The water level in a vertical glass tube 1.40 m long can be adjusted to any position in the tube. A tuning fork vibrating at 440 Hz is held just over the open top end of the tube, to set up a standing wave of sound in the air-filled top portion of the tube. (That air-filled top portion acts as a tube with one end closed and the other end open.)
(a) For how many different positions of the water level will sound from the fork set up resonance in the tube's air-filled portion, which acts as a pipe with one end closed (by the water) and the other end open?
(b) What is the least water height in the tube for resonance to occur?
(c) What is the second least water heights in the tube for resonance to occur?
Hi, I got the answer to part A but not B or C...
A) use F=(nV)/(2L)
F=frequency
n=number of nodes
V=speed of sound (usually 343 m/s)
L=length of tube
rearrange the equation to solve for n,
round down to the nearest integer...
in my notes it says the (2L) should actually be (4L), not sure why it worked for me when I used (2L)
To understand how sound resonance in a tube depends on the water level, we need to examine the fundamental concept of resonance in pipes. In a closed-end pipe, such as the air-filled portion of the tube in question, the air column can vibrate at specific frequencies when excited by an external sound source.
(a) To determine the number of different water levels that will result in resonance, we can use the formula for the fundamental frequency of a closed-end pipe:
f = (v / λ) * n
Where:
- f is the frequency of the sound wave (440 Hz in this case, as given)
- v is the speed of sound in air (approximately 343 m/s at room temperature)
- λ is the wavelength of the sound wave
- n is the harmonic number
For resonance to occur, the length of the air-filled portion of the tube must be equal to an integer multiple of half the wavelength (λ/2) of the sound wave. Considering that the tube has one closed end (at the water level) and one open end, the fundamental frequency corresponds to the first harmonic (n = 1).
Given that the tube is 1.40 m long, we can write:
1.40 m = λ/2
Rearranging the equation, we find:
λ = 2 * 1.40 m = 2.80 m
Now, we can calculate the number of different water levels that will result in resonance by considering the relationship between wavelength and tube length. Since the wavelength must be an integer multiple of the tube length, we have:
2.80 m = λ * n
Simplifying this equation, we find:
n = 2.80 m / 1.40 m = 2
Therefore, for resonance to occur, there will be two different water levels in the tube's air-filled portion.
(b) To determine the minimum water height for resonance to occur, we need to calculate the wavelength (λ) corresponding to the first harmonic (n = 1). Using the equation from part (a), we find:
1.40 m = λ/2
Solving for λ:
λ = 2 * 1.40 m = 2.80 m
Since the fundamental frequency (f) is given as 440 Hz, we can use the formula:
v = f * λ
To find the speed of sound (v) in air at room temperature. Rearranging the equation, we have:
v = 440 Hz * 2.80 m = 1232 m/s
Now, to determine the minimum water height, we need to calculate the distance between the closed end (at the water level) and the open end of the pipe. Since the tube length is given as 1.40 m, the minimum water height would be:
Minimum water height = 1.40 m - λ/2
Substituting the values, we find:
Minimum water height = 1.40 m - 2.80 m / 2 = 0.70 m
Therefore, the minimum water height required for resonance to occur is 0.70 m.
(c) To find the second least water height for resonance, we need to determine the wavelength (λ) corresponding to the second harmonic (n = 2). Using the same equation as in part (a), we find:
λ = 2 * 1.40 m = 2.80 m
Since the tube length, the speed of sound, and the fundamental frequency remain the same, the second least water height can be calculated similarly:
Second least water height = 1.40 m - λ/2
Substituting the values, we have:
Second least water height = 1.40 m - 2.80 m / 2 = 0.70 m
Interestingly, we arrive at the same minimum water height of 0.70 m for the second resonance position as well.
It is worth noting that these calculations assume ideal conditions and do not account for any factors that might affect the speed of sound or the behavior of the sound waves, such as variations in temperature or humidity.