If $n_1$, $n_2$ and $n_3$ are the fundamental frequencies of three segments into which a string is divided,…

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
  1. $\frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3}$
  2. $\frac{1}{\sqrt{n}} = \frac{1}{\sqrt{n_1}} + \frac{1}{\sqrt{n_2}} + \frac{1}{\sqrt{n_3}}$
  3. $\sqrt{n} = \sqrt{n_1} + \sqrt{n_2} + \sqrt{n_3}$
  4. $n = n_1 + n_2 + n_3$

Solution

If fundamental frequency of each part is calculated, then the fundamental frequency of complete wire can be calculated. A wire of total length $l$ is divided into three segments of lengths $l_1$, $l_2$, and $l_3$ using movable bridges. For 1st part, $n_1 = \frac{v}{2l_1} \Rightarrow l_1 = \frac{v}{2n_1}$ For 2nd part, $n_2 = \frac{v}{2l_2} \Rightarrow l_2 = \frac{v}{2n_2}$ For 3rd part, $n_3 = \frac{v}{2l_3} \Rightarrow l_3 = \frac{v}{2n_3}$ For the complete wire, $n = \frac{v}{2l} \Rightarrow l = \frac{v}{2n}$ Now, $l = l_1 + l_2 + l_3$ $\frac{v}{2n} = \frac{v}{2n_1} + \frac{v}{2n_2} + \frac{v}{2n_3}$ $\Rightarrow \frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3}$

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