A tuning fork is set into vibration above a vertical open tube filled with water. The water level is allowed to drop slowly. As it does so, the air in the tube above the water level is heard to resonate with the tuning fork when the distance from the tube opening to the water level is 0.145 m, and again at 0.345 m. What is the frequency (in hertz) of the tuning fork?
Well, well, let's calculate the frequency of that tuneful tuning fork, shall we?
So, we know that the distance between the tube opening and the water level where resonance occurs is 0.145 m and 0.345 m.
The fundamental frequency of a closed tube (like the one we have here) is given by the formula:
f = v / λ
where f is the frequency, v is the speed of sound in air, and λ is the wavelength of the sound wave.
Now, the wavelength of the sound wave is given by:
λ = 2L
where L is the length of the tube. In our case, L is the distance from the tube opening to the water level.
So, we have two values for L: 0.145 m and 0.345 m.
Using the formula λ = 2L, we can determine the wavelength for each case:
For L = 0.145 m, λ = 2(0.145) = 0.29 m
For L = 0.345 m, λ = 2(0.345) = 0.69 m
Now, since the speed of sound in air is approximately 343 m/s, we can calculate the frequency using the formula f = v / λ:
For L = 0.145 m: f = 343 / 0.29 ≈ 1182.76 Hz
For L = 0.345 m: f = 343 / 0.69 ≈ 497.83 Hz
So, the frequency of the tuning fork is approximately 1182.76 Hz when the water level is at 0.145 m, and approximately 497.83 Hz when the water level is at 0.345 m.
Now that tuning fork deserves a standing ovation! Bravo!
To calculate the frequency of the tuning fork, we need to use the formula:
v = fλ
Where:
- v is the speed of sound in air,
- f is the frequency of the tuning fork, and
- λ is the wavelength.
In this case, the distance between resonance points is equal to a half-wavelength of the sound wave produced by the tuning fork. So, we can use the following equation:
Δx = λ/2
Where:
- Δx is the difference in distance between the two resonance points.
Given that the difference in distance is 0.345 m - 0.145 m = 0.2 m, we can calculate the wavelength:
0.2 m = λ/2
Solving for λ, we find:
λ = 0.2 m * 2 = 0.4 m
Now, we need to find the speed of sound in air (v). The speed of sound depends on temperature. At room temperature, the speed of sound is approximately 343 m/s.
With the wavelength (λ) and the speed of sound (v), we can calculate the frequency (f):
v = fλ
343 m/s = f * 0.4 m
Solving for f, we find:
f = 343 m/s / 0.4 m ≈ 857.5 Hz
Therefore, the frequency of the tuning fork is approximately 857.5 Hz.
To find the frequency of the tuning fork, we need to use the formula for the speed of sound in air:
v = f * λ
Where:
v is the speed of sound in air,
f is the frequency of the sound wave,
and λ is the wavelength of the sound wave.
In this case, the resonances occur when the distance from the tube opening to the water level is 0.145 m and 0.345 m. The difference in these distances represents one-half of the wavelength (since it is the distance from a node to an antinode).
Let's calculate the wavelength first:
λ = 2 * (0.345 m - 0.145 m)
= 2 * 0.2 m
= 0.4 m
Now, to calculate the frequency, we can rearrange the formula:
f = v / λ
The speed of sound in air at room temperature is approximately 343 m/s, so substituting the values into the equation:
f = 343 m/s / 0.4 m
≈ 857.5 Hz
Therefore, the frequency of the tuning fork is approximately 857.5 Hz.
We have the standing sound wave in a pipe; an open end is the antinode for air column, a closed end (the water surface) is the node.
The distance between two adjacent nodes is the half of the wavelength:
λ/2 = 0.345 – 0.145 = 0.2 m, λ = 0.4 m.
λ = v•T = v/f.
In dry air at 20 °C the speed of sound is v = 343.2 m/s.
f =v/ λ = 343.2/0.4 = 858 Hz