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 relationship between the length of the tube and the frequency at which resonance occurs.
Resonance occurs when the length of the tube corresponds to half of a wavelength of the sound wave produced by the tuning fork. In other words, the length of the tube must be equal to an odd multiple of quarter wavelengths for resonance to occur.
Given that 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, we can determine the length of the tube in terms of quarter wavelengths.
First, we calculate the length of the tube when resonance occurs for the first time. Let's assume the speed of sound in air is 343 m/s (approximately at room temperature).
Length of the first resonance = Total length of the tube - Distance from water level to the top
= 54 cm - 18 cm
= 36 cm = 0.36 meters (converting to meters)
Now, we calculate the length of the second resonance.
Length of the second resonance = Total length of the tube - Distance from water level to the top
= 54 cm - 54 cm
= 0 cm = 0 meters
Since the length of the second resonance is 0 meters, we can conclude that resonance is achieved when the tube length is equal to a quarter wavelength.
Now, we have the lengths of the first and second resonance. To find the frequency of the tuning fork, we use the formula:
Frequency = Speed of sound / (2 * Length of tube)
Using the given speed of sound and the length of the first resonance, we can calculate the frequency:
Frequency = 343 m/s / (2 * 0.36 meters)
= 476.39 Hz
Therefore, the frequency of the tuning fork is approximately 476.39 Hz.