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
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.
$${\ell _1}\sqrt {\frac{{{\rho _1}}}{{{\rho _2}}}} $$
Answer :
$$\frac{4}{3}{\ell _1}\sqrt {\frac{{{\rho _1}}}{{{\rho _2}}}} $$
Solution :
Frequency of first overtone in closed pipe,
$$\nu = \frac{{3v}}{{4{\ell _1}}}\sqrt {\frac{P}{{{\rho _1}}}} \,\,\,.....\left( {\text{i}} \right)$$
Frequency of first overtone in open pipe,
$$\nu ' = \frac{1}{{{\ell _2}}}\sqrt {\frac{P}{{{\rho _2}}}} \,\,\,.....\left( {{\text{ii}}} \right)$$
From equation (i) and (ii)
$$ \Rightarrow \,\,{\ell _2} = \frac{4}{3}{\ell _1}\sqrt {\frac{{{\rho _1}}}{{{\rho _2}}}} $$