An organ pipe is 81 cm long and at a temperature of 20 degrees C. What is the fundamental (in Hertz) if the pipe is closed at one end?
To determine the fundamental frequency of a closed organ pipe, we can use the formula:
f = (v/λ) * 0.5
Where:
f is the fundamental frequency (in Hertz)
v is the speed of sound in air
λ is the wavelength of the fundamental mode
First, let's calculate the speed of sound in air at 20 degrees Celsius. The speed of sound can be approximated by the formula:
v = 331.4 + 0.6 * T
Where:
v is the speed of sound (in meters per second)
T is the temperature in degrees Celsius
Plugging in T = 20 into the equation, we have:
v = 331.4 + 0.6 * 20
v = 331.4 + 12
v = 343.4 m/s
Now, let's determine the wavelength of the fundamental mode. For a closed organ pipe, the fundamental mode has a wavelength equal to four times the length of the pipe, so:
λ = 4 * L
Where:
λ is the wavelength (in meters)
L is the length of the pipe (in meters)
Converting the length of the pipe from centimeters to meters, we have:
L = 81 / 100
L = 0.81 m
Plugging this value into the equation, we have:
λ = 4 * 0.81
λ = 3.24 m
Finally, let's substitute the values into the formula for the fundamental frequency:
f = (v/λ) * 0.5
f = (343.4/3.24) * 0.5
f = 105.98 Hz
Therefore, the fundamental frequency of the closed organ pipe is approximately 105.98 Hz.