Wave Motion 2 Question 18
18. 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 $(2004,2 M)$
(a) $\frac{L}{3}$
(b) $\frac{4 L}{3}$
(c) $\frac{4 L}{3} \sqrt{\frac{\rho _1}{\rho _2}}$
(d) $\frac{4 L}{3} \sqrt{\frac{\rho _2}{\rho _1}}$
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Answer:
Correct Answer: 18. (c)
Solution:
- $f _c=f _o$
(both first overtone)
$$ \begin{aligned} \text { or } & 3\left(\frac{v _c}{4 L}\right) & =2\left(\frac{v _o}{2 l _o}\right) \\ \therefore & l _o & =\frac{4}{3}\left(\frac{v _o}{v _c}\right) L=\frac{4}{3} \sqrt{\frac{\rho _1}{\rho _2}} L \quad \text { as } v \propto \frac{1}{\sqrt{\rho}} \end{aligned} $$
