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Q.
A pipe $30\, cm$ long, is open at both ends. Which harmonic mode of the pipe resonates a $1.1\, kHz$ source? (Speed of sound in air = $330 \,m \,s^{-1})$
According to the modes of vibration of air column in open organ pipe,
First mode of vibration
$v_{1}=\frac{V}{2l}\,\,\,\,\,\dots(i)$
where $v_{1}=$ frequency of vibration
v= speed of sound
l= length of organ pipe
The frequency of nth mode of vibration
$v_{n} =n \times v_{1}$
$=n \times\left(\frac{V}{2 l}\right)$
Given, $v_{n}=1.1\, kHz =1100\, Hz$
$V=330\, ms ^{-1}$
$l=30\, cm =0.30\, m$
$1100 =n \times\left(\frac{330}{2 \times 0.30}\right)$
$n =\frac{1100 \times 2 \times 0.30}{330}$
$n =\frac{660}{330}=2$
So, second harmonic mode of pipe resonates.