Question

Discuss different modes of vibrations in an air column of a pipe open at both the ends. State the cause of end correction. Find the end correction for the pipe open at both the ends in fundamental mode.
What should be tension applied to a wire of length 1m and maas 10 g, if it has to vibrate with fundamental frequency of 50 Hz?

Answer 1

Modes of vibration of air column in a pipe open at both ends : An open argan pipe is a cylindricla tube containing an air column open at both ends. A vibrating tuning fork is held horizontally near end of the pipe. Sound waves are sent by source (vibrating tuning fork) near one end. Refer to HSC paper, July 2018, Answer 4(i).
Cause of End correction : End correction arises because air particles in the plane of the open end of tube, are not free to move in all directions. Hence, reflection takes place at the plane, at small distance outside the tube.
End correction for the pipe open at both ends :
`v=2n_(1)L_(1)=2n_(2)L_(2)`
`n_(1)L_(1)=n_(2)L_(2)`
`n_(1)(l_(1)+2e)=n_(2)(l_(2)+2e)`
`e=(n_(1)l_(1)-n_(2)l_(2))/(2(n_(2)-n_(1)))`
or `=((n_(2)l_(2)-n_(1)l_(1)))/(2(n_(1)-n_(2)))`
Numerical :
We have, `l=1m,M=10g=10xx10^(-3)kg,n=50Hz`
Linear density of wire (m) `=(M)/(l)=(10xx10^(-3))/(1)`
`=10xx10^(-3)kgm^(-1)`.
`impliesn=(1)/(2)lsqrt((T)/(m))`
`implies50=(1)/(2)xx1xxsqrt((T)/(10xx10^(-3)))`
`implies(100)^(2)=(T)/(10xx10^(-3))`
`impliesT=100N`.
`:.` Tension to be applied to the wire is 100 N.