Problem 40 - College Physics

Question

\(\cdot\) Find the intensity \(\left(\) in \(W / m^{2}\right)\) of (a) a 55.0 dB sound, (b) a 92.0 dB sound, (c) a \(-2.0\) dB sound.

Step-by-Step Solution

Answer

The intensities are \( 3.1623 \times 10^{-7} \, W/m^2 \), \( 1.5849 \times 10^{-3} \, W/m^2 \), and \( 6.3096 \times 10^{-13} \, W/m^2 \) for 55.0 dB, 92.0 dB, and -2.0 dB respectively.

Step 1: Understanding Decibels

Decibels (dB) are a unit of measure for sound intensity, where the intensity level in decibels is calculated using the formula: \[ L = 10 imes ext{log}_{10} \left( \frac{I}{I_0} \right) \] Here, \( L \) is the sound level in decibels, \( I \) is the intensity of the sound in \( W/m^2 \), and \( I_0 = 10^{-12} \, W/m^2 \) is the reference intensity level, typically considered the threshold of hearing.

Step 2: Rearranging the Formula

To find the intensity \( I \) from a given decibel (dB) level, we rearrange the initial formula to solve for \( I \): \[ I = I_0 \times 10^{(L/10)} \] This equation allows us to calculate the intensity using the known decibel value.

Step 3: Calculating Intensity for 55.0 dB

Substitute \( L = 55.0 \, dB \) into the formula: \[ I = 10^{-12} \times 10^{(55/10)} \] Simplify the exponent: \[ I = 10^{-12} \times 10^{5.5} \] Calculate \( 10^{5.5} \) and multiply by \( 10^{-12} \). Using a calculator, \( 10^{5.5} \approx 316227.77 \), thus: \[ I \approx 3.1623 \times 10^{-7} \, W/m^2 \]

Step 4: Calculating Intensity for 92.0 dB

Substitute \( L = 92.0 \, dB \) into the formula: \[ I = 10^{-12} \times 10^{(92/10)} \] Simplify the exponent: \[ I = 10^{-12} \times 10^{9.2} \] Calculate \( 10^{9.2} \) and multiply by \( 10^{-12} \). Using a calculator, \( 10^{9.2} \approx 15848931.9 \), thus: \[ I \approx 1.5849 \times 10^{-3} \, W/m^2 \]

Step 5: Calculating Intensity for -2.0 dB

Substitute \( L = -2.0 \, dB \) into the formula: \[ I = 10^{-12} \times 10^{(-2/10)} \] Simplify the exponent: \[ I = 10^{-12} \times 10^{-0.2} \] Calculate \( 10^{-0.2} \) and multiply by \( 10^{-12} \). Using a calculator, \( 10^{-0.2} \approx 0.630957 \), thus: \[ I \approx 6.3096 \times 10^{-13} \, W/m^2 \]

Key Concepts

Decibel Scale

The decibel (dB) is a logarithmic unit used to measure sound intensity. It's not linear like other measurement units, such as centimeters or meters. Instead, each 10 dB increase represents a tenfold increase in intensity because decibels use a logarithmic scale. Understanding this concept is important because it explains why sounds that aren't linearly spaced in dB have dramatically different intensity levels. Decibels are calculated using a specific formula that relates the sound's actual intensity to a reference intensity level. This reference intensity, denoted as \( I_0 \), is typically \( 10^{-12} \, W/m^2 \), which is considered the threshold of hearing. By using this scale, very large or small sound intensities can be expressed in manageable numbers.

Intensity Formula

To find the intensity of a sound from its decibel level, we use a rearranged form of the decibel formula. The original formula for decibels is: \[ L = 10 \times \log_{10} \left( \frac{I}{I_0} \right) \] Where \( L \) represents the decibels, \( I \) is the intensity of the sound in watts per square meter (W/m²), and \( I_0 \) is the reference intensity. When we want to solve for \( I \), we manipulate the formula to isolate \( I \):\[ I = I_0 \times 10^{(L/10)} \] This formula allows us to convert dB values to intensity directly, making tasks like those in the original exercise straightforward once you understand the relationship. In practical terms, you substitute the decibel value (\( L \)) into the formula and compute to find the actual intensity in \( W/m^2 \). This method is crucial for solving various physics problems involving sound.

Physics Problem-Solving

When dealing with physics problems, such as calculating sound intensity from decibels, a structured approach makes problem-solving much easier. Here are some helpful tips:- **Understand the Basics**: Start by familiarizing yourself with core concepts, like the decibel scale and intensity formula.- **Use Appropriate Formulas**: Identify which formula is applicable, and ensure you know how to rearrange it if needed. For intensity problems, that would be: \[ I = I_0 \times 10^{(L/10)} \]- **Substitute Carefully**: Plug the given values into the formula carefully to avoid mistakes. Check each step as you go.- **Calculate Accurately**: Use a calculator for the exponents if necessary, as these can often involve non-integer values.- **Cross-Check Units**: Make sure your final answer is in the correct units, often \( W/m^2 \) for intensity.By understanding these steps, you can approach physics problems with confidence, ensuring you have both the comprehension and skills to find the correct solutions. Remember that practice is key to mastering these types of questions.