Question

If n₁, n₂ and n₃ 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,

$1 over straight n space equals space 1 over n subscript 1 plus 1 over n subscript 2 plus 1 over n subscript 3$
$fraction numerator 1 over denominator square root of straight n end fraction space equals space fraction numerator 1 over denominator square root of n subscript italic 1 end root end fraction plus fraction numerator 1 over denominator square root of n subscript italic 2 end root end fraction plus fraction numerator 1 over denominator square root of n subscript 3 end root end fraction$
$square root of straight n space equals square root of n subscript italic 1 end root space plus square root of n subscript italic 2 end root space plus space space square root of n subscript 3 end root$

n = n₁ + n₂ + n₃

Solution

1 over straight n space equals space 1 over n subscript 1 plus 1 over n subscript 2 plus 1 over n subscript 3

For first part, n₁ =

fraction numerator straight v over denominator 2 straight l subscript 1 end fraction space rightwards double arrow space l subscript 1 space equals space fraction numerator v over denominator 2 n subscript 1 end fraction

For second part, n₂ =

fraction numerator straight v over denominator 2 straight l subscript 2 end fraction space rightwards double arrow space l subscript 2 space equals space fraction numerator v over denominator 2 n subscript 2 end fraction

For third part, n₃ =

fraction numerator straight v over denominator 2 straight l subscript 3 end fraction space rightwards double arrow space l subscript 3 space equals space fraction numerator v over denominator 2 n subscript 3 end fraction

For the complete wire,

straight n space equals space fraction numerator straight v over denominator 2 straight l end fraction space rightwards double arrow space straight l space equals space fraction numerator straight v over denominator 2 straight n end fraction

We have, l = l₁ + l₂ + l₃
That is,

fraction numerator straight v over denominator 2 straight n end fraction space equals space fraction numerator v over denominator 2 n subscript 1 end fraction space plus space fraction numerator v over denominator 2 n subscript 2 end fraction plus fraction numerator v over denominator 2 n subscript 3 end fraction space 1 over n space equals space 1 over n subscript 1 space plus space 1 over n subscript 2 plus 1 over n subscript 3

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