MHT-CET : Physics Entrance Exam

### MHT - CET : Physics - Stationary Waves Page 6

‹‹ Previous  |  Page 1  |  Page 2  |  Page 3  |  Page 4  |  Page 5  |  Page 6  |  Page 7  |   Next ››

8.

Vibrations of Air Columns

An air column in the tubes or pipes closed at one end or open at both ends can be set into vibrations with tuning forks of suitable frequencies.

The air column in the tubes have a natural frequency of vibration. The vibrating tuning fork held at its mouth, sets the air column into forced vibrations.

If the frequency of the tuning fork equals the natural frequency of the tube, resonance occurs and a loud sound is emitted.

Pipe Closed at One End
In these pipes, there is a node at the closed end and an antinode at the open end.

• For the simplest mode of vibration, there is a single node and antinode.
\ Length of air column equals the distance between the node and its successive antinode.

l =

 l 1 4

• \ l1 = 4 l

fundamental frequency n1 =

 v l1

=

 v 4l

For the second mode of vibration, l =

 3l2 4

\ l2 =

 4l 3

\ frequency of first overtone n2 =

 v l2

=

 3v 4l

=3n1

Similarly, the frequencies of the subsequent modes are 5n1, 7n1, … etc.

• These frequencies consist of the fundamental frequency and its odd multiples.
• Only odd harmonics are present in the vibrations of an air column in a pipe closed at one end.

Pipe Open at Both Ends
In these pipes, there is an antinode formed at both the ends.

• At points of antinodes, there is no change in pressure.
• The air column in these tubes vibrate in different ways or modes, subject to the condition that there must be an antinode at each open end.
• In the simplest node of vibration,

n1=

 v l

=

 v 2l

= fundamental frequency or first harmonic

• For the subsequent nodes,

n2=

 v l1

= second harmonic (first overtone)

= 2n1

n3=

 v l3

=

 3v 2l

= 3n1 = third harmonic or second overtone.

• The air column in a pipe, open at both ends, can vibrate with frequencies n1, 2n1, 3n1, … etc., i.e. both the odd and even multiples of the fundamental frequency are present i.e. odd and even harmonics are present.
• End Correction: Since the air particles are not free to vibrate with maximum amplitude at the open end, the antinode is formed at a distance 'e' slightly above the open end. The distance 'e' is termed as the end correction.
e = 0.3 d
where
d: internal diameter of the pipe.
\ Corrected length l1 = l + e = l + 0.3 d

Relation between the fundamental frequencies in an open tube and closed tube:

Fundamental frequencies

For open tube f0 =

 V 4l

Closed tube fc =

 V 2l

\ fc = 2 f0

‹‹ Previous  |  Page 1  |  Page 2  |  Page 3  |  Page 4  |  Page 5  |  Page 6  |  Page 7  |   Next ››

 Career in India | Jobs in India