Question

Two organ pipes $P$ and $Q$ have the same length. Pipe $P$ is open at one end and is filled with oxygen. Pipe $Q$ is open at both ends and is filled with helium. Both gases are at the same temperature, then

• A. Ratio of velocity of sound in oxygen to helium pipe is $0.162$.
• B. Ratio of velocity of sound in oxygen to helium pipe is $0.324$.
• C. Ratio of fundamental frequency in pipe $P$ to pipe $Q$ is $0.162$.
• D. Ratio of fundamental frequency in pipe $P$ to pipe $Q$ is $0.081$.

Answer 1

Answer: (B), (C)

Ratio of fundamental frequency in pipe $P$ to pipe $Q$ is $0.162$.

For oxygen,

${\gamma }_o=7/5$ ($\because$ Oxygen is a diatomic)

Molecular weight, $M_o=32\text{g}$

and for helium

${\gamma }_{He}=\frac{5}{3}$ ($\because$ Helium is monoatomic)

Molecular weight, $M_{He}=4\text{g}$

The speed of sound in a gas is given by
$v=\sqrt{\frac{\gamma RT}{M}}$

So,
Speed of sound in oxygen is

$v_o=\sqrt{\frac{{\gamma }_{o}R{T}_o}{M_o}}=\sqrt{\frac{7R{T}_o}{5\times 32\times {10}^{-3}}}$

Speed of sound in helium is
$v_{He}=\sqrt{\frac{{\gamma }_{He}R{T}_{He}}{M_{He}}}=\sqrt{\frac{5R{T}_{He}}{3\times 4\times {10}^{-3}}}$

Both gases are at the same temperature.

So,
$\frac{v_o}{v_{He}}=\sqrt{\frac{7}{5}\times \frac{4\times {10}^{-3}}{32\times {10}^{-3}}\times \frac{3}{5}}=\sqrt{\frac{21}{25}\times \frac{1}{8}}=\frac{1}{10}\sqrt{\frac{21}{2}}=0.324$

Fundamental frequency in pipe $P$ is

$f_P=\frac{v_o}{4L}$ (Open at one end)

Fundamental frequency in pipe $Q$ is

$f_Q=\frac{v_{He}}{2L}$ (Both the ends are open)
So $\frac{f_P}{f_Q}=\frac{v_o}{4L}\times \frac{2L}{v_{He}}=\frac{1}{2}\left(\frac{v_o}{v_{He}}\right)=\frac{0.324}{2}=0.162$

Hence, options (b) and (c) are correct.