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.
(a) To find the frequency of the lowest note a piccolo can play, we can use the formula for the fundamental frequency of a pipe open at both ends:
f = (v / λ)
Where:
f = frequency
v = velocity of sound (343 m/s)
λ = wavelength
Since the length of the resonating air column in the piccolo is given as 30.6 cm, we need to convert it to meters:
L = 30.6 cm = 0.306 m
The fundamental frequency occurs when the wavelength is twice the length of the resonating air column:
λ = 2L = 2 * 0.306 = 0.612 m
Substituting the values into the formula, we can calculate the frequency:
f = (343 m/s) / (0.612 m) ≈ 561.44 Hz
Therefore, the frequency of the lowest note a piccolo can play is approximately 561.44 Hz.
(b) The distance between adjacent antinodes in a pipe open at both ends can be calculated using the formula:
d = λ / 2
Where:
d = distance between adjacent antinodes
λ = wavelength
In this case, we need to find the wavelength corresponding to the highest note a piccolo can sound, which is given as 41.0 kHz. We can convert it to Hz:
f = 41.0 kHz = 41.0 * 10^3 Hz
Now, we can calculate the wavelength:
λ = v / f = 343 m/s / (41.0 * 10^3 Hz) = 0.00837 m
Finally, we can calculate the distance between adjacent antinodes:
d = λ / 2 = 0.00837 m / 2 ≈ 0.00419 m
Therefore, the distance between adjacent antinodes for this mode of vibration is approximately 0.00419 m.
To find the frequency of the lowest note a piccolo can play, we need to determine the fundamental frequency of the resonating air column. In a pipe that is open at both ends, the fundamental frequency is given by the formula:
f = v / (2L)
Where:
f is the frequency
v is the speed of sound
L is the length of the air column
Given that the overall length of the piccolo is 30.6 cm (or 0.306 m) and the speed of sound is 343 m/s, we can substitute these values into the formula to find the frequency:
f = 343 m/s / (2 * 0.306 m) ≈ 561.44 Hz
So, the frequency of the lowest note a piccolo can play is approximately 561.44 Hz.
To determine the distance between adjacent antinodes for the mode of vibration corresponding to the highest note a piccolo can sound (41.0 kHz or 41000 Hz), we need to use the wavelength (λ) formula:
λ = v / f
Where:
λ is the wavelength
v is the speed of sound
f is the frequency
Rearranging the formula, we can solve for the wavelength:
λ = v / f = 343 m/s / (41000 Hz) ≈ 0.0084 m
Since the distance between adjacent antinodes is equal to half the wavelength (λ/2), we have:
Distance between adjacent antinodes = 0.0084 m / 2 ≈ 0.0042 m
So, the distance between adjacent antinodes for this mode of vibration is approximately 0.0042 m.