A) \[\frac{L}{3}\]
B) \[\frac{4L}{3}\]
C) \[\frac{4L}{3}\sqrt{\frac{{{\rho }_{1}}}{{{\rho }_{2}}}}\]
D) \[\frac{4L}{3}\sqrt{\frac{{{\rho }_{2}}}{{{\rho }_{1}}}}\]
Correct Answer: C
Solution :
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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