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)

$v=\frac{1}{2l}\sqrt{\frac{T}{\mu }}$
$=\frac{1}{2l}\sqrt{\frac{T}{\rho \cdot \pi r^2}}$
$=\frac{1}{2l\cdot r}\sqrt{\frac{T}{\rho \cdot \pi }}$
Here $l$ is same for both the wires, so
$\frac{v_1}{v_2}=\frac{r_2}{r_1}\cdot \sqrt{\frac{T_1}{r_1}\cdot \frac{{\rho }_2}{{\rho }_1}}$
$=\frac{1}{3}\sqrt{\frac{2}{1}\cdot \frac{2}{1}}$
$\Rightarrow \frac{v_1}{v_2}=\frac{2}{3}$