Final answer:
To achieve a fundamental frequency of 262 Hz, corresponding to middle C, a flute open at both ends and in a 20.0°C environment must be approximately 0.655 meters long.
Step-by-step explanation:
To determine how long a flute must be in order to have a fundamental frequency of 262 Hz, which is the frequency of middle C, we must use the physics of wave behavior in air columns.
A flute is an example of an open pipe, which means both ends of the flute are open. In an open pipe, the fundamental frequency (or the first harmonic) produces a wavelength that is twice the length of the pipe.
The speed of sound in air is dependent on the temperature of the air. At 20.0°C, the speed of sound in air is approximately 343 meters per second (m/s). We can use the formula for the fundamental frequency of an open pipe, which is:
f = ½ · (ν / L)
Where f is the frequency, ν is the speed of sound in the medium, and L is the length of the tube. Solving for L, we have:
L = ½ · (ν / f)
Substituting the values we have:
L = ½ · (343 m/s / 262 Hz)
By calculating this, we find that the length of the flute must be:
L ≈ 0.655 meters
Therefore, in order to achieve the fundamental frequency of 262 Hz, corresponding to middle C, at a temperature of 20.0°C, a flute open at both ends would need to be approximately 0.655 meters long.