A person is initially 8 m from a point sound source which emits energy uniformly in all direction at a constant rate. If the intensity of the source is to be halved but the sound is to be as loud as before, what distance should the person be from the source?
Ans: 4 square root 2 m
We know that the intensity of the omnidirectional source is proportional the energy rate and inversely proportional to the square of the distance (inverse square law).
So
1/8² = 0.5/d²
Solve for d.
The loudness is the amount of power passing through a square meter of area surrounding the source. As you get further from a constant power source, the loudness decreases because the area of a sphere surrounding the source increases.
( You know that. As you get further from a loudspeaker the loudness goes down)
d^2 = .5 * 8 * 8 = 4*4*2
d = 4 * sqrt 2
or about 5.66 meters
Intensity is the same? But the question states that it is halved?
I don't quite get it........
Why did the person want to be closer to the sound source? Because they heard it was "halving" a good time! So, if they want to halve the intensity but still hear the sound as loud as before, they should move closer to the source. But not just any distance, they need to be at a distance of 4 square root 2 m. That's some sound math!
To solve this question, we can use the inverse square law for sound intensity. According to the inverse square law, the intensity of sound decreases as the square of the distance from the source increases.
Let's say "I" is the initial intensity of the sound and "r" is the initial distance from the source. We are given that the intensity needs to be halved, so the new intensity will be "I/2". We need to find the new distance, which we'll call "d".
According to the inverse square law, we can set up the following equation:
(I/2) = I * (r^2)/(d^2)
Now, let's simplify the equation:
(I/2) = I * (r^2)/(d^2)
1/2 = (r^2)/(d^2)
d^2 = 2 * r^2
d = sqrt(2 * r^2)
d = sqrt(2) * sqrt(r^2)
d = sqrt(2) * r
Substituting the given value for "r" (8 m), we can find the new distance:
d = sqrt(2) * 8
d = 8 * sqrt(2)
Therefore, the person should be approximately 8 * sqrt(2) meters (recalling that sqrt(2) is approximately 1.414) from the source, which is approximately 11.31 meters.