MCQ Solution: Waves
A sonometre wire resonates with a given tuning forck forming standing waves with five antinodes between the two bridges when a mass of 9kg is suspended from the wire. When this mass is replaced by mass M, the wire resonates with the same positions of the bridges. Then find the value of square roof of M.
1) 5
2) 10
3) 25
4) none
Solution
The frequency of vibration of a string
$$n=\dfrac{p}{2l}\sqrt{\dfrac{T}{m}}$$
Also number of loops = Number of antinodes.
Hence with 5 antinodes and hanging mass of 9 kg. we have p=5 and T=9g
So,
$$n_1=\dfrac{5}{2l}\sqrt{\dfrac{9g}{m}}$$
With 3 antinodes and hanging mass M we have p=3 and T=Mg so,
$$n_2=\dfrac{3}{2l}\sqrt{\dfrac{Mg}{m}}$$
$$\because n_1=n_2$$
$$\dfrac{5}{2l}\sqrt{\dfrac{9g}{m}}=\dfrac{3}{2l}\sqrt{\dfrac{Mg}{m}}$$
Squaring both side we get
$$25\times9=9\times M$$
$$M=25\ kg$$
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