When a string of length ' $l$ ' is divided into three segments of length $l_1, l_2$ and $l_3$. The…

When a string of length ' $l$ ' is divided into three segments of length $l_1, l_2$ and $l_3$. The fundamental frequencies of three segments are $\mathrm{n}_1, \mathrm{n}_2$ and $\mathrm{n}_3$ respectively. The original fundamental frequency ' $n$ ' of the string is
  1. $\mathrm{n}=\mathrm{n}_1+\mathrm{n}_2+\mathrm{n}_3$
  2. $\sqrt{\mathrm{n}}=\sqrt{\mathrm{n}_1}+\sqrt{\mathrm{n}_2}+\sqrt{\mathrm{n}_3}$
  3. $\frac{1}{\mathrm{n}}=\frac{1}{\mathrm{n}_1}+\frac{1}{\mathrm{n}_2}+\frac{1}{\mathrm{n}_3}$
  4. $\frac{1}{\sqrt{\mathrm{n}}}=\frac{1}{\sqrt{\mathrm{n}_1}}+\frac{1}{\sqrt{\mathrm{n}_2}}+\frac{1}{\sqrt{\mathrm{n}_3}}$

Solution

The fundamental frequency of a string is given by Given: $l=l_1+l_2+l_3$... (i) $\mathrm{n}=\frac{1}{2 l} \sqrt{\frac{\mathrm{T}}{\mathrm{m}}}$ $\Rightarrow \mathrm{n} \propto \frac{1}{l}$ or $\mathrm{n} l=\mathrm{k}$ $\therefore \quad l_1=\frac{\mathrm{k}}{\mathrm{n}_1}, l_2=\frac{\mathrm{k}}{\mathrm{n}_2}$ and $l_3=\frac{\mathrm{k}}{\mathrm{n} 3}$... (ii) $\therefore \quad$ Original length $l=\frac{\mathrm{k}}{\mathrm{n}}$... (iii) Putting eq (ii) and (iii) into eq (i) $\frac{\mathrm{k}}{\mathrm{n}}=\frac{\mathrm{k}}{\mathrm{n}_1}+\frac{\mathrm{k}}{\mathrm{n}_2}+\frac{\mathrm{k}}{\mathrm{n}_3}$ $\therefore \quad \frac{1}{n}=\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3}$

Asked in: MHT CET 2023 (09 May Shift 1)

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