Answer:
To determine the number of waves inside the pipe and the length of the pipe at each resonance, we can use the formula:
L = (2n - 1) * λ / 4
Where L is the length of the pipe, n is the number of waves inside the pipe, and λ is the wavelength.
Given that the frequency of the tuning fork is 640 Hz, we can find the wavelength using the formula:
λ = v / f
Where v is the speed of sound in air, which is approximately 343 m/s at room temperature.
Converting the frequency to cycles per second (cps):
f = 640 Hz = 640 cps
Converting the speed of sound to cm/s:
v = 343 m/s = 34300 cm/s
Now we can calculate the wavelength:
λ = v / f = 34300 cm/s / 640 cps = 53.59375 cm
Using the formula mentioned earlier, we can find the length of the pipe at each resonance:
1st resonance:
L = (2 * 1 - 1) * 53.59375 cm / 4 = 0.25 * 53.59375 cm = 13.3984375 cm
2nd resonance:
L = (2 * 2 - 1) * 53.59375 cm / 4 = 0.75 * 53.59375 cm = 40.1953125 cm
3rd resonance:
L = (2 * 3 - 1) * 53.59375 cm / 4 = 1.25 * 53.59375 cm = 66.9921875 cm
4th resonance:
L = (2 * 4 - 1) * 53.59375 cm / 4 = 1.75 * 53.59375 cm = 93.7890625 cm
5th resonance:
L = (2 * 5 - 1) * 53.59375 cm / 4 = 2.25 * 53.59375 cm = 120.5859375 cm
17th resonance:
L = (2 * 17 - 1) * 53.59375 cm / 4 = 8.25 * 53.59375 cm = 442.3828125 cm
Therefore, the length of the pipe at each resonance is as follows:
1st resonance: 13.3984375 cm
2nd resonance: 40.1953125 cm
3rd resonance: 66.9921875 cm
4th resonance: 93.7890625 cm
5th resonance: 120.5859375 cm
17th resonance: 442.3828125 cm
Step-by-step explanation: