How Far Must You Be from a Sound Source for It to Become Painful?

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Homework Statement

The loudness of a sound heard at a point, a distance d = 10m from a small source of 125Hz, is 60phons. Provided that the intensity level of the sound heard at the point given is 70dB.

The threshold of pain is 120dB for all frequencies. How far from the source will a normal ear feel painful?

Homework Equations

For a small source, the intensity of sound, I is inversely proportional to d^2.
And intensity level = 10*log(I1 / I2)

The Attempt at a Solution

Let the required distance be x.

so, I1 / I2 = d2^2 / d1^2

70/120 = x^2 / 10^2

x = 7.64m

But the solution is:

Since intensity level difference = 120 -70 = 50dB

So, 50 = 10*log(10^2 / x^2)

x = 0.032m

What's wrong with my answer? Thank you so much!

 

[anathema]

Thank you for your post and for your attempt at solving this problem. Your solution is almost correct, but there are a few small errors that led to your incorrect answer.

Firstly, when using the equation for intensity level, it is important to note that the intensity levels being compared must be in the same units. In this case, both intensity levels are given in decibels (dB), so there is no need to convert them. Therefore, the correct equation would be:

70 = 10*log(10^2 / x^2)

Secondly, your mistake lies in the calculation of the intensity level difference. Instead of subtracting the two intensity levels, we need to use the ratio of the two intensity levels. In this case, it would be:

50 = 10*log(10^2 / x^2)

Solving for x, we get:

x = 0.1m

Therefore, a normal ear would feel pain at a distance of 0.1m from the source.

Hope this helps clarify your doubts. Keep up the good work!

Scientist

 

[m2003]

Your answer is incorrect because you have used the incorrect formula for finding the distance at which the sound will be painful for a normal ear. The correct formula to use in this case is the inverse square law for sound intensity, which states that the intensity of sound is inversely proportional to the square of the distance from the source. This means that as the distance from the source increases, the intensity of the sound decreases by a factor of the square of the distance. In your attempt at a solution, you have used the formula for finding the distance at which the intensity level of the sound will be 70dB, which is not the same as the distance at which the sound will be painful for a normal ear. To find the distance at which the sound will be painful, you need to use the formula for finding the distance at which the intensity level of the sound will be 120dB (the threshold of pain). This will give you the correct answer of 0.032m.