Problem 22 - University Physics with Modern Physics

Question

The Sacramento City Council recently 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 of the new intensity to the old intensity is approximately 0.0032.

Step 1: Understanding Decibel Scale

The decibel (dB) scale is a logarithmic scale used to measure sound intensity. A change of 10 dB represents a tenfold change in intensity.

Step 2: Finding the Intensity Levels

The initial intensity level is 95 dB, and the reduced intensity level is 70 dB. We need to convert these decibel levels to their corresponding intensity levels.

Step 3: Calculating Intensity from Decibels

The formula to convert decibels to intensity is: \( I = 10^{(L/10)} \), where \(L\) is the level in decibels. Thus, the initial intensity \( I_1 = 10^{(95/10)} \) and the new intensity \( I_2 = 10^{(70/10)} \).

Step 4: Simplifying the Ratios

Calculate \( I_1 = 10^{9.5} \) and \( I_2 = 10^{7} \). Thus, the ratio \( \frac{I_2}{I_1} = \frac{10^7}{10^{9.5}} \).

Step 5: Simplifying the Final Expression

Use the laws of exponents to simplify the fraction: \( \frac{10^7}{10^{9.5}} = 10^{7-9.5} = 10^{-2.5} \).

Step 6: Converting to a Decimal Number

Calculate \(10^{-2.5}\) using a calculator. This results in approximately \(0.0032\).

Key Concepts

Decibel Scale

The decibel scale is a critical concept when working with sound intensity. It's a unit of measurement used to express the ratio of two values of a physical quantity, particularly power or intensity. The decibel scale is logarithmic. This means each increment of 10 dB represents a tenfold change in intensity.
For example:

• A sound that is 10 dB higher than another sound is ten times more intense.
• A sound 20 dB higher is 100 times more intense.
• In our exercise, reducing the sound intensity from 95 dB to 70 dB involves significant reduction, showcasing the effectiveness of the logarithmic scale in representing large variations in intensity levels.

This scale helps to manage the wide range of sound intensities our ears can handle. It also illustrates how even small changes in dB can effectively manage noise pollution.

Logarithmic Scale

A logarithmic scale is crucial in many scientific disciplines, including physics, because it can simplify the representation of vastly different numbers. A linear scale would not effectively represent the wide range of sound intensities that exist.
With a logarithmic scale:

• Each step on the scale is a constant factor above or below the previous step. For example, with base-10 logarithms, each step up is a tenfold increase.
• This makes it much easier to comprehend changes in sound intensity, which often involve orders of magnitude changes.

In the case of sound intensity measured in decibels, the logarithmic nature of the scale leverages the calculation simplification, allowing us to use exponents to compute intensity ratios as in the given problem. Understanding logs helps grasp concepts around exponential growth or decay effectively.

Exponents in Physics

Exponents are a fundamental concept in physics and maths, facilitating the representation and calculation of very large or very small numbers. In the exercise to find the sound intensity ratio, exponents play a key role in simplification.
Consider the following:

• When calculating intensity from decibels, the formula used is often represented in terms of powers of ten, e.g., \( I = 10^{(L/10)} \).
• Using the laws of exponents, significant simplifications can occur, streamlining complex calculations.
  For instance, the ratio \( \frac{10^7}{10^{9.5}} \) simplifies to \( 10^{-2.5} \) using properties like \( a^m/a^n = a^{m-n} \).

Exponents allow the concise expression of calculations, especially in physics problems involving large quantities. Understanding these principles is essential to solving a variety of physics problems efficiently.