A closed organ pipe of length L and an open organ pipe contain gases… | JEE Main 2021 (16 Mar, Shift 2)

Q90JEE Main 2021 · 16 MAR · SHIFT 2Medium

Organ Pipes & Resonance Tube

A closed organ pipe of length L and an open organ pipe contain gases of densities $\rho$ 1 and $\rho$ 2 respectively. The compressibility of gases are equal in both the pipes. Both the pipes are vibrating in their first overtone with same frequency. The length of the open pipe is $\frac{x}{3} L \sqrt{\frac{\rho_1}{\rho_2}}$ where x is ___________. (Round off to the Nearest Integer)

Numerical answer type. Work it out, then open the answer.

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Answer: 4

Speed $v = \sqrt{\frac{B}{\rho}}$ ; same $B$ , so $\frac{v_2}{v_1} = \sqrt{\frac{\rho_1}{\rho_2}}$
Closed pipe first overtone: $\frac{3v_1}{4L}$ ; open pipe first overtone: $\frac{2v_2}{2L_o} = \frac{v_2}{L_o}$
Equate: $L_o = \frac{4L}{3}\frac{v_2}{v_1} = \frac{4}{3}L\sqrt{\frac{\rho_1}{\rho_2}}$
So $x = 4$ .

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