Question

A sonometer consists of two wires of lengths $1.5m$ and $1m$ made up of different materials whose densities are $5g/cc$ and $8g/cc$, and their respective radii are in the ratio $4:3$. The ratio of tensions in these two wires, if their fundamental frequencies are equal is

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

Answer 1

Answer: (B)

$5:2$

Given :$\frac{l_1}{l_2}=\frac{1.5}{1}$ $\frac{{\rho }_1}{{\rho }_2}=\frac{5}{8}$
$\frac{r_1}{r_2}=\frac{4}{3}$
${\nu }_1={\nu }_2$
$⟹\frac{\sqrt{\frac{T_1}{{\rho }_1\pi {r_1}^2}}}{2l_1}=\frac{\sqrt{\frac{T_2}{{\rho }_2\pi {r_2}^2}}}{2l_2}$
$\frac{T_1}{{\rho }_1{r_1}^2{l_1}^2}=\frac{T_2}{{\rho }_2{r_2}^2{l_2}^2}$
$\frac{T_1}{T_2}=\frac{\left({\rho }_1\right)}{{\rho }_2}{\left(\frac{r_1}{r_2}\right)}^2{\left(\frac{l_1}{l_2}\right)}^2$
$=\left(\frac{5}{8}\right){\left(\frac{4}{3}\right)}^2{\left(\frac{1.5}{1}\right)}^2$
$=\frac{5}{2}.$