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
Text Solution
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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 ⇒ ⇒ $n_1 = \frac{5}{21} \sqrt{\frac{9g}{m}}$
With 3 antinodes and hanging mass M
We have p = 3 and T = Mg ⇒ ⇒ $n_2 = \frac{3}{21} \sqrt{\frac{Mg}{m}}$
: n 1 = n 2 ⇒ ⇒ $\frac{5}{21} \sqrt{\frac{9g}{m}} = \frac{3}{21} \sqrt{\frac{Mg}{m}}$ ⇒ ⇒ $\mathbf{M} = 25$ kg.
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