Question

Two organ pipes are the same size. Hydrogen and oxygen gases are filled in them. Taking the elasticity of gases to be same, the ratio of their fundamental frequencies will be:
A) 1:4
B) 4:1
C) 8:1
D) 16:1

Answer 1

Hint: The speed of sound in an ideal gas is given by the relationship
v = $\sqrt {\dfrac{\gamma RT}{{M}}}$ where R= gas constant, T= absolute temperature, M= molecular mass and $ \gamma$= adiabatic gas constant.

Complete step by step solution:
Velocity of sound gas is v
$v=\sqrt{\dfrac{\gamma RT}{M}}$ Where M is the molecular mass of the gas
Keeping T and $\gamma $ to be constant.
\[\begin{align}
  & \dfrac{V_{{H}_{2}}}{V_{{O}_{2}}}=\sqrt{\dfrac{M_{{0}_{2}}}{M_{{H}_{2}}}} \\
 & \Rightarrow \dfrac{V_{{H}_{2}}}{V_{{O}_{2}}}=\sqrt{\dfrac{32}{2}} \\
 & \Rightarrow \dfrac{V_{{H}_{2}}}{V_{{O}_{2}}}=\sqrt{16} \\
 & \therefore \dfrac{V_{{H}_{2}}}{V_{{O}_{2}}}=4 \\
 & \\
\end{align}\]
Now, fundamental frequency $v$ = $\dfrac{v}{{2L}}$
For the same length of the pipes $\text{frequency }\!\!\propto\!\!\text{ v}$
Therefore ratio of frequency will be same as of the ratio of velocity
\[\begin{align}
  & \Rightarrow \dfrac{V_{H_2}}{V_{O_2}} = 4 \\
 & \therefore \dfrac{\text{frequency }{{\text{H}}_{\text{2}}}}{\text{frequency }{{\text{O}}_{\text{2}}}}=\dfrac{4}{1} \\
\end{align}\]

So option B is correct.

Additional Information: The organ pipe is a musical instrument used to produce musical sound by blowing air into the pipe. Limb pipes are of two types (a) closed organ pipes, closed at one end (b) open organ pipes, open at both ends.
Closed Organ Pipe: A hollow wooden or metal tube that is used to produce sound is called an organ pipe. If both ends of the pipe are open, it is called an open organ pipe; The flute is an example pipe but if one end is closed then this organ pipe is closed. The closed end is constrained to be a node of the wave and the open end is definitely an antinode. This makes the fundamental mode such that the wavelength is four times the length of the air column. The closure of the closed end prevents the column from producing symmetry.
Open Organ Pipe: An open pipe is one that is opened at both ends. When air is blown into the pipe from one end, a wave travels through the tube to the next end from where it is reflected.

Note: Musical instruments such as flute, clarinet etc. are based on the principle of vibration of air columns. Due to the superposition of the incident wave and the reflected wave, longitudinal stationary waves are formed in the pipe.