Question

The ends of two wires of radii $r$ and $2r$ are mutually joined. The compound wire is used in a sonometer. The joint is kept at the centre between two bridges and tension $T$ is produced into it. If the joint is a node, then the ratio of the number of loops produced in two wires will be

• A. $1:4$
• B. $1:5$
• C. $3:1$
• D. $1:2$

Answer 1

The correct option is D $1:2$

Frequency of a sonometer is given by
$\nu =\frac{1}{2l}\sqrt{\frac{T}{m}}$, where m is the mass per unit length.
$m=\pi r^2\rho$, where $\rho$ is density.
Ratio of frequencies of the two rods can be give by
$\frac{{\nu }_2}{{\nu }_1}=\frac{\frac{1}{2l}\sqrt{\frac{T}{\pi (2r{)}^2\rho }}}{\frac{1}{2l}\sqrt{\frac{T}{\pi r^2\rho }}}=\frac{1}{2}$.
The number of loops produced in the wires will be $1:2$.