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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 $v_1,\,v_2$ and $v_3$ respectively. The original fundamental frequency $(v)$ of the string is
$\frac{1}{v} = \frac{1}{{{v_1}}} + \frac{1}{{{v_2}}} + \frac{1}{{{v_3}}}$
$\frac{1}{{\sqrt v }} = \frac{1}{{\sqrt {{v_1}} }} + \frac{1}{{\sqrt {{v_2}} }} + \frac{1}{{\sqrt {{v_3}} }}$
$\sqrt v = \sqrt {{v_1}} + \sqrt {{v_2}} + \sqrt {{v_3}} $
$v = v_1 + v_2 + v_3$
Solution
$V=\frac{1}{2 l} \sqrt{\frac{T}{m}}$
$v_{1} l_{1}=v_{2} l_{2}=v_{3} l_{3}=k$
from eq. $(1)$
$l_{1}=\frac{k}{v_{1}}, l_{2}=\frac{k}{v_{2}}, l_{3}=\frac{k}{v_{3}}$
original length
$l=\frac{k}{v}$
Here$, l=l_{1}+l_{2}+l_{3}$
$\frac{k}{v}=\frac{k_{1}}{v_{1}}+\frac{k_{2}}{v_{2}}+\frac{k_{3}}{v_{3}}$
$\frac{1}{v}=\frac{1}{v_{1}}+\frac{1}{v_{2}}+\frac{1}{v_{3}}$
