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
IIT 2004, Diffcult
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(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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