The ability to hear a "pin drop" is the sign of sensitive hearing. Suppose a 0.40 pin is dropped from a height of 30 , and that the pin emits sound for 1.4 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. (This is the ideal maximum distance, but atmospheric absorption and other factors will make the actual maximum distance considerably smaller.)

maximum distance, r = ?

To find the maximum distance from which a person can hear the pin drop, we need to calculate the sound intensity at that distance. Sound intensity is the amount of sound energy passing through a unit area perpendicular to the direction of sound propagation per unit time.

First, let's calculate the mechanical energy of the falling pin. We know that the height (h) from which the pin is dropped is 30 m and the mass (m) of the pin is 0.40 kg. The potential energy (PE) of the pin at that height is given by PE = mgh, where g is the acceleration due to gravity (approximately 9.8 m/s^2).

PE = (0.40 kg)(9.8 m/s^2)(30 m) = 117.6 J

Since all of the mechanical energy is converted to sound energy, the sound energy will be 117.6 J.

Next, we need to calculate the sound intensity at the maximum distance. Sound intensity (I) is related to the sound power (P) and the spherical surface area (A) by the formula I = P/A.

Since the sound radiates uniformly in all directions, the sound energy propagates spherically as it expands from the source. The surface area of a sphere is given by 4πr^2, where r is the distance from the source (maximum distance in this case).

To determine the sound power (P), we can use the time (t) the sound radiates for when the pin lands. The power (P) is given by P = E/t, where E is the sound energy and t is the time.

P = 117.6 J / 1.4 s ≈ 84 J/s

Now, let's substitute the values back into the formula for sound intensity:

I = P / A

Since the sound propagates spherically, the surface area (A) is given by 4πr^2:

I = P / (4πr^2)

We want to find the maximum distance (r) at which the sound can still be heard, so we rearrange the formula to solve for r:

r = √(P / (4πI))

Substituting the values into the equation:

r = √(84 J/s / (4πI))

The maximum distance (r) from which a person can hear the pin drop would be the value of r calculated using the above formula, where I is the sound intensity threshold for a person to perceive the sound of a pin drop. Keep in mind that this is the ideal maximum distance, but in the real world, atmospheric conditions and other factors will reduce the actual maximum distance considerably.