Calculating Energy Received by Ear

[FeDeX_LaTeX]

Hello;

This is not a homework question, but it's from a physics olympiad paper.

Homework Statement

A pin dropped on a hard floor on the far side of a quiet room can be heard by the human ear.

(i) If the pin has a mass of 0.2 g, and is dropped from a height 1 m onto a hard floor, with 10% of the energy being converted into sound, calculate the sound energy released.

(ii) If the eardrum (which we can assume is circular) has a diameter of 6 mm, and one human ear can detect the sound of a pin at a distance of 5 m, estimate the energy received by the ear. State any assumptions you make.

Homework Equations

Surface area of sphere = 4 \pi r^{2}

\bigtriangleup E = mgh

Take g = 9.8 m/s^2

The Attempt at a Solution

I think I can do the first one. Since 0.2g = 0.0002 kg, then since

(i) change in GPE = mgh

then ΔE = 0.0002*9.8*1 = 0.00196 J

Since 10% of this energy is converted to sound, multiply this by 0.1 to get 0.000196 J, or 1.96 x 10^-4 J.

The second question I'm not sure about.

(ii) If we're assuming the eardrum is circular, then the energy is spread over a hemisphere at the ear, whose area is 2 \pi r^{2}. Do we assume also that the energy is spread from the pin to the ear in a cone-like fashion? I'm not sure where to go from here.

Thanks.

 

[cepheid]

The sound propagates from the source in all directions, so a spherical sound wave seems natural (rather than a cone like you suggested).

As the distance from the source increases, the same amount of energy is spread out over a larger and larger surface area (the area of the propagating spherical wavefront). So you could figure out how much energy per unit area is received at the distance of the ear. Combining that with the total area of the ear, you could figure out how much total energy is received.