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  4. 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

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Q. If $n_1, n_2$ 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

AIPMTAIPMT 2014Waves

Solution:

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$n_1=\frac{1}{2l_1}\sqrt{\frac{T}{\mu}}...(i)$
$n_2=\frac{1}{2l_2}\sqrt{\frac{T}{\mu}} ...(ii)$
$n_3=\frac{1}{2l_3}\sqrt{\frac{T}{\mu}} ...(iii)$
$n=\frac{1}{2l}\sqrt{\frac{T}{\mu}}...(iv)$
From eqn. (iv), we get
$\frac{1}{n}=\frac{2l}{\sqrt{\frac{T}{\mu}}}$
As $l=l_1+l_2+l_3$
$\therefore \frac{1}{n}=\frac{2(l_1+l_2+l_3)}{\sqrt{\frac{T}{\mu}}}$
$=\frac{2l_1}{\sqrt{\frac{T}{\mu}}}+\frac{2l_2}{\sqrt{\frac{T}{\mu}}}+\frac{2l_3}{\sqrt{\frac{T}{\mu}}}$
$=\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3}$ [Using (i), (ii) and (iii)]