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 find the frequency of the tuning fork, we need to understand the concept of resonance and how it relates to the length of the tube.
Resonance occurs when the frequency of a vibrating object matches the natural frequency of another object. In this case, the tuning fork is producing sound waves that can create resonance in the tube.
The length of the tube affects the wavelength of the sound waves that can resonate. When the length of the tube is an integer multiple of half the wavelength, resonance occurs.
Given that resonance is heard when the water level has dropped 18 cm and then again after 54 cm, we can deduce that the wavelength of the sound wave must have changed from one resonance to the other.
Let's analyze the situation:
1. When the water level has dropped 18 cm, the length of the air column above the water is equivalent to an odd number of half wavelengths (λ/2).
2. When the water level has dropped 54 cm, the length of the air column above the water is equivalent to an even number of half wavelengths (λ/2).
Since the difference between the lengths of the air column in the two cases is 54 cm - 18 cm = 36 cm, it represents a whole wavelength (λ) of the sound wave.
Now, we need to calculate the frequency using the speed of sound. The speed of sound in air is approximately 343 m/s at room temperature (20°C).
Given that the wavelength (λ) is 36 cm (or 0.36 meters), we can use the formula:
speed of sound = frequency × wavelength
343 m/s = frequency × 0.36 m
Solving for frequency:
frequency = 343 m/s / 0.36 m
frequency ≈ 953.89 Hz
Therefore, the frequency of the tuning fork is approximately 953.89 Hz.