The first overtone of the vibrating air colum | vedclass

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(B) The frequency of an open pipe of length $L_O$ is given by $f_n = \frac{n v}{2 L_O}$,where $n = 1, 2, 3, \dots$ is the harmonic number. The second

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$0.75$

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(B) The frequency of an open pipe of length $L_O$ is given by $f_n = \frac{n v}{2 L_O}$,where $n = 1, 2, 3, \dots$ is the harmonic number. The second overtone corresponds to the $3^{rd}$ harmonic $(n=3)$. Thus,$f_{O,2} = \frac{3 v}{2 L_O}$.For a closed pipe of length $L_C$,the frequency is given by $f_m = \frac{(2m-1) v}{4 L_C}$,where $m = 1, 2, 3, \dots$ is the overtone number. The first overtone corresponds to $m=2$ (the $3^{rd}$ harmonic). Thus,$f_{C,1} = \frac{3 v}{4 L_C}$.Given that $f_{O,2} = f_{C,1}$,we have:$\frac{3 v}{2 L_O} = \frac{3 v}{4 L_C}$$\frac{1}{2 L_O} = \frac{1}{4 L_C}$$L_C = \frac{L_O}{2} = \frac{1.5 \ m}{2} = 0.75 \ m$.

The first overtone of the vibrating air column of a closed pipe has the same frequency as the second overtone of an open pipe $1.5 \ m$ long. The length of the closed pipe is (in $m$)

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