A trombone player stands at the end zone (x=0) of a football field and begins to play its fundamental tone. Assume the trombone is a half open tube that is 3 m long. How much would the trombone player have to move the slider in cm to play 25 hertz?
To calculate how much the trombone player would have to move the slider to play at a frequency of 25 hertz, we first need to determine the wavelength of the fundamental tone.
The wavelength of the fundamental tone in a half-open tube is equal to four times the length of the tube. Therefore, the wavelength (λ) is:
λ = 4 * length = 4 * 3 m = 12 m
The formula for the frequency of a wave is:
frequency = speed of sound / wavelength
The speed of sound in air at room temperature is approximately 343 m/s. Therefore, the frequency of the fundamental tone is:
25 Hz = 343 m/s / λ
Rearranging the formula to solve for λ:
λ = 343 m/s / 25 Hz = 13.72 m
Since the trombone player needs to adjust the length of the tube to match the wavelength of 25 Hz, the new length (L') is:
L' = λ / 4 = 13.72 m / 4 = 3.43 m
The trombone player would have to move the slider by the difference between the new length and the original length:
ΔL = L' - 3 m = 3.43 m - 3 m = 0.43 m = 43 cm
Therefore, the trombone player would have to move the slider by 43 cm to play the fundamental tone at a frequency of 25 hertz.