A Loudspeaker Emits Sound With A Power Output Of 400 Watts . If The Speaker Is Placed At The Center Of A Spherical Dome With A Radius Of 15 Meters , What Is The Intensity Of Sound On The Surface Of The Spherical Dome?

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**A loudspeaker emits sound with a power output of 400 watts . If the speaker is placed at the center of a spherical dome with a radius of 15 meters , what is the intensity of sound on the surface of the spherical dome?**

Understanding the Basics of Sound Intensity

Before we dive into the calculation, let's understand the basics of sound intensity. Sound intensity is a measure of the power per unit area of a sound wave. It's typically measured in watts per square meter (W/m²). The intensity of sound is affected by the distance from the source, with the intensity decreasing as the distance increases.

Calculating Sound Intensity

To calculate the intensity of sound on the surface of the spherical dome, we need to use the formula for sound intensity:

I = P / (4 * π * r²)

Where:

  • I is the intensity of sound (in W/m²)
  • P is the power output of the loudspeaker (in watts)
  • r is the radius of the spherical dome (in meters)

Plugging in the Values

Now, let's plug in the values given in the problem:

P = 400 watts r = 15 meters

I = 400 / (4 * π * (15)²) I = 400 / (4 * 3.14159 * 225) I = 400 / 2827.43 I ≈ 0.1417 W/m²

What does this mean?

The intensity of sound on the surface of the spherical dome is approximately 0.1417 W/m². This means that the sound wave is spreading out over the surface of the dome, with a power of 0.1417 watts per square meter.

Q&A

Q: What is the unit of measurement for sound intensity? A: The unit of measurement for sound intensity is watts per square meter (W/m²).

Q: How does the distance from the source affect the intensity of sound? A: The intensity of sound decreases as the distance from the source increases.

Q: What is the formula for calculating sound intensity? A: The formula for calculating sound intensity is I = P / (4 * π * r²), where I is the intensity of sound, P is the power output of the loudspeaker, and r is the radius of the spherical dome.

Q: What is the power output of the loudspeaker in this problem? A: The power output of the loudspeaker is 400 watts.

Q: What is the radius of the spherical dome in this problem? A: The radius of the spherical dome is 15 meters.

Q: What is the intensity of sound on the surface of the spherical dome? A: The intensity of sound on the surface of the spherical dome is approximately 0.1417 W/m².

Real-World Applications

Understanding sound intensity is crucial in various real-world applications, such as:

  • Acoustics: Sound intensity is used to design and optimize sound systems, such as concert halls and auditoriums.
  • Noise pollution: Sound intensity is used to measure and mitigate noise pollution in urban areas.
  • Medical applications: Sound intensity is used in medical imaging techniques, such as ultrasound.

Conclusion

In conclusion, calculating sound intensity is a crucial aspect of understanding how sound waves propagate through space. By using the formula I = P / (4 * π * r²), we can calculate the intensity of sound on the surface of a spherical dome. This knowledge has various real-world applications, from acoustics to medical imaging.