When you \"crack\" a knuckle, you cause the knuckle cavity to widen rapidly. Thi

Question

When you "crack" a knuckle, you cause the knuckle cavity to widen rapidly. This, in turn, allows the synovial fluid to expand into a larger volume. If this expansion is sufficiently rapid, it causes a gas bubble to form in a process known as cavitation. This is the mechanism responsible for the cracking sound. If a "crack" produces a sound with an intensity level of 57 dB at your ear, which is 18 cm from the knuckle, how far from your knuckle can the "crack" be heard? Assume the sound propagates uniformly in all direction, with no reflections or absorption.

Explanation / Answer

Given that,

Intensity = 57 dB ; distance = 18 cm

We need to find the distance to which the "crack" can be heard. Let it be D.

We know that,

dB = 10 log ( I / Io) where, Io = 10-12 W/m2

57 = 10 log ( I / 10-12)

57/10 = log ( I / 10-12)

105.7 = I 10-12 =>I = 10(-12+5.7) = 10-6.3 = 5 x 10-7

I/Io = 5 x 10-7 / 10-12 = 5 x 105 .

We know that the intensity of the sound varies with inverse of square of the distance that it 1/x2

5 x 105/x2 = 1 => x = sqrt( 5 x 105 ) = 707.11

So the sound can be heard till, D = x * d = 707.11 x 18 cm = 12728 cm.

Hence, D = 12728 cm = 127.28 m.

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