If n1, n2 and n3 are the fundamental frequencies of three...

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 (A) $n=n_{1}+n_{2}+n_{3}$ (B) $\frac{1}{n}=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\frac{1}{n_{3}}$ (C) $\frac{1}{\sqrt{n}}=\frac{1}{\sqrt{n_{1}}}+\frac{1}{\sqrt{n_{2}}}+\frac{1}{\sqrt{n_{3}}}$ (D) $\sqrt{n}=\sqrt{n_{1}}+\sqrt{n_{2}}+\sqrt{n_{3}}$

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
(A) $n=n_{1}+n_{2}+n_{3}$
(B) $\frac{1}{n}=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\frac{1}{n_{3}}$
(C) $\frac{1}{\sqrt{n}}=\frac{1}{\sqrt{n_{1}}}+\frac{1}{\sqrt{n_{2}}}+\frac{1}{\sqrt{n_{3}}}$
(D) $\sqrt{n}=\sqrt{n_{1}}+\sqrt{n_{2}}+\sqrt{n_{3}}$

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Key Concepts

Fundamental Frequency of a Vibrating String

The fundamental frequency of a vibrating string, when under uniform tension and with constant mass per unit length, is given by a relation where the frequency is inversely proportional to the length of the string. This concept shows that if the length increases, the frequency decreases, and vice versa, provided other parameters like tension remain constant.

Relating Segment Lengths to Overall Frequency

When a string is divided into segments, each segment behaves as an independent vibrating string with its own fundamental frequency determined by its length. Since the full string is assembled by these segments end?to?end, its total length is the sum of the individual segment lengths. By expressing the frequencies in terms of lengths, one can deduce that the inverse of the full string's frequency is equal to the sum of the inverses of the segments' frequencies. This relationship arises from the proportionality between frequency and the inverse of length.

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