An organ pipe of length $L$ open at both ends is found to vibrate in its first harmonic when sounded with a tuning fork of $480\, Hz$. What should be the length of a pipe closed at one end, so that it also vibrates in its first harmonic with the same tuning fork ?
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For open organ pipe, frequency of first harmonic,
$v=\frac{v}{2 L}=480 \mathrm{\,Hz}$
For a pipe closed at one end, frequency of first harmonic,
${v^{\prime}=\frac{v}{4 L^{\prime}}=480 \mathrm{\,Hz}}$
$\therefore 4{{\rm{L}}^\prime } = 2{\rm{L}}$ or ${{\rm{L}}^\prime } = \frac{{2{\rm{L}}}}{4} = \frac{{\rm{L}}}{2}$
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