Workers around jet aircraft typically wear protective...

Workers around jet aircraft typically wear protective devices over their ears. Assume that the sound level of a jet airplane engine, at a distance of $30 \mathrm{m},$ is 130 $\mathrm{dB}$ ,and that the average human ear has an effective radius of 2.0 $\mathrm{cm} .$ What would be the power intercepted by an unprotected ear at a distance of 30 $\mathrm{m}$ from a jet airplane engine?

Workers around jet aircraft typically wear protective devices over their ears. Assume that the sound level
of a jet airplane engine, at a distance of $30 \mathrm{m},$ is 130 $\mathrm{dB}$ ,and that the average human ear has an effective radius of 2.0 $\mathrm{cm} .$ What would be the power intercepted by
an unprotected ear at a distance of 30 $\mathrm{m}$ from a jet airplane engine?

Key Concepts

Decibel Scale

The decibel (dB) scale is a logarithmic measure of sound intensity relative to a reference value. It compresses a wide range of sound power levels into manageable numbers, allowing us to express large variations in intensity in a compact form.

Reference Intensity

The reference intensity for sound in air, typically taken as 10?¹² W/m², represents the threshold of human hearing. This baseline is used in converting dB values into absolute intensities using a logarithmic relationship.

Sound Intensity Conversion

To convert a sound level in decibels to a linear intensity (in watts per square meter), the formula I = I_ref × 10^(L_dB/10) is used. This process translates a logarithmic dB scale value into the actual power density of the sound energy.

Inverse Square Law

The inverse square law states that the intensity of a sound energy radiating from a point source decreases as the square of the distance from the source. This principle is key when determining how sound spreads and how its intensity diminishes over distance.

Effective Area of a Receiver

The effective area of a receiver, such as a human ear, quantifies the physical region over which sound energy is intercepted. For a circular area, this is calculated using the area formula for a circle, which helps in determining the total power striking the receiver based on the local intensity.

Transcript

00:01 Our question that workers around jet aircrafts typically wear protective ear devices over their ears.

00:08 Assume the sound level of a jet airplane engine at a distance of 30 meters is 130 decibels, and that the average human ear has an effective radius of 2 centimeters.

00:19 What would be the power interpreted by an unprotected ear at a distance of 30 meters from a jet airplane? so we're told here right now what we're told.

00:27 We're told that the distance from the jet airplane is 30 meters, which i write as d.

00:31 The decibel level, which i read as beta, is 130 decibels.

00:35 And the effective radius of an ear, which i call r, is two centimeters.

00:40 Now, rewrite that in meter in terms of si units, and two centimeters is two times 10 to the minus two meters.

00:50 And that's because there are 100 centimeters in every meter.

00:55 Okay? so to find the power, we can use the equation for the power output is equal to the intensity times the area being affected.

01:05 So this would be the intensity of the sound times four, in this case, times pi, times the radius of the ear squared.

01:14 So r squared.

01:16 Okay.

01:17 Well, we don't know i, but we can find i using the definition of decibels, which says that beta is equal to 10.

01:25 Times log base 10.

01:29 So this is log base 10 of the ratio of the intensity to i -0.

01:37 Okay...

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