A friend in another city tells you that she has two organ pipes of different lengths, one open at both ends, the other open at one end only. In addition, she has determined that the beat frequency caused by the second-lowest frequency of each pipe is equal to the beat frequency caused by the third-lowest frequency of each pipe. Her challenge to you is to calculate the length of the organ pipe that is open at both ends, given that the length of the other pipe is 1.20 m .
I found the answer to be 2.4, but that is incorrect so I need help
Correct answer is 1.5 m
lam = wavelength
second lowest with open ends:
lam = Lo
second lowest with closed end
lam = 4/3 Lc
f proportional to 1/Lam
so
f open = k/Lo and f closed = 3k/4Lc
difference frequency = k (1/Lo -3/4Lc)
third lowest open
Lam = (2/3) Lo
third lowest closed
Lam = (4/5) Lc
f open = k (3/2Lo)
fclosed = k(5/4Lc)
difference frequency = k(3/2Lo -4/5Lc)
so
3/2Lo - 4/5Lc = 1/Lo - 3/4Lc
but Lc = 1.2
3/2Lo - 4/6 = 1/Lo - 3/4.8
3/2Lo - 2/2Lo = .6667-.625
1/2Lo = .0417
2 Lo = 24
Lo = 12
It says that is still wrong
To solve this problem, we can start by considering the characteristic frequencies of each organ pipe.
For an organ pipe with both ends open, the natural frequencies are given by the following formula:
f = (n * v) / (2 * L)
where:
- f is the frequency
- n is the mode number (1, 2, 3, ...)
- v is the speed of sound (approximately 343 m/s at standard temperature and pressure)
- L is the length of the pipe
For an organ pipe open at one end, the natural frequencies are given by a slightly different formula:
f = (2n - 1) * v / (4 * L)
It is given that the beat frequency caused by the second-lowest frequency of each pipe is equal to the beat frequency caused by the third-lowest frequency of each pipe. Let's call this beat frequency f_beat.
To find the length of the pipe that is open at both ends, we can set up the following equation:
f_beat = (2 * v) / λ
Here, λ is the wavelength, and since the two pipes have different lengths, the wavelengths for their respective second- and third-lowest frequencies are different. We can express the wavelengths as:
λ1 = 2L1
λ2 = 4L2
where L1 is the length of the pipe open at both ends, and L2 is the length of the pipe open at one end.
Now, let's solve the equation for f_beat using the given information for the lengths of the pipes:
f_beat = (2 * v) / λ1 = (2 * v) / λ2
Substituting the expressions for the wavelengths:
(2 * v) / 2L1 = (2 * v) / 4L2
Simplifying the equation:
1 / L1 = 1 / (2L2)
Now, we know that L2 = 1.20 m, so we can substitute it into the equation:
1 / L1 = 1 / (2 * 1.20)
Simplifying further:
1 / L1 = 1 / 2.40
Finally, we can find L1 by taking the reciprocal of both sides of the equation:
L1 = 2.40 m
Therefore, the correct length of the organ pipe open at both ends is 2.40 m.
It seems that you found the correct answer. Make sure to double-check your calculations to identify any possible error.