The ability to hear a "pin drop" is the sign of sensitive hearing. Suppose a 0.50 g pin is dropped from a height of 27 cm, and that the pin emits sound for 1.4 s 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.)
Energy release = mgh = 1.32*10^-3 J
Avg. sound power = mgh/1.4s
= 0.95*10^-4 Watts
Avg. Sound intensity at distance R
= 0.95*10^-4 Watts/(4*pi*R^2)
= 7.5*10^-6/R^2 W/m^2
Set this equal to the minimum sound intensity level detectable by the human ear (about 10^-12 W/m^2), and solve for R.
should be 7.5*10^-5/R^2 W/m^2
To find the maximum distance from which a person can hear the pin drop, we need to calculate the sound intensity at that distance using the given information.
First, let's find the initial potential energy of the pin at a height of 27 cm.
Potential Energy (PE) = mass (m) * gravitational acceleration (g) * height (h)
PE = 0.50 g * 9.8 m/s^2 * 0.27 m
PE = 0.1323 J
Since all the mechanical energy is converted into sound energy, this potential energy will be equal to the sound energy emitted by the pin (assume no other losses).
Sound Energy (E) = 0.1323 J
The sound energy is related to the sound intensity (I) by the equation:
I = E / A * t
Where A is the surface area of the imaginary spherical shell through which the sound is radiated, and t is the time for which the sound is emitted.
To calculate the maximum distance (R), we can use the inverse square law for sound intensity:
I = P / (4πR^2)
Where P is the sound power (watts) and R is the distance from the source.
Rearranging the equation:
R = sqrt(P / (4πI))
Since P is equal to sound Energy divided by the time t:
R = sqrt((E / t) / (4πI))
Let's plug in the given values:
R = sqrt((0.1323 J / 1.4 s) / (4πI))
Now, we need to calculate the sound intensity (I). For a spherically symmetric sound wave, the sound intensity (I) decreases equally in all directions. Thus, if the maximum distance (R) is reached, the intensity is equal to the threshold of hearing (I0), which is approximately 1 x 10^-12 W/m^2.
R = sqrt((0.1323 J / 1.4 s) / (4π * 1 x 10^-12 W/m^2))
Simplifying further:
R = sqrt((0.0945 x 10^12 W) / (4π * 1 x 10^-12 W/m^2))
R = sqrt(7.362 x 10^24 m^2)
R ≈ 8.59 x 10^12 m
Therefore, the maximum distance from which a person can hear the pin drop (assuming ideal conditions) is approximately 8.59 x 10^12 meters.
To find the maximum distance from which a person can hear the pin drop, we need to first calculate the total sound energy emitted by the pin when it lands.
The mechanical energy of the pin can be calculated using the formula:
E = mgh
where E is the mechanical energy, m is the mass of the pin, g is the acceleration due to gravity, and h is the height from which the pin is dropped.
In this case, the mass of the pin is given as 0.50 g, which is equivalent to 0.50 × 10^(-3) kg (since 1 g = 10^(-3) kg). The height from which the pin is dropped is given as 27 cm, which is equivalent to 27 × 10^(-2) m. The acceleration due to gravity, g, is approximately 9.8 m/s^2.
So, substituting the given values into the formula, we have:
E = (0.50 × 10^(-3) kg) × (9.8 m/s^2) × (27 × 10^(-2) m)
Simplifying the expression, we get:
E = 0.1323 J (Joules)
Now, since we know that all of the mechanical energy of the pin is converted to sound energy, we can say that this is the total sound energy emitted by the pin.
The sound energy received by a listener decreases with distance due to the spreading of sound waves in all directions. The spreading of sound waves follows the inverse square law, which states that the intensity of sound decreases as the square of the distance from the source.
The sound intensity, I, is given by the formula:
I = P / (4πr^2)
where I is the sound intensity, P is the power of the sound source, and r is the distance from the sound source.
We are interested in finding the maximum distance, so we rearrange the above formula to solve for r:
r = √(P / (4πI))
However, the power of the sound source is equal to the total sound energy emitted divided by the time for which the sound is emitted:
P = E / t
where P is the power, E is the sound energy, and t is the time.
Substituting the values into the formula, we have:
P = 0.1323 J / 1.4 s
Simplifying the expression, we get:
P ≈ 0.0945 W (Watts)
Now, substituting the power value and the total sound energy into the distance formula, we have:
r = √((0.1323 J / (1.4 s)) / (4πI))
Since the problem doesn't provide a specific value for the sound intensity (I), we can use a reference value. For example, the minimum sound intensity that a human can hear is usually taken to be 1.0 × 10^(-12) W/m^2.
Substituting this value into the formula, we get:
r = √((0.1323 J / (1.4 s)) / (4π(1.0 × 10^(-12) W/m^2)))
Simplifying the expression, we can approximate π as 3.14:
r ≈ √((0.1323 J / (1.4 s)) / (4 × 3.14 × 1.0 × 10^(-12) W/m^2))
Evaluating the expression, we find:
r ≈ 24.7 m
Therefore, the maximum distance from which a person can hear the pin drop, assuming ideal conditions, is approximately 24.7 meters. However, it's important to note that actual maximum distance will be smaller due to factors like atmospheric absorption and other sound-related factors.