Problem 26 - University Physics with Modern Physics

Question

Threshold of Pain. You are investigating the report of a UFO landing in an isolated portion of New Mexico, and you encounter a strange object that is radiating sound waves uniformly in all directions. Assume that the sound comes from a point source and that you can ignore reflections. You are slowly walking toward the source. When you are 7.5 \(\mathrm{m}\) from it, you measure its intensity to be 0.11 \(\mathrm{W} / \mathrm{m}^{2}\) . An intensity of 1.0 \(\mathrm{W} / \mathrm{m}^{2}\) is often used as the "threshold of pain." How much closer to the source can you move before the sound intensity reaches this threshold?

Step-by-Step Solution

Answer

You can move about 5.01 meters closer.

Step 1: Review the Formula for Intensity

Sound intensity from a point source is calculated using the formula \( I = \frac{P}{4\pi r^2} \), where \( I \) is the intensity, \( P \) is the power of the source, and \( r \) is the distance from the source. We'll use this formula to understand how the intensity changes as we move closer to the source.

Step 2: Understand Relationship Between Intensity and Distance

The initial intensity \( I_0 \) at a distance \( r_0 = 7.5 \) m is 0.11 \( \text{W/m}^2 \). We need to find the new distance \( r_1 \) where the intensity \( I_1 \) is 1.0 \( \text{W/m}^2 \). Using the relationship \( I \propto \frac{1}{r^2} \), we can write \( \frac{I_1}{I_0} = \left(\frac{r_0}{r_1}\right)^2 \).

Step 3: Substitute Known Values into the Intensity Relationship

We know \( I_1 = 1.0 \), \( I_0 = 0.11 \), and \( r_0 = 7.5 \). Substitute these values into \( \frac{1.0}{0.11} = \left(\frac{7.5}{r_1}\right)^2 \).

Step 4: Solve for the New Distance

Firstly, calculate \( \frac{1.0}{0.11} \). This gives approximately 9.09. So, \( 9.09 = \left(\frac{7.5}{r_1}\right)^2 \). Taking the square root of both sides, we get \( \sqrt{9.09} = \frac{7.5}{r_1} \). Calculate \( \sqrt{9.09} \) which is approximately 3.01. Now, solving for \( r_1 \), we find \( r_1 = \frac{7.5}{3.01} \approx 2.49 \).

Step 5: Calculate How Much Closer You Can Move

Subtract the new distance from the initial distance to find how much closer you can get: \( 7.5 \text{ m} - 2.49 \text{ m} = 5.01 \text{ m} \).

Key Concepts

Threshold of Pain

The threshold of pain is a level of sound intensity that is extremely uncomfortable and can cause pain to the human ear. This threshold is typically measured at an intensity level of around 1.0 \( \text{W/m}^2 \), although it can vary slightly from person to person. The concept is important because it guides us in understanding safe levels of sound exposure. Sound at or above this intensity can not only be painful but may also cause immediate harm to one's hearing.

When studying sound intensity, especially in scenarios involving loud noise or potential hearing damage, the threshold of pain serves as a critical reference point. It's used in fields like acoustics to design safer sound environments in public spaces such as concerts or factories, where sound levels can reach dangerous heights.

Point Source

A point source in acoustics refers to an idealized point from which sound waves radiate equally in all directions. Imagine it like a tiny sphere emanating sound all around it. This simplification helps in solving problems related to sound propagation, as it allows us to focus on the geometry of sound waves rather than the complexities introduced by the shape of the source.

In real-world scenarios, most sound sources aren't perfect point sources. However, many sources can be approximated as point sources when the distance from the source is large compared to the size of the sound-emitting object. For example:

• A small speaker or a person speaking in a large open area can be approximated as a point source.
• This approximation simplifies mathematical calculations, making it easier to predict how sound travels through a space.

Understanding point sources helps in designing acoustic environments, predicting noise pollution, and even in scientific investigations like studying UFO sightings, as in the exercise.

Inverse Square Law

The inverse square law is a fundamental principle governing how sound intensity diminishes as it travels away from a point source. According to this law, sound intensity decreases proportionally to the square of the distance from the source. Mathematically, this is expressed as:\[I \propto \frac{1}{r^2}\]where \( I \) is the intensity, and \( r \) is the distance from the sound source.This principle explains why sound becomes quieter as you move away from its origin. For example, if you double the distance from a sound source, the intensity becomes one-fourth of what it was at the initial distance.

The inverse square law is crucial in understanding the acoustic environment and sound design. It helps predict how sound levels decrease with distance, allowing architects and engineers to design spaces with appropriate sound distribution, ensuring comfort and safety.

Acoustics

Acoustics is the science of sound, encompassing its generation, transmission, and effects. It is a broad field that touches various aspects of our daily lives, from designing concert halls to improving sound quality in automobiles.

In practical applications, acousticians study how sound behaves in different environments and work on:

• Reducing noise pollution in urban areas.
• Enhancing sound clarity in theaters and auditoriums.
• Improving speech privacy in offices and public buildings.

Understanding acoustics can also involve the study of sound wave propagation through different mediums, absorption by materials, and reflection and refraction behaviors. Acoustics embraces both the pleasant and problematic aspects of sound, and experts in this field aim to enhance human experiences by controlling and harnessing sound effectively. This knowledge becomes especially significant in areas where sound intensity and its effects on human hearing are critical considerations, like understanding the threshold of pain in acoustics design.