Question

How long must a flute be in order to have a fundamental frequency of 262 Hz (this frequency corresponds to middle C on the evenly tempered chromatic scale) on a day when air temperature is 20.0ºC? It is open at both ends.

a) 0.65 m
b) 0.80 m
c) 1.30 m
d) 2.60 m

Answer 1

The length of the flute needed to have a fundamental frequency of 262 Hz at 20.0°C is approximately 0.65 meters, which makes the answer option (a) the closest match.

Step-by-step explanation:

The length of a flute needed to produce a fundamental frequency of 262 Hz, which corresponds to middle C on the evenly tempered chromatic scale, can be calculated using the physics of standing waves in an open pipe. The speed of sound in air at a temperature of 20.0°C is approximately 343 m/s. Since the flute is open at both ends, the fundamental mode of vibration will have a node at each end and an antinode in the middle, which is equivalent to ½ of a wavelength.

To find the wavelength (λ) for the given frequency (f), we use the formula λ = v / f, where v is the speed of sound. Plugging in the given values, we have λ = 343 m/s / 262 Hz ≈ 1.31 meters. However, since the flute is open at both ends and the fundamental frequency corresponds to ½ wavelength, the length of the flute (L) must be ½λ. Therefore, L = ½ × 1.31 meters = 0.655 meters.

The closest option to 0.655 meters is 0.65 meters, making this the answer to the question. To play the fundamental frequency of middle C at 262 Hz, the flute should be approximately 0.65 meters long.