Question

Two wires are kept tight between the same pair of supports. The tensions in the wires are in the ratio 2: 1 , the radii are in the ratio 3: 1 and the densities are in the ratio $1: 2 .$ The ratio of their fundamental frequencies is

• A. 2: 3
• B. 2: 4
• C. 2: 5
• D. 2: 6

Answer 1

Answer: (A)

$ v=\frac{1}{2 l} \sqrt{\frac{T}{\mu}}$
$=\frac{1}{2 l} \sqrt{\frac{T}{\rho \cdot \pi r^{2}}}$
$=\frac{1}{2 l \cdot r} \sqrt{\frac{T}{\rho \cdot \pi}}$
Here $l$ is same for both the wires, so
$\frac{v_{1}}{v_{2}}=\frac{r_{2}}{r_{1}} \cdot \sqrt{\frac{T_{1}}{r_{1}} \cdot \frac{\rho_{2}}{\rho_{1}}}$
$=\frac{1}{3} \sqrt{\frac{2}{1} \cdot \frac{2}{1}} $
$\Rightarrow \frac{v_{1}}{v_{2}}=\frac{2}{3}$