A vertical tube with a tap at the base is filled with water, and a tuning fork vibrates over its mouth. As the water level is lowered in the tube, resonance is heard when the water level has dropped 18 cm, and again after 54 cm of distance exists from the water to the top of the tube. What is the frequency of the tuning fork?
To determine the frequency of the tuning fork, we need to understand the concept of resonance in a tube.
Resonance occurs when the frequency of an external source matches the natural frequency of a system. In this case, when the tuning fork vibrates, it produces sound waves with a specific frequency. By adjusting the water level in the tube, we can find the point where the sound waves produced by the tuning fork match the natural frequency of the tube, resulting in resonance.
In this problem, resonance is heard when the water level has dropped 18 cm and again after 54 cm of distance exists from the water to the top of the tube.
The distance between these two resonances is 54 cm - 18 cm = 36 cm.
Now, let's break down the problem step by step to find the frequency of the tuning fork:
Step 1: Convert the distances to wavelengths.
We know that the distance between two consecutive resonance points is half a wavelength. Therefore, we need to find the wavelength of the sound waves produced by the tuning fork at these two resonance points.
The first resonance occurs at 18 cm. Since this is half a wavelength, the wavelength at this point is 2 * 18 cm = 36 cm.
The second resonance point occurs at 54 cm. Again, since this is half a wavelength, the wavelength at this point is 2 * 54 cm = 108 cm.
Step 2: Calculate the frequency using the formula: frequency = speed of sound / wavelength.
The speed of sound in air is approximately 343 meters per second (m/s). However, we need to convert the wavelengths from centimeters to meters to match the units.
For the first resonance:
Frequency = 343 m/s / (36 cm / 100 cm/m) = 952 Hz.
For the second resonance:
Frequency = 343 m/s / (108 cm / 100 cm/m) = 317 Hz.
Therefore, the frequency of the tuning fork is approximately 317 Hz.