A vibrating string of certain length $\ell$ under a tension $\mathrm{T}$ resonates with a mode corresponding to the first overtone (third harmonic) of an air column of length $75 \mathrm{~cm}$ inside a tube closed at one end. The string also generates $4$ beats per second when excited along with a tuning fork of frequency $\mathrm{n}$. Now when the tension of the string is slightly increased the number of beats reduces $2$ per second. Assuming the velocity of sound in air to be $340 \mathrm{~m} / \mathrm{s}$, the frequency $\mathrm{n}$ of the tuning fork in $\mathrm{Hz}$ is
IIT 2008, Diffcult
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$ \mathrm{n}_5=\frac{3}{4}\left(\frac{340}{0.75}\right)=\mathrm{n}-4 $
$ \therefore \mathrm{n}=344 \mathrm{~Hz}$
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