The overall length of a piccolo is 30.6 cm. The resonating air column vibrates as in a pipe that is open at both ends. Assume the speed of sound is 343 m/s. (a) Find the frequency of the lowest note a piccolo can play, assuming the speed of sound in air is 343 m/s. (b) Opening holes in the side effectively shortens the length of the resonant column. If the highest note a piccolo can sound is 41.0 kHz, find the distance between adjacent antinodes for this mode of vibration. Assume the speed of sound is 343 m/s.
To find the frequency of the lowest note a piccolo can play, we can use the formula:
f = v / λ
where:
f = frequency
v = speed of sound
λ = wavelength
Since the resonating air column vibrates as in a pipe that is open at both ends, the wavelength for the lowest note will be twice the length of the piccolo (L).
(a) Frequency of the lowest note:
Given: L = 30.6 cm = 0.306 m
v = 343 m/s
λ = 2L = 2 * 0.306 = 0.612 m
f = v / λ = 343 / 0.612 = 560.46 Hz
Therefore, the frequency of the lowest note a piccolo can play is approximately 560.46 Hz.
To find the distance between adjacent antinodes for the highest note a piccolo can sound, we can use the formula:
λ = 2L / n
where:
λ = wavelength
L = length of the piccolo
n = number of antinodes
(b) Distance between adjacent antinodes:
Given: L = 0.306 m
f = 41.0 kHz = 41,000 Hz
v = 343 m/s
First, we need to find the wavelength (λ) for the highest note:
f = v / λ
λ = v / f = 343 / 41000 = 0.00839 m
Since there are n antinodes in one wavelength, we can write:
0.00839 = 2L / n
n = 2L / 0.00839 = 2 * 0.306 / 0.00839 = 73.02
Therefore, the distance between adjacent antinodes for this mode of vibration is approximately 73.02 cm.
To find the frequency of the lowest note a piccolo can play, we need to determine the fundamental frequency of the resonating air column.
(a) The fundamental frequency is given by the formula:
f = v / λ
where f represents the frequency, v is the speed of sound, and λ is the wavelength.
In this case, the resonating air column vibrates as in a pipe that is open at both ends. For a pipe open at both ends, the wavelength of the fundamental frequency is equal to twice the length of the pipe:
λ = 2 * L
Given that the overall length of the piccolo is 30.6 cm, we convert it to meters:
L = 30.6 cm = 0.306 m
Substituting these values into the formula, we have:
f = v / λ = v / (2 * L)
Plugging in the speed of sound in air, v = 343 m/s, and the length of the piccolo, L = 0.306 m, we can calculate the frequency of the lowest note:
f = 343 / (2 * 0.306)
(b) To find the distance between adjacent antinodes for the highest note a piccolo can sound, we need to determine the wavelength of this mode of vibration.
Given that the highest note frequency is 41.0 kHz, we convert it to Hz:
f = 41.0 kHz = 41.0 × 10^3 Hz
The wavelength of this frequency can be calculated using the formula:
λ = v / f
Substituting the values of the speed of sound in air, v = 343 m/s, and the frequency, f = 41.0 × 10^3 Hz, we can find the wavelength:
λ = 343 / (41.0 × 10^3)
Finally, to find the distance between adjacent antinodes, we can use the relationship between wavelength and distance between adjacent antinodes. In a pipe open at both ends, the distance between adjacent antinodes is equal to half the wavelength:
Distance between adjacent antinodes = λ / 2
By substituting the calculated wavelength into the formula, we can find the distance between adjacent antinodes.