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An open organ pipe of length 30 cm and a closed organ pipe of length 25 cm are filled with the same gas. Which of the following pairs of overtones from the two pipes will have the exact same frequency?

Options

  1. A1^ st overtone of the open pipe and 2^ nd overtone of the closed pipe
  2. B2^ nd overtone of the open pipe and 1^ st overtone of the closed pipe
  3. C3^ rd overtone of the open pipe and 5^ th overtone of the closed pipe
  4. D2^ nd overtone of the open pipe and 2^ nd overtone of the closed pipe

Correct answer

D. 2^ nd overtone of the open pipe and 2^ nd overtone of the closed pipe

Step-by-step solution

Let the speed of sound in the gas be v . For the open organ pipe of length L_o = 30 cm , the frequency of the n^ th harmonic is given by: f_n = nv 2L_o = nv 60 (where n = 1, 2, 3, ) For the closed organ pipe of length L_c = 25 cm , the frequency of the m^ th harmonic is given by: f_m = mv 4L_c = mv 100 (where m = 1, 3, 5, ) Equating the two frequencies: nv 60 = mv 100 n 3 = m 5 5n = 3m We need the smallest integer values for n and m such that m is an odd integer. The smallest valid solution is m = 5 and n = 3 . For

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