An open vertical tube is filled with water and a tuning fork vibrates over the top near the open end. As the water level is lowered in the tube, the first resonance is heard when the water level is at 31 cm from the top of the tube. What is the frequency of the tuning fork?
The speed of sound in air is 343 m/s.
Answer in units of Hz
To determine the frequency of the tuning fork, we need to use the first resonance condition in a closed tube. In this case, the tube is opened at the top and closed at the bottom by the water level.
The general formula for the frequency of a closed tube in resonance is given by:
f = (N * v) / (2L)
where:
f is the frequency of the sound wave,
N is the harmonic number (in this case, N = 1 for the first resonance),
v is the speed of sound in air (343 m/s), and
L is the length of the tube.
In this problem, the length of the tube is the distance from the water level to the top, which is 31 cm or 0.31 m.
Now, let's calculate the frequency using the given values:
f = (1 * 343 m/s) / (2 * 0.31 m)
f = 343 m/s / 0.62 m
f ≈ 553.23 Hz
Therefore, the frequency of the tuning fork is approximately 553.23 Hz.
To find the frequency of the tuning fork, we need to apply the concept of resonance.
Resonance occurs when the frequency of the tuning fork matches the natural frequency of the air column in the tube. In this case, the tube is open at one end and closed at the other, so it behaves like an open-closed pipe.
For a pipe with an open end, the fundamental frequency (the lowest resonant frequency) is given by the equation:
f = v / (4L)
where:
- f is the frequency
- v is the speed of sound in air (343 m/s)
- L is the length of the air column
In this problem, the water level is at 31 cm from the top, so the length of the air column is equal to the height of the tube minus the water level:
L = tube height - water level = (entire tube height) - 31 cm
Given that the speed of sound in air is 343 m/s, we can convert cm to meters and calculate the frequency:
L = (entire tube height - 31 cm) / 100 (convert cm to meters)
f = 343 m/s / (4L) (substitute values)
Now we have the formula to find the frequency. We just need to know the entire tube height to calculate it.
1/2 of the wave in the tube
so wavelength = 2*31 = 62cm = .62 m
distance = speed * T
T = .62/343
so f = 1/T = 343/.62 = 553 Hz