Wave Mechanics Question 664
Question: A pipe of length $ {\ell _{1}} $ , closed at one end is kept in a chamber of gas of density $ {\rho _{1}} $ . A second pipe open at both ends is placed in a second chamber of gas of density $ {\rho _{2}} $ . The compressibility of both the gases is equal Calculate the length of the second pipe if frequency of first overtone in both the cases is equal
Options:
A) $ \frac{4}{3}{\ell _{1}}\sqrt{\frac{{\rho _{2}}}{{\rho _{1}}}} $
B) $ \frac{4}{3}{\ell _{1}}\sqrt{\frac{{\rho _{1}}}{{\rho _{2}}}} $
C) $ {\ell _{1}}\sqrt{\frac{{\rho _{2}}}{{\rho _{1}}}} $
D) $ {\ell _{1}}\sqrt{\frac{{\rho _{1}}}{{\rho _{2}}}} $
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Answer:
Correct Answer: B
Solution:
[b] Frequency of first overtone in closed pipe, $ v=\frac{3v}{4{\ell _{1}}}\sqrt{\frac{P}{{\rho _{1}}}} $ ?. (i) Frequency of first overtone in open pipe, $ v’=\frac{1}{{\ell _{2}}}\sqrt{\frac{P}{{\rho _{2}}}} $ ?. (ii) From equation (i) and (ii)
$ \Rightarrow {\ell _{2}}=\frac{4}{3}{\ell _{1}}\sqrt{\frac{{\rho _{1}}}{{\rho _{2}}}} $
BETA
