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 relationship between the height of the water column in the tube and the corresponding resonant frequencies.
First, let's write down the given information:
- Resonance occurs when the water level is 18 cm from the base of the tube.
- Resonance occurs again when the water level is 54 cm from the top of the tube.
To understand the pattern of resonance in a tube, we can consider the fundamental frequency and its overtones. The fundamental frequency is the lowest resonant frequency produced in the tube, and the overtones are higher resonant frequencies.
In a closed tube like this one, the fundamental frequency can be determined using the formula:
f1 = (v/2L1)
where:
- f1 is the fundamental frequency,
- v is the speed of sound in air (which is around 343 m/s at room temperature),
- L1 is the length of the air column in the tube when resonance occurs.
Similarly, for the second resonance (an overtone), we can calculate the frequency using the formula:
f2 = (v/2L2)
where:
- f2 is the frequency of the second resonance,
- L2 is the length of the air column in the tube when the second resonance occurs.
Let's substitute the given values into the formulas:
For the first resonance, L1 = 18 cm = 0.18 m:
f1 = (343 / (2 * 0.18))
For the second resonance, L2 = total length of the tube - distance of the water from the top.
The total length of the tube is equivalent to L1 + L2 (since the water level is at the bottom during resonance).
L2 = 54 cm = 0.54 m (since there's 54 cm from the water to the top).
So, L2 = (L1 + L2) - L1 = 0.54 - 0.18 = 0.36 m.
Now, we can calculate the frequency of the second resonance:
f2 = (343 / (2 * 0.36))
Simplifying the equations will give us the values of f1 and f2.
By finding these two frequencies, we can then determine the frequency of the tuning fork, as it will match one of these resonant frequencies.