Question

Two wires of radii $r$ and $2r$ respectively are welded together end to end. This combination is used as a sonometer wire kept under tension $T$. The welded point is midway between two bridges. What would be the ratio of the number of loops formed in the wires such that the joint is a node when stationary waves are set up in the wire?

• A. $2:1$
• B. $1:2$
• C. $1:4$
• D. $4:1$

Answer 1

The correct option is B $1:2$

Let $n_1$ and $n_2$ be the no of loops formed in the wires of radii $r$ and $2r$ respectively.
$f_1=\frac{n_1}{2l}\sqrt{\frac{T}{{\mu }_1}}$
$f_2=\frac{n_2}{2l}\sqrt{\frac{T}{{\mu }_2}}$
where ${\mu }_1$ and ${\mu }_2$ are mass per unit lengths of the wires.
Since the welded wire is continuous, $f_1=f_2$
$\therefore \frac{n_1}{n_2}=\sqrt{\frac{{\mu }_1}{{\mu }_2}}=\sqrt{\frac{\pi r^2}{\pi (2r{)}^2}}=\sqrt{\frac{1}{4}}=\frac{1}{2}$