The ability to hear a “pin drop” is the sign of sensitive hearing. Suppose a 0.55-g pin is dropped from a height of 28 cm, and that the pin emits sound for 1.5s when it lands. Assuming all of the mechanical energy of the pin is converted to sound energy, and that
the sound radiates uniformly in all directions, find the maximum distance from which a person can hear the pin drop. (The sound intensity at the threshold of hearing is Io = 10-12 W/m2.)
I = P/4pi r^2 (for spherical dissipation)
solve for r:
r = sqrt(P/I4pi)
In this case power is given by mgh/t.
Watch your units, kg and m only.
To find the maximum distance from which a person can hear the pin drop, we need to calculate the sound intensity at that distance. Here's how you can do it:
1. Calculate the initial potential energy (PE) of the pin when it is at a height of 28 cm. Use the formula PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height. In this case, m = 0.55 g, g = 9.8 m/s^2, and h = 0.28 m.
PE = (0.55 g) * (9.8 m/s^2) * (0.28 m)
2. Since all the mechanical energy is converted into sound energy, the initial potential energy is equal to the sound energy emitted. Convert this to watts by using the conversion factor 1 J = 1 W·s.
Sound energy emitted = PE = (0.55 g) * (9.8 m/s^2) * (0.28 m) * (1 J / 1 W·s)
Note: We assume the energy emitted is spread over the entire 1.5 s of sound emission.
3. Calculate the sound power (P) using the formula P = E / t, where E is the sound energy emitted and t is the time for which the pin emits sound. In this case, E is the energy emitted calculated in step 2, and t = 1.5 s.
P = (0.55 g) * (9.8 m/s^2) * (0.28 m) * (1 J / 1 W·s) / (1.5 s)
4. Calculate the sound intensity (I) at a distance r using the formula I = P / (4πr^2), where r is the distance from the source. Rearrange the formula to solve for r:
r = √(P / (4πI))
In this case, P is the sound power calculated in step 3, and I is the sound intensity at the threshold of hearing (I0 = 10^-12 W/m^2).
r = √(((0.55 g) * (9.8 m/s^2) * (0.28 m) * (1 J / 1 W·s) / (1.5 s)) / (4π * (10^-12 W/m^2)))
Now, you can calculate the maximum distance from which a person can hear the pin drop.