Two wires of same material of radius ' $r$ ' and ' $2 r$ ' respectively are welded together end to end. The…
Two wires of same material of radius ' $r$ ' and ' $2 r$ ' respectively are welded together end to end. The combination is then used as a sonometer wire under tension ' $\mathrm{T}$ '. The joint is kept midway between the two bridges. The ratio of the number of loops in the wires with that the joint is a node is
$1: 5$
$1: 2$
$1: 4$
$1: 3$
Solution
Frequency of vibration will be same for both the segments. If $\mathrm{p}_1$ and $\mathrm{p}_2$ are the number of loops for the wires of radius $r$ and $2 r$ then we have
$\begin{aligned}
& \mathrm{n}=\frac{\mathrm{p}_1}{2 \ell} \sqrt{\frac{\mathrm{T}}{\pi \mathrm{r}^2 \rho}}=\frac{\mathrm{p}_2}{4 \ell} \sqrt{\frac{\mathrm{T}}{\pi(2 \mathrm{r})^2 \rho}} \\
& \therefore \frac{\mathrm{p}_1}{\mathrm{p}_2}=\frac{1}{2}
\end{aligned}$