Homework # 12
Due: 10:00pm on Thursday, December 10, 2009
Note:
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Ear Damage from a Small Firecracker
Description:
Find the sound intensity level one meter away from a firecracker. Then determine at what distance the sound of the
firecracker would rupture be intense enough to rupture an eardrum.
Learning Goal:
To understand how to convert between different sound intensity scales and how the decibel intensity of a sound
changes with distance.
The decibel scale is logarithmic in intensity:
.
In this formula,
is a reference intensity, which, for sound waves, is taken to be
. This constant must be used to convert a
particular physical intensity into a sound intensity level measured in decibels.
Once we know the sound intensity level (in decibels) at a certain reference distance from a sound source, the
decrease of intensity
with distance can be accounted for by
subtracting
the decibel value appropriate to the ratio of the new distance to the reference distance.
In this problem you will use the decibel scale to analyze a small firecracker that emits 1200
of peak power. To avoid confusion,
intensities denoted by
are in units of watts per meter squared; intensities denoted by
are in units of decibels.
Part A
What is the peak intensity
in decibels at a distance of 1 m from the firecracker?
Hint A.1
Find
at 1 m
Find the peak intensity
in watts per meter squared of the sound from the firecracker at a distance of 1
.
Hint A.1.1
Start with geometry
All of the power from the firecracker has to pass through a sphere 1
in radius. What is the area
of this sphere?
Express your answer numerically, in square meters, to two decimal places.
ANSWER:
=
Express your answer numerically to one decimal place.
ANSWER:
=
Hint A.2
Converting intensity to dB
Use the formula in the introduction to convert
into decibels. Note that the reference intensity is
.
Express
in decibels to the nearest integer.
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12/12/09 3:47 AM
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ANSWER:
=
dB
Part B
It takes a sound intensity of about 160 dB to rupture the human eardrum. How close must the firecracker described in the introduction
be to the ear to rupture the eardrum?
Hint B.1
How to approach the problem
This problem can be worked from first principles. To do so, figure out the intensity in watts per meters squared corresponding to 160
, then divide this quantity into the 1200W peak power of the firecracker. The result will be the surface of a sphere, centered on
the firecracker, whose radius is the distance at which the sound intensity is 160
.
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 Spring '10
 physics
 Physics, Work, sound intensity level

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