Question 401: Waves

Question

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

Accepted Answer

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 $$\mathrm{n} l=\mathrm{k}$$
$$	herefore \quad l_1=\frac{\mathrm{k}}{\mathrm{n}_1}, l_2=\frac{\mathrm{k}}{\mathrm{n}_2} 	ext { and } l_3=\frac{\mathrm{k}}{\mathrm{n} 3}$$ .... (ii)
$$	herefore \quad$$ Original length $$l=\frac{\mathrm{k}}{\mathrm{n}}$$ .... (iii)
Putting eq (ii) and (iii) into eq (i)
$$\begin{aligned}
& \frac{\mathrm{k}}{\mathrm{n}}=\frac{\mathrm{k}}{\mathrm{n}_1}+\frac{\mathrm{k}}{\mathrm{n}_2}+\frac{\mathrm{k}}{\mathrm{n}_3} \
	herefore \quad \frac{1}{\mathrm{n}} & =\frac{1}{\mathrm{n}_1}+\frac{1}{\mathrm{n}_2}+\frac{1}{\mathrm{n}_3}
\end{aligned}$$

Answer

$$\mathrm{n}=\mathrm{n}_1+\mathrm{n}_2+\mathrm{n}_3$$

Answer

$$\sqrt{\mathrm{n}}=\sqrt{\mathrm{n}_1}+\sqrt{\mathrm{n}_2}+\sqrt{\mathrm{n}_3}$$

Answer

$$\frac{1}{\mathrm{n}}=\frac{1}{\mathrm{n}_1}+\frac{1}{\mathrm{n}_2}+\frac{1}{\mathrm{n}_3}$$

Answer

$$\frac{1}{\sqrt{\mathrm{n}}}=\frac{1}{\sqrt{\mathrm{n}_1}}+\frac{1}{\sqrt{\mathrm{n}_2}}+\frac{1}{\sqrt{\mathrm{n}_3}}$$

Waves

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

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