Question

When a string is divided into three segments of length $l_1,l_2$ and $l_3$, the fundamental frequencies of these three segments are ${\nu }_1,{\nu }_2$ and ${\nu }_3$ respectively. The original fundamental frequency $\left(\nu \right)$ of the string is

• A. $\frac{1}{\nu }=\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}+\frac{1}{{\nu }_3}$
• B. $\frac{1}{\sqrt{\nu }}=\frac{1}{\sqrt{{\nu }_1}}+\frac{1}{\sqrt{{\nu }_2}}+\frac{1}{\sqrt{{\nu }_3}}$
• C. $\sqrt{\nu }=\sqrt{{\nu }_1}+\sqrt{{\nu }_2}+\sqrt{{\nu }_3}$
• D. $\nu ={\nu }_1+{\nu }_2+{\nu }_3$

Answer 1

The correct option is A

$\frac{1}{\nu }=\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}+\frac{1}{{\nu }_3}$

Let fundamental frequency $\left(\nu \right)=\frac{v}{2l}$
$l=l_1+l_2+l_3...\left(1\right)$
we can write
${\nu }_1=\frac{v}{2l_1},{\nu }_2=\frac{v}{2l_2},{\nu }_3=\frac{v}{2l_3}$
$l_1=\frac{v}{2{\nu }_1},l_2=\frac{v}{2{\nu }_2},l_3=\frac{v}{2{\nu }_3}$
Put this value in equation (1)
$\frac{v}{2\nu }=\frac{v}{2{\nu }_1}+\frac{v}{2{\nu }_2}+\frac{v}{2{\nu }_3}$
$\frac{1}{\nu }=\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}+\frac{1}{{\nu }_3}$