Question

A string of length $l$ fixed at both ends vibrates in resonance with a tuning fork of frequency $\text{'}f\text{'}$ at two successive values of tension $T_1$ and $T_2$ in the string. Find the specific mass (mass per unit length) of the string.

• A. $\frac{4T_1{T}_2}{f^2{l}^2{(\sqrt{T_1}+\sqrt{T_2})}^2}$
• B. $\frac{T_1{T}_2}{4f^2{l}^2{(\sqrt{T_1}-\sqrt{T_2})}^2}$
• C. $\frac{4T_1{T}_2}{f^2{l}^2{(\sqrt{T_1}-\sqrt{T_2})}^2}$
• D. $\frac{2T_1{T}_2}{f^2{l}^2{(\sqrt{T_1}+\sqrt{T_2})}^2}$

Answer 1

The correct option is B $\frac{T_1{T}_2}{4f^2{l}^2{(\sqrt{T_1}-\sqrt{T_2})}^2}$

$f=\frac{n}{2l}\sqrt{\frac{T_1}{\mu }}.....\left(1\right)f=\frac{n+1}{2l}\sqrt{\frac{T_2}{\mu }}...\left(2\right)$
from equations (1) and (2) eliminate $\text{'}n\text{'}$
to get $\mu$
$\mu =\frac{T_1{T}_2}{4f^2{l}^2{(\sqrt{T_1}-\sqrt{T_2})}^2}$