“Vibrations of air column in pipe” MHT-CET Class 12 Physics Crash Course | Chapter – Superposition of Waves
In this session G.S. Khairnar discussed the theory and MCQ’s of “Vibrations of air column in pipe”. The session will be useful for all the aspirants of MHT-CET 2022. The session will be in Marathi and English.
Vibrations of air column in pipe:-
Hello everyone, we have learnt in previous articles about the concept of superposition of waves and formation of stationary waves. When two waves travelling along same line and in opposite direction, it gets overlapped and form loops. These loops are known as stationary waves. Loop of stationary waves consists of alternate node and antinode.
We also studied that the frequencies can be formed in vibrations of stretched string, air column. The frequency ranges can be explained with harmonics and overtones.
- Harmonics:-
The integral multiple of fundamental frequency including fundamental frequency are called harmonics
- Overtone:-
The integral multiple of fundamental frequency, excluding the fundamental frequency which are actually present are called overtones.
Let’s discuss the modes of vibration of air column in pipe……!
- Pipe closed at one end
Consider an air column of length ‘L’ in a pipe closed at one end. Longitudinal waves (sound waves) travel through air column & reflected from closed end of pipe. Due to interference between incident & reflected waves, longitudinal stationary waves are produced node is formed at closed end & Antinodes is formed at open end.
Conditions are shown in the fig below
Case 1. First Harmonic
Case 2. Third harmonic
Case 3. Fifth harmonic
Fundamental frequency or 1stharmonic:-
In 1st harmonic one node is formed at closed end while antinode at open end.
From fig (1),
First overtone or 3rd harmonic:-
In the first overtone or 3rdharmonic air column vibrates with two nodes & two Antinodes shown in fig (2)
Let n1& λ1 be the frequency & wavelength of air column then,
Second overtone or 5th harmonic:-
In the 2nd overtone or 5thharmonic air column three nodes & three antinodes are formed.
Let n2 & λ2 be the frequency & wavelength
From above equations it is clear that only odd harmonics are present in vibrations of air column when pipe is closed at one end.
2. Pipe opens at both ends.
Consider an air column of length ‘L’ in a pipe open at both ends. When air column is vibrating in a pipe open at both ends antinodes are formed at open end and nodes are formed within the length of pipe as shown in fig. below
- Fundamental frequency or First harmonic:-
In case mode of vibration, antinodes are formed at open ends and node is formed at the middle of the pipe. So that the length of pipe is divided in to two antinodes.
From figure (a)
- First overtone or Second harmonic:-
In the first overtone or second harmonic air column vibrates with three antinodes & two nodes.
Let n1&λ1 be the frequency& wavelength then
- Second overtone or third harmonic:-
In the second overtone or third harmonic, four antinodes & three nodes are formed.
Let n2&λ2 be the frequency wavelength then,
Vibrations of air column in pipe:-
Let’s solve some MCQ’s …..
Q.1) In case of vibrations of air column in a pipe closed at one end, the fundamental frequency is 350 Hz. The frequency of 3rd overtone is (neglecting end correction)..
Q.2) Ratio of fundamental frequencies for pipe closed at one end and open at the both ends is 3:4 with same speed. The ratio of lengths of pipes respectively is (neglecting end correction)
a) 3/2
b) 2/3
c) 4/3
d) 4/3
Answer
b) 2/3
Q.3) Ratio of fundamental frequencies for pipe closed at one end and open at the both ends is 2:3 and the ratio of lengths of the pipe is 3:2. The ratio of velocities of sound will be….
Q.4) Which among the following statement is correct for the 3rd mode of vibrations of air column in pipe closed at one end….
Q.7) Air columns are formed in the organ pipe of length 24 cm closed at one for 2nd mode. If the velocity of sound in the medium is 400 m/s, the frequency of 2nd overtone is (neglecting end correction)..
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