Question

When a string is divided into three segments of lengths ${\text{l}}_1$ , ${\text{l}}_2$ and ${\text{l}}_3$ 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. $\sqrt{\nu }=\sqrt{{\nu }_1}+\sqrt{{\nu }_2}+\sqrt{{\nu }_3}$
• B. $\nu ={\nu }_1+{\nu }_2+{\nu }_3$
• C. $\frac{1}{\nu }=\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}+\frac{1}{{\nu }_3}$
• D. $\frac{1}{\sqrt{\nu }}=\frac{1}{\sqrt{{\nu }_1}}+\frac{1}{\sqrt{{\nu }_2}}+\frac{1}{\sqrt{{\nu }_3}}$

Answer 1

Answer: (C)

Let $\text{l}$ be the length of the string.
The fundamental frequency is given by
$\nu =\frac{1}{2l}\sqrt{\frac{T}{\mu }}$
$\text{or}\nu \propto \frac{1}{l}$ ( $\because T$ and $\mu$ are constant )
$\text{or}\nu \propto \frac{k}{l}$ ( where $k$ is a constant )
Here, $l_1=\frac{k}{{\nu }_1}$ , $l_2=\frac{k}{{\nu }_2}$ , $l_3=\frac{k}{{\nu }_3}$ and $l=\frac{k}{\nu }$
But $\text{l}={\text{l}}_1+{\text{l}}_2+{\text{l}}_3$
$\therefore \frac{1}{\nu }=\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}+\frac{1}{{\nu }_3}$