If n _ 1 , n _ 2 and n _ 3 are the fundamental frequencies of three...

{n}_{1}, {n}_{2}

and

{n}_{3}

are the fundamental frequencies of three segments into which a string is divided, then the original fundamental frequency

n

of the string is given by: If n _ 1 , n _ 2 and n _ 3 are the fundamental frequencies of three segments into which a string is divided, then the

Options

A $\dfrac{1}{n} = \dfrac{1}{{n}_{1}} + \dfrac{1}{{n}_{2}} + \dfrac{1}{{n}_{3}}$
B $\dfrac{1}{\sqrt{n}} = \dfrac{1}{\sqrt{{n}_{1}}} + \dfrac{1}{\sqrt{{n}_{2}}} + \dfrac{1}{\sqrt{{n}_{3}}}$
C $\sqrt{n} = \sqrt{{n}_{1}} + \sqrt{{n}_{2}} + \sqrt{{n}_{3}}$
D $n = {n}_{1} + {n}_{2} + {n}_{3}$

A. $\dfrac{1}{n} = \dfrac{1}{{n}_{1}} + \dfrac{1}{{n}_{2}} + \dfrac{1}{{n}_{3}}$

If n _ 1 , n _ 2 and n _ 3 are the fundamental frequencies of three segments into which a string is divided, then the
For first segment we have,

${n}_{1} = \dfrac{1}{2{l}_{1}}\sqrt{\dfrac{T}{\mu}}$

For second segment we have,
${n}_{2} = \dfrac{1}{2{l}_{2}}\sqrt{\dfrac{T}{\mu}}$

For third segment we have,
${n}_{3} = \dfrac{1}{2{l}_{3}}\sqrt{\dfrac{T}{\mu}}$

For the whole string we have,
$n = \dfrac{1}{2l}\sqrt{\dfrac{T}{\mu}}$

Total length of the string is,
$l = {l}_{1} + {l}_{2} + {l}_{3}$

Thus, the fundamental frequency for the whole string is,
$\dfrac{1}{n} = \dfrac{1}{{n}_{1}} + \dfrac{1}{{n}_{2}} + \dfrac{1}{{n}_{3}}$