A closed organ pipe of length 15 cm and an open organ pipe of length 20 cm are filled with two different gases. Both gases have the same bulk modulus but different densities _c and _o respectively. If the 5^ th harmonic of the closed pipe is identical to the 3^ rd harmonic of the open pipe, the ratio of the density of the gas in the closed pipe to that in the open pipe ( _c : _o ) is :

Step-by-step solution

5^ th harmonic of closed pipe = 5 v_c 4 L_c 3^ rd harmonic of open pipe = 3 v_o 2 L_o Given that they are identical: 5 v_c 4 15 = 3 v_o 2 20 5 v_c 60 = 3 v_o 40 v_c 12 = 3 v_o 40 v_c v_o = 36 40 = 9 10 The speed of sound in a gas is given by v = B . Since the bulk modulus B is the same for both gases: v_c v_o = _o _c _o _c = 9 10 _o _c = 81 100 Therefore, the required ratio is _c _o = 100 81 . Answer: 100 81