A train sounds its horn as it approaches an intersection. The horn can just be heard at a level of 53 dB by an observer 11 km away.
(a) What is the average power generated by the horn?
A train sounds its horn as it approaches an intersection. The horn can just be heard at a level of 42.0 dB by an observer 10.0 km away.
(a) What is the average power generated by the horn?
W
(b) What intensity level of the horn's sound is observed by someone waiting at an intersection 60.0 m from the train? Treat the horn as a point source and neglect any absorption of sound by the air.
dB
To find the average power generated by the train's horn, we can use the equation:
Power = Intensity × Area
Where:
Power is the average power generated by the horn (in watts),
Intensity is the sound intensity (in watts per square meter), and
Area is the area over which the sound spreads (in square meters).
First, let's calculate the sound intensity at the observer's location.
The sound intensity (I) is given by the formula:
I = (P / 4πr^2)
Where:
P is the power transmitted by the sound source (in watts), and
r is the distance from the source of sound (in meters).
In this case, the sound intensity is given as 53 dB, which we need to convert to watts per square meter.
The formula to convert decibels to watts per square meter is:
I (watts per square meter) = 10^(dB/10) × I0
Where:
I0 is the reference intensity level, which is 10^(-12) watts per square meter.
Using the given information, we have:
I = 10^(53/10) × (10^(-12)) = 10^(53/10 - 12) watts per square meter
Next, we need to calculate the area over which the sound spreads.
The area of a sphere (A) is defined by the formula:
A = 4πr^2
In this case, the distance from the observer to the sound source is given as 11 km. To convert kilometers to meters, we multiply by 1000:
r = 11 km × 1000 m/km = 11,000 meters
Substituting the values into the equation, we have:
A = 4π(11,000)^2 square meters
Finally, we can calculate the average power generated by the horn using the formula mentioned earlier:
Power = Intensity × Area
Substituting the values we have calculated, we get:
Power = (10^(53/10 - 12)) watts per square meter × 4π(11,000)^2 square meters
We can now calculate this expression to find the average power generated by the horn.