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An air column in a pipe, which is closed at one end will be in resonance with a vibrating tuning fork of frequency $264 \mathrm{~Hz}$ for various lengths. Which one of the following lengths is not possible? $(\mathrm{V}=330 \mathrm{~m} / \mathrm{s})$
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The correct answer is:
$62.50 \mathrm{~cm}$
$$
\mathrm{n}=264 \mathrm{~Hz}, \mathrm{~V}=300 \mathrm{~m} / \mathrm{s}
$$
For fundamental frequency, $\mathrm{n}=\frac{\mathrm{V}}{4 \ell}$
$$
\ell=\frac{\mathrm{V}}{4 \mathrm{n}}=\frac{330}{4 \times 264}=0.3125 \mathrm{n}=31.25 \mathrm{~cm}
$$
For fundamental mode, $\ell=\frac{\lambda}{4}$ other possible lengths are $\frac{3 \lambda}{4}, \frac{5 \lambda}{4}, \ldots \ldots$ Hence, $62.50 \mathrm{~m}$ is not possible.
\mathrm{n}=264 \mathrm{~Hz}, \mathrm{~V}=300 \mathrm{~m} / \mathrm{s}
$$
For fundamental frequency, $\mathrm{n}=\frac{\mathrm{V}}{4 \ell}$
$$
\ell=\frac{\mathrm{V}}{4 \mathrm{n}}=\frac{330}{4 \times 264}=0.3125 \mathrm{n}=31.25 \mathrm{~cm}
$$
For fundamental mode, $\ell=\frac{\lambda}{4}$ other possible lengths are $\frac{3 \lambda}{4}, \frac{5 \lambda}{4}, \ldots \ldots$ Hence, $62.50 \mathrm{~m}$ is not possible.
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