Problem 65 - Physics

Question

Two identical rifles are shot at the same time, and the sound intensity level is \(80.0 \mathrm{~dB}\). What would be the sound intensity level if only one rifle were shot? (Hint: The answer is not \(40.0 \mathrm{dB.}\) )

Step-by-Step Solution

Answer

The sound intensity level for one rifle is approximately 77 dB.

Step 1: Understanding Decibels and Intensity

Decibels (dB) measure the intensity level of sound. The decibel scale is logarithmic, which means a small change in the dB level corresponds to a large change in intensity. In sound intensity, a difference of 10 dB is a tenfold increase in intensity.

Step 2: Initial Condition with Two Rifles

We are given that two rifles produce a combined sound intensity level of 80 dB. Since both rifles are identical and shot simultaneously, this means the intensity level is doubled compared to one rifle.

Step 3: Using the Decibel Formula

The formula for sound intensity level in decibels, when comparing two intensities, is \( L_2 - L_1 = 10 \log \left( \frac{I_2}{I_1} \right) \). Here, \( L_2 = 80 \text{ dB} \) for two rifles and \( L_1 \) is what we need to find for one rifle, where \( I_2 = 2I_1 \).

Step 4: Setting Up the Equation

Substitute \( I_2 = 2I_1 \) into the decibel formula: \( 80 - L_1 = 10 \log \left( 2 \right) \). Simplifying this gives us \( 10\log(2) \approx 3.01 \), resulting in \( 80 - L_1 = 3.01 \).

Step 5: Solving for One Rifle's Level

Rearrange the equation to find \( L_1 \): \( L_1 = 80 - 3.01 \). Calculating this gives \( L_1 = 76.99 \text{ dB} \), which can be approximated as 77 dB.

Key Concepts

Decibel Scale

Decibels are a standard unit for measuring the intensity of sound. The scale is unique because it is logarithmic, not linear. This means that every 10 dB increase represents a tenfold increase in sound intensity. For example, a sound at 70 dB is ten times more intense than one at 60 dB.

The decibel scale is practical because it condenses the wide range of human hearing into a more manageable scale. Sounds ranging from the faintest whisper to a jet engine take up a span of 0 to around 140 dB.

When you see a value in decibels, it's essential to remember that it relates logarithmically to actual sound intensity, not straightforwardly. A change that might seem small in dB could mean a big change in what you actually hear.

Logarithmic Scale

The logarithmic scale, on which the decibel system is based, compresses vast ranges of numbers into a smaller, more understandable range. In the case of sound, it helps manage the immense variability in sound intensity that the human ear can perceive.

Mathematically, a logarithmic scale uses powers of a base number—in this case, base 10—to express numbers. On this scale, every step represents a multiplication of the previous step by a set amount. For sound, each 10 dB represents a tenfold increase in intensity. But this means that

• 20 dB is not twice as loud as 10 dB.
• Instead, it's 10 times 10, or 100 times as intense.

This characteristic enables easier comprehension of very high numbers, such as the large variations in auditory intensities.

Sound Intensity Formula

To convert between sound intensity levels and actual sound intensity measurements, you use the sound intensity formula. The specific formula employed in decibel calculations compares two intensities: \[ L_2 - L_1 = 10 \log \left( \frac{I_2}{I_1} \right) \] where:

• \( L_2 \) and \( L_1 \) are the dB levels of the two intensities being compared,
• \( I_2 \) and \( I_1 \) are the respective sound intensities.

In our exercise, this formula helps us find the sound level of a single rifle when two together produce an 80 dB level. We are given that two rifles mean twice the intensity of one: \( I_2 = 2I_1 \). Substituting values, we get an important step: calculating the logarithm of 2, which is approximately 3.01. This indicates that the sound intensity from a single rifle is

• 77 dB, not half the original 80 dB.

Understanding this formula is crucial to solving problems involving comparisons of sound.