A stretched string under certain load has fundamental frequency $f$ . When the string is divided into two parts, fundamental frequencies of both parts are $f_{1}$ and $f_{2}$ under same tension. Which of the following relation is correct ?

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$f = \frac{f_{1} f_{2}}{f_{1} - f_{2}}$

$f = \frac{f_{1}^{} f_{2}^{}}{f_{1} + f_{2}}$

$f = \frac{f_{1}^{} + f_{2}^{}}{2}$

$f = \frac{\sqrt{f_{1}^{2} + f_{2}^{2}}}{2}$

answer is A.

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Detailed Solution

$f = \frac{1}{2 l} \sqrt{\frac{T}{μ}} \Rightarrow l = \frac{1}{2 f} \sqrt{\frac{T}{μ}}$

$\begin{array}{l} but l = l_{1} + l_{2} \\ \frac{1}{2 f} \sqrt{\frac{T}{μ}} = \frac{1}{2 f_{1}} \sqrt{\frac{T}{μ}} + \frac{1}{2 f_{2}} \sqrt{\frac{T}{μ}} \\ f = (\frac{f_{1} f_{2}}{f_{1} + f_{2}}) \end{array}$

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