An organ pipe A , open at both ends, has a length of 80 cm . Another organ pipe B is closed at one end. If the first overtone of pipe A has the same frequency as the first overtone of pipe B , the length of pipe B is ________ cm .

Step-by-step solution

For an open organ pipe, the frequency of the n -th harmonic is f_n = nv 2L_A . The first overtone of an open pipe is its second harmonic ( n=2 ). Frequency of the first overtone of pipe A is f_A = 2v 2L_A = v L_A . For a closed organ pipe, the allowed frequencies are odd harmonics, given by f_m = mv 4L_B where m = 1, 3, 5, The first overtone of a closed pipe corresponds to the third harmonic ( m=3 ). Frequency of the first overtone of pipe B is f_B = 3v 4L_B . According to the problem, f_A = f_B : v L_A = 3v 4L_B S