Sound is detected when a sound wave causes the tympanic membrane (the eardrum) to vibrate (see the figure ). Typically, the diameter of this membrane is about 8.40 {\rm mm} in humans.

a.)how much energy is delivered to the eardrum each second when someone whispers (20.0 {\rm 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 2.00 {\rm mg} mosquito would have to fly (in {\rm mm/s}) to have this amount of kinetic energy.

a) Energy delivered per second to the tympanic membrane = 5.54 × 10⁻¹⁵ J/s

b) velocity of mosquito that will generate that amount of energy, v = 0.0000744 m/s = 0.0744 mm/s.

Explanation:

a) [D] = 10 log (I/I₀)

I₀ = 10⁻¹² W/m²

Given the sound intensity level in decibels, we need to obtain the corresponding sound intensity.

20 = 10 log (I/(10⁻¹²))

2 = log (I/(10⁻¹²))

100 = (I/(10⁻¹²))

I = 10⁻¹⁰ W/m²

Power experienced by the tympanic membrane of the ear due to the sound intensity = Intensity × Area of the membrane

Area of the membrane = πD²/4 = π(8.4 × 10⁻³)²/4 = 5.54 × 10⁻⁵ m²

Power = 10⁻¹⁰ × 5.54 × 10⁻⁵ = 5.54 × 10⁻¹⁵ W

Energy delivered per second to the tympanic membrane = 5.54 × 10⁻¹⁵ J/s

b) Kinetic energy = mv²/2

5.54 × 10⁻¹⁵ = (2 × 10⁻⁶)v²/2

v² = (2 × 5.54 × 10⁻¹⁵)/(2 × 10⁻⁶)

v = 0.0000744 m/s = 0.0744 mm/s.