Question

A sonometer wire resonates with a given tuning fork, forming standing waves with five antinodes between the two bridges when a mass of $9\text{kg}$ is suspended from the wire. When a mass of $M\text{kg}$ is suspended from the wire, the same tuning fork causes the wire to form three antinodes for the same positions of the bridges. What is the value of $M$?

• A. $25\text{kg}$
• B. $5\text{kg}$
• C. $12.5\text{kg}$
• D. $\frac{1}{25}\text{kg}$

Answer 1

The correct option is A $25\text{kg}$

For vibrating string, frequency
$f_n=\frac{n}{2l}\sqrt{\frac{T}{\mu }}$
where, $n$ = no of loops, $T$ = Tension, $\mu$ = mass per unit length
For five antindoes, $n=5$


$\therefore f_5=\frac{5}{2l}\sqrt{\frac{9g}{\mu }}$
For, three antinodes, $n=3$


$f_3=\frac{3}{2l}\sqrt{\frac{Mg}{\mu }}$
Since, $f_5=f_3$ (for sonometer)
$\Rightarrow \frac{5}{2l}\sqrt{\frac{9g}{\mu }}=\frac{3}{2l}\sqrt{\frac{Mg}{\mu }}$
$(\because \mu =\text{constant})$
$\Rightarrow 5\sqrt{9}=3\sqrt{M}$
$\Rightarrow M={\left(\frac{5}{3}\right)}^2\times 9=25\text{kg}$