It expresses the intensity of a sound wave compared to a reference intensity, typically the quietest sound a human can hear, which is set at \(1 imes 10^{-12} \, \text{W/m}^2 \).
When solving problems involving sound levels, you often use the formula: \[L = 10 \log_{10} \left( \frac{I}{I_0} \right)\]

• \(L\) is the sound level in decibels.
• \(I\) is the sound intensity in watts per meter squared (W/m²).
• \(I_0\) is the reference intensity \(1 \times 10^{-12} \, \text{W/m}^2 \).

In the context of this problem, understanding sound level helps us determine how far away one must be from the chainsaw for the sound to drop to desired levels of 100 dB or 60 dB.

Intensity

Intensity is a measure of how much sound power is passing through a certain area and is expressed in watts per square meter (W/m²). In simpler terms, it represents the strength or loudness of a sound at a specific location.
The intensity of sound radiated by a source like a chain saw is inversely related to the square of the distance from the source. This relationship is due to the sound energy spreading out uniformly in all directions.
The formula to calculate intensity from power is:\[I = \frac{P}{4\pi r^2}\]

• \(I\) stands for intensity.
• \(P\) is the acoustic power (in this case, 20 W).
• \(r\) is the distance from the source in meters.

Intensity decreases as you move away from the source. This formula helps us find the intensity at a certain distance and calculate the corresponding sound level using decibels.

Decibels

Decibels (dB) are the units used to measure sound level. Decibels rely on a logarithmic scale, which compresses the range of sound intensities into a manageable scale.
Understanding decibels can help grasp why small changes in dB levels can mean large differences in sound intensity.
For example, a sound that measures 100 dB is ten times more intense than 90 dB and 100 times more intense than 80 dB. Due to the logarithmic nature, each increase of 10 dB represents a tenfold increase in perceived loudness.
In practice, this makes decibels a convenient way to express sound levels because:

• They allow for the easy comparison of different sound levels.
• They are better suited to reflect human perception of sound than linear units.

For our chainsaw problem, we use decibels to determine how far one needs to be so that the sound level decreases to a safer or more comfortable level.

Uniform Distribution

When we talk about uniform distribution in the context of sound, we refer to the even spreading out of sound waves from the source. For a chainsaw, which emits sound uniformly, the acoustic power is distributed equally in all directions.
This distribution allows us to use formulas that assume the sound intensity decreases consistently with distance.
Understanding uniform distribution helps in:

• Using formulas to relate power, intensity, and distance correctly.
• Predicting sound levels at different distances from the source.

In the chainsaw example, because the sound is emitted uniformly, the intensity formula \( I = \frac{P}{4\pi r^2} \) is valid. It ensures calculations give accurate readings of how far you need to be to experience specific sound levels, based on uniform spreading of the sound power.