A sonometer wire resonates with a given tuning fork forming standing waves with five antinodes between the two bridges when a mass of $9 kg$ is suspended from the wire. When this mass is replaced by a mass $M$, the wire resonates with the same tuning fork forming three antinodes for the same positions of the bridges. The value of $M$ is ... $kg$
IIT 2002, Diffcult
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(a) The frequency of vibration of a string $n = \frac{p}{{2l}}\sqrt {\frac{T}{m}} $
Also number of loops = Number of antinodes.
Hence, with $5$ antinodes and hanging mass of $9 kg$.
We have $p = 5$ and $T = 9g$ ==> ${n_1} = \frac{5}{{2l}}\sqrt {\frac{{9g}}{m}} $
With $3$ antinodes and hanging mass $M$
We have $p = 3$ and $T = Mg$
==> ${n_2} = \frac{3}{{2l}}\sqrt {\frac{{Mg}}{m}} $
$ \because n_1 = n_2$
==> $\frac{5}{{2l}}\sqrt {\frac{{9g}}{m}} = \frac{3}{{2l}}\sqrt {\frac{{Mg}}{m}} $==> $M = 25\, kg.$
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