A sonometer wire resonates with 4 antinodes between two bridges for a given tuning fork, when $1…
A sonometer wire resonates with 4 antinodes between two bridges for a given tuning fork, when $1 \mathrm{~kg}$ mass is suspended from the wire. Using same fork, when mass $M$ is suspended, the wire resonates producing 2 antinodes between the two bridges (distance between two bridges is as before). The value of $M$ is
$2.5 \mathrm{~kg}$
$3.5 \mathrm{~kg}$
$4 \mathrm{~kg}$
$1 \mathrm{~kg}$
Solution
If $\mathrm{p}$ is the number of loops (or antinodes) then we have,
$\begin{aligned}
& \mathrm{f}=\frac{\mathrm{p}}{2 \ell} \sqrt{\frac{\mathrm{T}}{\mu}} \\
& \text { or } \quad \mathrm{Tp}^2=\text { constant } \\
& \therefore \mathrm{T}_1 \mathrm{p}_1^2=\mathrm{T}_2 \mathrm{p}_2^2 \\
& \text { or } \frac{\mathrm{T}_2}{\mathrm{~T}_1}=\frac{\mathrm{p}_1^2}{\mathrm{p}_2^2}=\left(\frac{4}{2}\right)^2=4 \\
& \therefore \mathrm{T}_2=4 \mathrm{~T}_1=4 \mathrm{~kg}
\end{aligned}$
^