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 are –

• A. ½
• B. 2/3
• C. 3/4
• D. 4/9

Answer 1

Answer: (D)

4/9

Answer 2

$\frac{\nu_{1}}{\nu_{2}}$ = $\frac{\frac{1}{2\mathcal{l}}\sqrt{\frac{T_{1}}{\mu_{1}}}}{\frac{1}{2\mathcal{l}}\sqrt{\frac{T_{2}}{\mu_{2}}}}$ = $\sqrt{\frac{T_{1}}{T_{2}}\frac{\mu_{2}}{\mu_{1}}}$

= $\sqrt{\frac{T_{1}}{T_{2}}\frac{\rho_{2}A_{2}}{\rho_{1}A_{1}}}$ = $\sqrt{\frac{T_{1}}{T_{2}}\frac{\rho_{2}\pi{r_{2}}^{2}}{\rho_{1}\pi{r_{1}}^{2}}}$

= $\left( \frac{T_{1}}{T_{2}} \right)^{1/2}$ $\left( \frac{\rho_{2}}{\rho_{1}} \right)^{1/2}$ $\left( \frac{r_{2}}{r_{1}} \right)$