Unless indicated otherwise, assume the speed of sound in air to be v=344ms. Sound is...

Salvatore Boone

Salvatore Boone

Answered

2022-01-10

Unless indicated otherwise, assume the speed of sound in air to be v=344ms. Sound is detected when a sound wave causes the tympanic membrane (the eardrum) to vibrate. Typically, the diameter of this membrane is about 8.4 mm in humans.
(a) How much energy is delivered to the eardrum each second when someone whispers (20 dB) a secret in your ear?
(b) To comprehend how sensitive the ear is to very small amounts of energy, calculate how fast a typical 20-mg mosquito would have to fly (in mms) to have this amount of kinetic energy.

Answer & Explanation

Navreaiw

Navreaiw

Expert

2022-01-11Added 34 answers

a) Our goal is to find the intensity of the sound wave, then to make use of the value of the intensity to determine the energy being delivered to the eardrum each second.
We can calculate the intensity of the sound wave using the relation that describes the intensity level of a sound wave, which is
β=10log(II0) (1)
Where β=20dB,I0=1012Wm2 and I is the intensity of the sound wave that we would like to determine.
Hence
20dB=10log(I1012Wm2)
2=log(I1012Wm2)
Remember that 10log(x)=x, hence
102=10log(I1012Wm2)
I1012Wm2=102
I=1010Wm2
But we know that intensity is energy per unit time per unit area (Power per unit area), which could be represented mathematically as follows
E=IA (2)
Where A is the area of the tympanic membrane and it can be calculated as follows
A=πr2=π×(4.2×103m)2=5.54×105m2
Now, substitute into (2) with 5.54×105m2 for A,1010Wm2 for I and 1 s for Notice that t=1s since we want to calculate the energy is delivered each second
E=(1010Wm2)×(5.54×105m2)×(1s)
E=5.54×1015J
le(1,0){370}
Deufemiak7

Deufemiak7

Expert

2022-01-12Added 34 answers

b) The velocity of a mosquito with a mass of 2.0 mg and has a kinetic energy of 5.54×1015J can be calculated as follows
KE=12mv2
v=2×KEm
To get v, substitute with 5.54×1015J tor KE and 2×106kg
v=2×(5.54×1015J)2×106kg
v=0.074×103ms
v=0.074mms

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