An organ pipe P_1 closed at one end has a vib | vedclass

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(A) Since both pipes are in resonance with the same tuning fork,they must have the same frequency of vibration,$f$.For an organ pipe closed at one end

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$3$:$8$

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(A) Since both pipes are in resonance with the same tuning fork,they must have the same frequency of vibration,$f$.For an organ pipe closed at one end,the frequency of the $n$-th overtone is given by $f = \frac{(2n+1)v}{4l_1}$,where $n$ is the overtone number. For the first overtone $(n=1)$,the frequency is $f = \frac{3v}{4l_1}$.For an organ pipe open at both ends,the frequency of the $n$-th overtone is given by $f = \frac{(n+1)v}{2l_2}$. For the third overtone $(n=3)$,the frequency is $f = \frac{(3+1)v}{2l_2} = \frac{4v}{2l_2} = \frac{2v}{l_2}$.Equating the frequencies: $\frac{3v}{4l_1} = \frac{2v}{l_2}$.Rearranging to find the ratio $\frac{l_1}{l_2}$: $\frac{l_1}{l_2} = \frac{3v}{4l_1} \cdot \frac{l_2}{2v} = \frac{3}{8}$.

An organ pipe $P_1$ closed at one end has a vibrating air column in its first overtone,and another pipe $P_2$ open at both ends has a vibrating air column in the third overtone. Both are in resonance with a given tuning fork. The ratio of the length of pipe $P_1$ to that of $P_2$ is

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