Question

Two strings A and B of lengths, $L_A=80cm$ and $L_B=xcm$ respectively are used separately in a sonometer. The ratio of their densities $\left(d_A/d_B\right)$ is $0.81$. The diameter of B is half of that of A. If the strings have the same tension and fundamental frequency, the value of $x$ is

• A. $33$
• B. $102$
• C. $144$
• D. $130$

Answer 1

The correct option is C $144$

frequency $f=\frac{1}{2l}\sqrt{\frac{T}{\mu }}=\frac{1}{2l}\sqrt{\frac{T}{\rho \left(A\right)}}$
$\therefore \frac{f_1}{f_2}=1\Rightarrow \frac{l_2}{l_1}=\sqrt{\frac{{\rho }_2{A}_2}{{\rho }_1{A}_1}}$
$=\sqrt{\frac{1}{0.81}\times {\left(\frac{1}{2}\right)}^2}$
$\Rightarrow l_2=\frac{l_1}{0.9\times 2}$
$l_1=0.9\times l_2\times 2$
$=0.9\times 80\times 2$
$=144cm$