Question

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

• A. 25 kg
• B. 5 kg
• C. 12.5 kg
• D. 1/25 kg

Answer 1

The correct option is A 25 kg

The frequency of vibration of a string $n=\frac{p}{2l}\sqrt{\frac{T}{\mu }}$
Also number of loops = Number of antinodes.
Hence, with 5 antinodes and hanging mass of 9 kg.
We have p=5 and T=9g $\Rightarrow n_1=\frac{5}{2l}\sqrt{\frac{9g}{\mu }}$
with 3 antinodes and hanging mass M
We have p=3 and $T=Mg\Rightarrow n_2=\frac{3}{2l}\sqrt{\frac{Mg}{\mu }}$

$\because n_1=n_2\Rightarrow \frac{5}{2l}\sqrt{\frac{9g}{\mu }}=\frac{3}{2l}\sqrt{\frac{Mg}{\mu }}\Rightarrow M=25kg.$