Problem 60 - Pearson Physics

Question

One hundred violins combine to give an intensity level of \(76 \mathrm{~dB}\). What is the intensity level of just one violin by itself?

Step-by-Step Solution

Answer

The intensity level of one violin is 56 dB.

Step 1: Understand the Intensity Level Formula

The intensity level in decibels is given by the formula: \( L = 10 \log_{10} \left( \frac{I}{I_0} \right) \), where \( L \) is the intensity level in decibels (dB), \( I \) is the intensity of the sound, and \( I_0 \) is the reference intensity typically set as \( 10^{-12} \text{ W/m}^2 \). In our problem, we know \( L = 76 \text{ dB} \) for 100 violins.

Step 2: Relate Intensity Level to Intensity

For 100 violins, the intensity level is 76 dB. This means for 100 violins combined, \( L_{100} = 76 \text{ dB} \). We can express the intensity for 100 violins as \( I_{100} \) where \( L_{100} = 10 \log_{10} \left( \frac{I_{100}}{I_0} \right) \).

Step 3: Calculate the Intensity of One Violin

Assume each violin emits the same intensity \( I_{1} \). Since intensity is additive, \( I_{100} = 100 \times I_{1} \). Substitute this into the equation: \( 76 = 10 \log_{10} \left( \frac{100 I_{1}}{I_0} \right) \).

Step 4: Solve for One Violin's Intensity Level

Using the logarithm property that \( \log_{10} (ab) = \log_{10} (a) + \log_{10} (b) \), we rewrite \( 76 = 10 \log_{10} (100) + 10 \log_{10} \left( \frac{I_{1}}{I_0} \right) \). Since \( 10 \log_{10} (100) = 20 \), we have \( 76 = 20 + 10 \log_{10} \left( \frac{I_{1}}{I_0} \right) \).

Step 5: Solve for the Intensity Level of One Violin

Rearrange to find the intensity level of one violin: \( 76 - 20 = 10 \log_{10} \left( \frac{I_{1}}{I_0} \right) \). Thus, \( 56 = 10 \log_{10} \left( \frac{I_{1}}{I_0} \right) \), and dividing both sides by 10 gives \( \log_{10} \left( \frac{I_{1}}{I_0} \right) = 5.6 \).

Step 6: Compute the Intensity Level for One Violin

Raise 10 to both sides of the equation to remove the logarithm: \( \frac{I_{1}}{I_0} = 10^{5.6} \). This shows the intensity level of one violin alone is \( 56 \text{ dB} \).

Key Concepts

Decibels

When discussing sound, decibels (dB) are a crucial unit of measurement. They help us quantify how intense a sound is. This scale is not linear; instead, it gives us a relative intensity of sound. This means a small change in decibels can represent a large change in actual sound intensity. The formula used for calculating the intensity level in decibels is: \[ L = 10 \log_{10} \left( \frac{I}{I_0} \right) \] where \( L \) is the intensity level in decibels, \( I \) is the sound intensity, and \( I_0 \) is the reference intensity \( (10^{-12} \text{ W/m}^2) \). This reference intensity is the threshold of human hearing in a quiet environment. It's important to remember that the decibel scale involves a logarithm, which means it compresses a large range of numbers into a more manageable one. Relating this to our violin problem: if 100 violins measure 76 dB, then one violin measures 56 dB.

Logarithmic Scale

The logarithmic scale is a mathematical concept used to compress a large range of values into a smaller, more manageable range. This property is especially useful in dealing with quantities that can vary over many orders of magnitude, such as sound levels or earthquake intensities. Logarithms have some essential properties:

• The logarithm of a product is the sum of the logarithms: \( \log_{10}(ab) = \log_{10}(a) + \log_{10}(b) \).
• The logarithm of a quotient is the difference of the logarithms: \( \log_{10}\left( \frac{a}{b} \right) = \log_{10}(a) - \log_{10}(b) \).

This mathematical trickery means that if you know the sound intensity level for a group of instruments, you can find out what one instrument's sound intensity would be. In the violin problem, since we know the combined intensity level and the number of violins, we can calculate the intensity level for a single instrument using logarithms.

Physics Problem Solving

Solving physics problems often involves understanding and manipulating formulas. It's like solving a puzzle; each piece of information fits together to provide a complete picture. In our sound intensity problem, it's important to note several things:

• Identify the formula needed. Understanding how decibels relate to sound intensity helps us pick the correct formula \(L = 10 \log_{10} \left( \frac{I}{I_0} \right) \).
• Set up the relationships. Recognize that for 100 violins, the intensity is the sum of 100 individual intensities: \( I_{100} = 100 \times I_1 \).
• Manipulate the equations correctly. Doing so allows us to isolate the variable we need—in this case, the intensity level for a single violin.
• Use logarithmic properties to simplify the calculations. This involves breaking down the logarithmic operation into smaller, manageable steps to achieve the solution.

This problem-solving method is widely applicable in physics. It involves carefully deciphering information, logically setting equations, and solving for the unknown.