A tuning fork is held above an air column filled with water. The water level can be raised and lowered to
change the length of the column. The shortest length of air column that produces a resonance is 0.30 m, and
the next resonance is heard when the air column is 0.80 m long.
a. Calculate the speed of sound at 25°C. [A2]
b. Calculate the wavelength of the sound wave produced by the tuning fork, along with the frequency of
the tuning fork. [A4]
c. If the water level is lowered again until a third resonance is heard, what is the length of the column at
that point? [A2]
To solve this problem, we can use the formula for the speed of sound in air, which is given by:
v = λf
Where:
v = speed of sound in air
λ = wavelength of the sound wave
f = frequency of the sound wave
a. To calculate the speed of sound at 25°C, we need to use the known wavelength and frequency of the sound wave. The wavelength can be found by subtracting the length of the air column for the second resonance from the length for the first resonance:
λ = 0.80 m - 0.30 m = 0.50 m
The frequency of the tuning fork can be found by using the formula:
f = v/λ
We need to calculate the speed of sound first, so we rearrange the formula:
v = λf
Substituting the known values gives:
v = (0.50 m)(f)
Now, we need to determine the frequency of the tuning fork. Since we don't have that information given in the problem, we can't solve this part of the problem.
b. To find the wavelength of the sound wave produced by the tuning fork, we know that the shortest length of the air column that produces resonance is 0.30 m. This length is half of a wavelength, so we can determine the wavelength:
λ = 2(0.30 m) = 0.60 m
Since we still don't have the frequency of the tuning fork, we can't solve this part of the problem.
c. If the water level is lowered again until a third resonance is heard, we need to determine the length of the air column at that point. Since the first resonance occurred at 0.30 m and the second resonance occurred at 0.80 m, the distance between these two resonances is equal to one wavelength. Therefore, the length of the air column for the third resonance would be:
Length for third resonance = 0.80 m + (0.80 m - 0.30 m) = 1.30 m
a. To calculate the speed of sound at 25°C, we can use the formula:
speed of sound = frequency × wavelength
We can start by finding the frequency of the sound wave produced by the tuning fork. Since we know the shortest length of the air column that produces resonance is 0.30 m, we can consider this as the length of half a wavelength. Therefore, the full wavelength would be twice this length, which is 0.60 m.
To find the frequency, we can use the formula:
frequency = speed of sound / wavelength
First, let's convert the given lengths to the same unit. Since the lengths are already in meters, we don't need to do any conversion.
Plugging in the values, we have:
frequency = speed of sound / (0.60 m)
Now, we need to find the speed of sound. However, the problem does not provide any information about the tuning fork frequency or the given resonances being the fundamental or overtones. Therefore, we need further information to proceed with this calculation.
b. To calculate the wavelength of the sound wave produced by the tuning fork, we can use the formula:
wavelength = 2 × length of air column (resonance)
Using the first resonance (0.30 m), the wavelength would be:
wavelength = 2 × 0.30 m
To find the frequency of the tuning fork, we can rearrange the formula:
frequency = speed of sound / wavelength
Since we don't have the speed of sound yet, we cannot calculate the frequency or the wavelength without additional information.
c. To find the length of the column at the third resonance, we need to know the relationship between the lengths of the resonances. If the resonances are consecutive harmonics (overtones), we can use the formula:
length of air column (n) = (n × wavelength) / 4
Where "n" is the resonance number (1, 2, 3, ...).
With only the given information, we cannot calculate the length of the column at the third resonance without knowing the relationship between the lengths of the resonances.