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A pipe, 30.0 cm long. Is open at both en...

A pipe, 30.0 cm long. Is open at both ends. Which harmonic mode of the pipe resonates a 1.1 kHz source? Will resonance with the same source be observed if one end of the pipe is closed ? Take the speed of sound in air as `330 m s^(-1)`.

Text Solution

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For an open pipe, `n^(th)` harmonic is `f _(n) = (nv)/(2L)`
`therefore 1.1 xx 10^(3) = (n xx 330)/(2xx 0.3) implies n=2`
`implies` Resonance will be produced with second harmonic.
Now, when one end of open pipe is closed, it becomes a closed pipe for which first harmonic
is `f _(1) =(v)/(4L) = (330)/(4 xx 0.3) =275 Hz`
Now in case of cloed pipe, only odd multiples of first harmonic are possible. So, here possible harmonics are `f ._(n) = n.f._(1)=(2n-1_ f._(1)`
(where `n = 1,2,3,...)`
`(n.. =2n-1=` order of harmonic)
`therefore` Different values of possible harmonics will be
`f ._(n) =f._(1) , 3f._(1) 5g._(1) 7f._(1),....`
`=275 Hz, (3 xx 275) Hz,`
`(5xx 275) Hz, (7xx 275) Hx,...`
`=275Hz, 825Hz, 1375Hz, 1925 Hz,...`
Thus we can see that none of above frequency values mathches with the frequency `1.1kHz=1100 Hz` of source and so resonance will not take place at the time of closing one end of an open pipe.
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