A stereo speaker emits sound waves with a power output of 100W. (a) find the intensity 10.0m from the source. B) find the intensity level in decibels, at this distance. C) at what distance would you experience the sound at the threshold of pain,120dB?
To answer these questions, we can use the formulas related to sound intensity and decibels (dB). Let's go through each question step by step:
a) Finding the intensity at a distance of 10.0m from the source:
The intensity of sound waves follows an inverse square law. The formula for intensity is I = P/A, where I is the intensity, P is the power output, and A is the area through which the sound waves pass.
In this case, the power output is given as 100W. To find the intensity at 10.0m, we need to calculate the area of a sphere centered at the source with a radius equal to the distance from the source to the point of interest.
The area of a sphere is given by the formula A = 4πr^2, where r is the radius.
So, in this case, r = 10.0m. Plugging the values into the formula, we get:
A = 4π(10.0)^2 = 400π m^2.
Now, we can find the intensity:
I = P/A = 100W/400π m^2 = 0.07958 W/m^2 (rounded to 5 decimal places).
Therefore, the intensity at a distance of 10.0m from the source is approximately 0.07958 W/m^2.
b) Finding the intensity level in decibels (dB) at the same distance:
The formula to calculate the intensity level (L) in decibels is L = 10 log10(I/I0), where I is the intensity and I0 is the reference intensity.
The reference intensity I0 is the threshold of hearing, which is generally considered to be 10^(-12) W/m^2.
Plugging in the values, we can calculate the intensity level at 10.0m:
L = 10 log10(0.07958/10^(-12)) = 96.9 dB (rounded to one decimal place).
Therefore, the intensity level at a distance of 10.0m from the source is approximately 96.9 dB.
c) Finding the distance at which the sound would reach the threshold of pain (120 dB):
To find the distance at which the sound is at the threshold of pain, we can rearrange the formula from part b) to solve for distance (D):
L = 10 log10(I/I0)
120 = 10 log10(I/10^(-12))
12 = log10(I/10^(-12))
Converting the logarithmic equation to exponential form:
I/10^(-12) = 10^12
I = 10^0 = 1 W/m^2
Now, we can calculate the distance D:
I = P/A = 100/4πD^2
Plugging in the values, we find:
1 = 100/4πD^2
D^2 = 100/4π
D = √(100/4π) ≈ 2.52 m (rounded to two decimal places).
Therefore, you would experience the sound at the threshold of pain (120 dB) at a distance of approximately 2.52 meters from the source.