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Waves and Sound — JEE physics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceJEE physicsWaves and Sound1 Mark
MCQ
A sonometer wire resonates with a given tuning fork forming standing waves with five antinodes between the two bridges when a mass of $9 \mathrm{~kg}$ is suspended from the wire. When this mass is replaced by a mass $\mathrm{M}$, the wire resonates with the same tuning fork forming three antinodes for the same positions of the bridges. The value of $M$ is
  • ✓
    $25 \mathrm{~kg}$
  • B
    $5 \mathrm{~kg}$
  • C
    $12.5 \mathrm{~kg}$
  • D
    $1 / 25 \mathrm{~kg}$
✓

Answer

Correct option: A.
$25 \mathrm{~kg}$
The frequency of vibration of a string $n=\frac{p}{2 l} \sqrt{\frac{T}{m}}$
Also number of loops $=$ Number of antinodes.
Hence, with 5 antinodes and hanging mass of $9 \mathrm{~kg}$.
We have $p=5$ and $T=9 g \Rightarrow n_1=\frac{5}{2 l} \sqrt{\frac{9 g}{m}}$With $3$ antinodes and hanging mass $M$
We have $p=3$ and $T=M g \Rightarrow n_2=\frac{3}{2 l} \sqrt{\frac{M g}{m}}$
$\because \quad n=n \Rightarrow \frac{5}{2 l} \sqrt{\frac{9 g}{m}}=\frac{3}{2 l} \sqrt{\frac{M g}{m}} \Rightarrow M=25 \mathrm{~kg} .$

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