A closed organ pipe is maintained at an absolute temperature T₁ , and an open organ pipe of the exact same length is maintained at an absolute temperature T₂ . Both pipes are filled with the same gas. If the fundamental frequencies of the two pipes are equal, what is the ratio of their absolute temperatures T₁ : T₂ ?

Step-by-step solution

The fundamental frequency of a closed organ pipe of length L is given by f_c = v₁ 4L . The fundamental frequency of an open organ pipe of the same length L is given by f_o = v₂ 2L . The speed of sound in a gas is v = RT M . Since the gas is the same, v T . Given that the fundamental frequencies are equal, we have: v₁ 4L = v₂ 2L v₁ 4 = v₂ 2 v₁ v₂ = 4 2 = 2 Substituting the proportionality of speed with temperature: T₁ T₂ = 2 Squaring both sides yields: T₁ T₂ = 4 Thus, the ratio T₁ : T₂ is 4:1 . Answer: 4:1