A tuning fork of frequency 558 Hz is placed near the top of the pipe shown in the Figure. The water level is lowered so that the length L slowly increases from an initial value of 20.0 cm. Determine the next two values of L that correspond to resonant modes. (Assume that the speed of sound in air is 343 m/s.)
I found the wavelenght:
0.2m x 4 = 0.8m. Then used the wavelength to calculate the other lengths: L3 = [3(.8)/4] x (343/558). This was incorrect. What am I doing wrong?
The wavelength of the sound is
(sound speed)/(frequency)
Resonances of an open pipe occur when the pipe length is an odd number of quarter wavelengths.
l=lambda/4
456
To determine the next two values of L that correspond to resonant modes, use the formula:
L = (2n - 1) * λ/4
Where:
L is the length of the pipe
n is the mode number (1 for the first mode, 2 for the second mode, etc.)
λ is the wavelength of the sound
Since you have already calculated the wavelength as 0.8 m using the formula (speed of sound)/(frequency), you can now substitute the values into the formula to find the lengths for the resonant modes.
For the first mode (n = 1):
L1 = (2(1) - 1) * (0.8)/4
L1 = 0.4 m
For the second mode (n = 2):
L2 = (2(2) - 1) * (0.8)/4
L2 = 0.9 m
Therefore, the next two values of L that correspond to resonant modes are 0.4 m and 0.9 m.
To find the next two values of L that correspond to resonant modes, you need to consider the quarter-wavelength resonance condition for an open pipe.
The first step is to calculate the wavelength of the sound using the formula:
wavelength = sound speed / frequency.
Given that the sound speed in air is 343 m/s and the frequency of the tuning fork is 558 Hz, we can calculate the wavelength:
wavelength = 343 m/s / 558 Hz = 0.615 m.
Now, for an open pipe, resonances occur when the pipe length is an odd number of quarter wavelengths. This means:
For the first resonance mode (n = 1), the pipe length is a quarter wavelength:
L1 = (1/4) * wavelength = (1/4) * 0.615 m = 0.154 m.
For the second resonance mode (n = 3), the pipe length is three-quarters of a wavelength:
L2 = (3/4) * wavelength = (3/4) * 0.615 m = 0.461 m.
Therefore, the next two values of L that correspond to resonant modes are L1 = 0.154 m and L2 = 0.461 m.