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The fundamental frequency of an air column in pipe ' $\mathrm{A}$ ' closed at one end coincides with second overtone of pipe ' $\mathrm{B}$ ' open at both ends. The ratio of length of pipe ' $\mathrm{A}$ ' to that of pipe ' $\mathrm{B}$ ' is
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The correct answer is:
$1: 6$
Fundamental frequency of pipe closed at one end
$$
\mathrm{n}=\frac{\mathrm{v}}{4 \ell}
$$
Second overtone of pipe open at both the ends
$$
\begin{aligned}
& \mathrm{n}^{\prime}=\left(\frac{3 \mathrm{v}}{2 \ell^{\prime}}\right) \\
& \mathrm{n}^{\prime}=\mathrm{n} \\
& \therefore \frac{3 \mathrm{v}}{2 \ell^{\prime}}=\frac{\mathrm{v}}{4 \ell} \\
& \therefore \frac{\ell}{\ell^{\prime}}=\frac{1}{6}
\end{aligned}
$$
$$
\mathrm{n}=\frac{\mathrm{v}}{4 \ell}
$$
Second overtone of pipe open at both the ends
$$
\begin{aligned}
& \mathrm{n}^{\prime}=\left(\frac{3 \mathrm{v}}{2 \ell^{\prime}}\right) \\
& \mathrm{n}^{\prime}=\mathrm{n} \\
& \therefore \frac{3 \mathrm{v}}{2 \ell^{\prime}}=\frac{\mathrm{v}}{4 \ell} \\
& \therefore \frac{\ell}{\ell^{\prime}}=\frac{1}{6}
\end{aligned}
$$
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