Question

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 m from it, you measure its intensity to be 0.11 W/m${}^2$. An intensity of 1.0 W/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?

Short answer

You can move 5.01 meters closer to reach the threshold of pain.

Step-by-step solution

Understand the Intensity Formula for Point Sources

The intensity of sound from a point source can be described by 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.

Solve for the Source Power

Use the given intensity at 7.5 m to find the power of the source. Rearrange the intensity formula: $P=I\cdot 4\pi r^2$. Substitute $I=0.11{\text{ W/m}}^2$ and $r=7.5\text{ m}$: $$P=0.11\times 4\pi (7.5{)}^2$$.

Calculate Source Power

Calculate the power using the values: $$P=0.11\times 4\pi \times 56.25=77.8217\text{ W}$$.

Determine New Distance for Intensity Threshold

Set the intensity to the threshold of pain: $I=1.0{\text{ W/m}}^2$. Use $P=77.8217\text{ W}$ and solve for $r$ using $I=\frac{P}{4\pi r^2}$: $$1.0=\frac{77.8217}{4\pi r^2}$$.

Solve for New Distance $r$

Rearrange to find $r^2=\frac{77.8217}{4\pi }$ and then $r=\sqrt{\frac{77.8217}{4\pi }}$.

Calculate and Compare Distances

Calculate $r$: $$r=\sqrt{\frac{77.8217}{12.5664}}\approx 2.49\text{ m}$$. Determine how much closer you can get: 7.5 m - 2.49 m = 5.01 m.

Key concepts

Point Source

When studying sound or any wave phenomena, a point source is an essential concept. Imagine a tiny, single point in space from which sound radiates out uniformly in all directions. This is a point source. Although real-world sources might not be perfect points, many can be approximated as such for simplicity.
Point sources create sound waves that spread out spherically. That means as you move away, the sound becomes less intense because the energy spreads over a larger area. Think of it as ripples on water widening with distance.
Key features of point sources include:

• Sounds decrease in intensity with distance.
• The spread is uniform, like an expanding sphere.
• Simple calculations with the surface area of a sphere equation.

In the exercise, the UFO's sound source is treated as a point source, making it easier to calculate changes in sound intensity.

Intensity Formula

Sound intensity is a measure of power per unit area, showing how strong or weak a sound is at a certain distance. The formula for intensity from a point source is given by:$$I=\frac{P}{4\pi r^2}$$where

• $I$ is the sound intensity, measured in watts per square meter (W/m${}^2$).
• $P$ is the power of the sound source, in watts (W).
• $r$ is the distance from the sound source, in meters (m).

This formula reflects how intensity decreases with the square of the distance from the source, known as the inverse square law.
In practical terms, if you double the distance, the sound intensity drops to a quarter of its original value.
To find out how close you can get before reaching a specific intensity level, like the threshold of pain, you can rearrange the formula to solve for $r$ with a known power and intensity.

Threshold of Pain

The threshold of pain is a critical concept in acoustics. It's the maximum sound intensity level humans can typically tolerate without experiencing pain. The commonly accepted value for this threshold is about 1.0 W/m${}^2$.
Experiencing sounds above this level can cause discomfort or even damage to one's hearing over time. Therefore, knowing how sound intensity changes helps you gauge safe listening distances from loud sources.
In the context of our exercise, the threshold of pain acts as the limit for how close you can approach the sound source. Once this threshold is reached, it's crucial to maintain a safe distance to avoid discomfort, hence calculating the exact distance using the intensity formula is important.