A closed organ pipe of length L and an open organ pipe contain ga… - Vidyadip

MCQ

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 organ pipe is

A
$\frac{L}{3}$
B
$\frac{{4L}}{3}$
✓
$\frac{{4L}}{3}\sqrt {\frac{{{\rho _1}}}{{{\rho _2}}}} $
D
$\frac{{4L}}{3}\sqrt {\frac{{{\rho _2}}}{{{\rho _1}}}} $

✓

Answer

Correct option: C.

$\frac{{4L}}{3}\sqrt {\frac{{{\rho _1}}}{{{\rho _2}}}} $

c
(c) Frequency of first over tone of closed pipe $=$ Frequency of first over tone of open pipe

$ \Rightarrow \frac{{3v}}{{4{L_1}}} = \frac{v}{{{L_2}}}$

$ \Rightarrow \frac{3}{{4{L_1}}}\sqrt {\frac{{\gamma P}}{{{\rho _1}}}} $$ = \frac{1}{{{L_2}}}\sqrt {\frac{{\gamma P}}{{{\rho _2}}}} $

$\left[ {\because v = \sqrt {\frac{{\gamma P}}{\rho }} } \right]$

$ \Rightarrow {L_2} = \frac{{4{L_1}}}{3}\sqrt {\frac{{{\rho _1}}}{{{\rho _2}}}} = \frac{{4L}}{3}\sqrt {\frac{{{\rho _1}}}{{{\rho _2}}}} $

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