At T= 20 degrees C, how long (in meters) must an open organ pipe be if it is to have a fundamental frequency of 240.0 Hz?
To determine the length of an open organ pipe required to produce a specific frequency, we can use the formula for the fundamental frequency of an organ pipe:
f = (v/2L) * n
Where:
f represents the frequency (in Hz)
v represents the speed of sound in the medium (which depends on temperature)
L represents the length of the organ pipe (in meters)
n represents the harmonic number (1 for the fundamental frequency)
In this case, we need to find the length of the organ pipe (L), so we can rearrange the formula:
L = (v/2f) * n
Now, we need to consider the speed of sound (v) at the given temperature T = 20 degrees Celsius. The speed of sound in air can be approximated using the formula:
v = 331.4 + (0.6 * T)
Substituting the given temperature into the equation, we get:
v = 331.4 + (0.6 * 20)
v = 331.4 + 12
v = 343.4 m/s
Next, we substitute the values into the formula for the length (L) of the organ pipe:
L = (343.4 / (2 * 240)) * 1
L = (171.7) * 1
L = 171.7 meters
Therefore, the open organ pipe must be approximately 171.7 meters long to produce a fundamental frequency of 240.0 Hz at a temperature of 20 degrees Celsius.