One method for measuring the speed of sound uses standing waves. A cylindrical tube is open at both ends, and one end admits sound from a tuning fork. A movable plunger is inserted into the other end at a distance L from the end of the tube where the tuning fork is. For a fixed frequency, the plunger is moved until the smallest value of L is measured that allows a standing wave to be formed. Suppose that the tuning fork produces a 456-Hz tone, and that the smallest value observed for L is 0.269 m. What is the speed of sound in the gas in the tube?
Open at one end? Is that not a quarter of a wavelength in the pipe, a node at the closed end and a maximum at the entrance?
So
L = .25 * wavelength = .269 meter
so wavelength = 4*.269
period time = 1/456
speed = distance/time
= 4*.269 /(1/456)
= 456 * 4 *.269 meters/ second
To find the speed of sound in the gas in the tube, we can use the formula:
v = f λ
where:
v is the speed of sound,
f is the frequency of the tuning fork,
and λ is the wavelength of the sound wave.
In this case, the frequency is given as 456 Hz. To find the wavelength, we need to calculate it using the distance L.
The standing wave in the tube has nodes at the ends (open ends), so the length of the tube (2L) must be equal to a whole number of half-wavelengths.
Since the smallest value of L is measured, the standing wave pattern that corresponds to this condition is the fundamental frequency, where the length of the tube is equal to a quarter of the wavelength.
Thus, we have:
2L = (1/4)λ
Rearranging the equation, we get:
λ = (8L)
Where L is 0.269 m, given in the question.
λ = 8 * 0.269 m
λ = 2.152 m
Now, we can substitute the values of f and λ into the speed of sound formula to find the speed of sound (v):
v = (456 Hz) * (2.152 m)
v = 981.312 m/s
Therefore, the speed of sound in the gas in the tube is approximately 981.312 m/s.
To find the speed of sound in the gas in the tube, we can use the formula:
v = f * λ
where v is the speed of sound, f is the frequency, and λ is the wavelength.
First, let's find the wavelength. In a closed-open tube, the fundamental mode of resonance occurs when the length of the tube is equal to one-fourth of the wavelength.
Since the smallest value observed for L is given as 0.269 m, we can set this as one-fourth of the wavelength:
L = λ/4
Rearranging the equation, we find:
λ = 4L
Substituting the given value of L, we have:
λ = 4 * 0.269 m = 1.076 m
Now that we have the wavelength, we can use the formula v = f * λ to find the speed of sound.
Substituting the given frequency f = 456 Hz and the calculated wavelength λ = 1.076 m, we get:
v = 456 Hz * 1.076 m = 491.456 m/s
Therefore, the speed of sound in the gas in the tube is approximately 491.456 m/s.