A 71.4cm long tube has a 71.4cm long insert that can be pulled in and out. A vibrating tuning fork is held next to the tube. As the insert is slowly pulled out, the sound from the tuning fork creates standing waves in the tube when the total length L is 77.5cm, 126.0cm, and 174.5cm. What is the frequency of the tuning fork? Take the speed of sound in air to be 343m/s.
To find the frequency of the tuning fork, we need to use the concept of standing waves. In a closed tube, standing waves are formed when the length of the tube matches certain multiples of half-wavelengths. In this case, we have three different lengths of the tube: L = 77.5 cm, L = 126.0 cm, and L = 174.5 cm.
The first step is to determine the wavelength corresponding to each length of the tube. We can use the relationship between wavelength, speed, and frequency:
wavelength = speed / frequency
Given that the speed of sound in air is 343 m/s, we need to convert the lengths of the tube into meters:
L1 = 77.5 cm = 0.775 m
L2 = 126.0 cm = 1.260 m
L3 = 174.5 cm = 1.745 m
Now we can calculate the wavelength for each length of the tube:
wavelength1 = 0.775 m
wavelength2 = 1.260 m
wavelength3 = 1.745 m
Since the tube is closed on one end and open on the other, the fundamental frequency (first harmonic) occurs when the tube length is equal to one-fourth of the wavelength (L = λ/4). We can use this relationship to find the frequency:
frequency1 = speed / (4 * wavelength1)
frequency2 = speed / (4 * wavelength2)
frequency3 = speed / (4 * wavelength3)
Now we can substitute the speed of sound in air and the calculated wavelengths into the equations:
frequency1 = 343 m/s / (4 * 0.775 m)
frequency2 = 343 m/s / (4 * 1.260 m)
frequency3 = 343 m/s / (4 * 1.745 m)
Performing the calculations:
frequency1 ≈ 111.6 Hz
frequency2 ≈ 68.0 Hz
frequency3 ≈ 49.4 Hz
Therefore, the frequency of the tuning fork is approximately 111.6 Hz for the first length of the tube, 68.0 Hz for the second length, and 49.4 Hz for the third length.