Problem 22 - University Physics with Modern Physics

Question

The Sacramento City Council adopted a law to reduce the allowed sound intensity level of the much-despised leaf blowers from their current level of about 95 \(\mathrm{dB}\) to 70 \(\mathrm{dB}\) . With the new law, what is the ratio of the new allowed intensity to the previously allowed intensity?

Step-by-Step Solution

Answer

The ratio is approximately 0.00316.

Step 1: Understand Sound Intensity Level

The sound intensity level is measured in decibels (dB), a logarithmic scale. An increase of 10 dB corresponds to a tenfold increase in sound intensity.

Step 2: Apply the Decibel Formula

The formula to calculate the intensity level in decibels is: \( L = 10 \log_{10}\left(\frac{I}{I_0}\right) \), where \( I \) is the sound intensity and \( I_0 \) is the reference intensity. We can use this formula to understand how changes in decibels relate to changes in intensity.

Step 3: Calculate the Intensity Ratio

When changing from 95 dB to 70 dB, we use the formula for the difference in decibels: \( \Delta L = 10 \log_{10}\left(\frac{I_2}{I_1}\right) \). Plugging in the values, we get 25 dB = \( 10 \log_{10}\left(\frac{I_2}{I_1}\right) \).

Step 4: Solve for the Intensity Ratio

Rearrange and solve the equation for the intensity ratio: \( \frac{I_2}{I_1} = 10^{\frac{25}{10}} = 10^{2.5} \). Calculate \( 10^{2.5} \), which equals approximately 316.23.

Step 5: Compute the New to Old Intensity Ratio

The ratio of the new allowed intensity to the previously allowed intensity is \( \frac{1}{316.23} \) since the sound is being reduced, not increased. Compute this ratio to get approximately 0.00316.

Key Concepts

Decibels

Decibels (dB) are a unit used to measure the intensity of sound. A decibel does not measure sound directly. Instead, it measures the intensity level on a logarithmic scale. This scale takes a wide range of sounds and compresses it into a manageable number. This way of measuring is very effective because human hearing perceives sound intensity logarithmically rather than linearly.

This means each increase of 10 decibels is experienced as a doubling of the perceived volume to the human ear. If a sound increases from 60 dB to 70 dB, it is perceived to be twice as loud, though its actual intensity has increased by ten times.

• 0 dB is near-total silence, the faintest sound the human ear can hear.
• Each 10 dB increment means that a sound has become ten times more intense.

Intensity Ratio

The intensity ratio is a way to express how one sound intensity compares to another. In context of sound, it shows how much a sound's power level changes. This is especially important when laws or regulations limit sound levels, like in the case of leaf blower noise limits set by the Sacramento City Council.

To find the intensity ratio between two different sound levels, you can compare their intensities using the decibel scale. The formula for the decibel difference between two sound levels is: \[ \Delta L = 10 \log_{10}\left(\frac{I_2}{I_1}\right) \] where \(I_2\) is the new intensity and \(I_1\) is the original intensity. A positive difference indicates an increase in intensity, while a negative difference indicates a decrease.

In our case, moving from 95 dB to 70 dB, the intensity ratio \(\frac{I_2}{I_1}\) is approximately \(10^{-2.5}\), which equals roughly 0.00316. This indicates the new intensity allowed is only a fraction of the original intensity.

Logarithmic Scale

A logarithmic scale is a nonlinear scale used for a large range of quantities, where each step on the scale is a consistent multiple of the previous one. This kind of scale is used when dealing with quantities that can cover a huge range of values, such as sound intensity.

The need for a logarithmic scale arises because our hearing is extremely sensitive and can detect a very wide range of sound intensities. Measuring sound intensity in direct units would be unwieldy. Therefore, the logarithmic decibel scale simplifies these measurements.

• On a logarithmic scale, every 10-fold increase in intensity adds 10 dB to the sound level.
• This is why the decibel formula includes a logarithm: \[ L = 10 \log_{10}\left(\frac{I}{I_0}\right) \]which compares the sound intensity \(I\) to a reference intensity \(I_0\).

By compressing large ranges into manageable numbers, the logarithmic scale makes things easier to comprehend, similar to how the Richter scale measures seismic activity.