Question

When a string is divided into three segments of lengths $\textit{l}_{1}$ , $\textit{l}_{2}$ and $\textit{l}_{3}$ 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 $\textit{l}$ be the length of the string.
The fundamental frequency is given by
$\nu=\frac{1}{2 l}\sqrt{\frac{T }{\mu }}$
$\text{or}\nu \propto \frac{1}{l }$ ( $\because T$ 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 $\textit{l}=\textit{l}_{1}+\textit{l}_{2}+\textit{l}_{3}$
$\therefore \frac{1}{\nu}=\frac{1}{\nu_{1}}+\frac{1}{\nu_{2}}+\frac{1}{\nu_{3}}$