A string is divided into three parts having lengths ℓ1, ℓ2 and ℓ3 each. If the fundamental frequ

step 1 In this question we will learn about some basic concept of the oscillation and waves. The questi

A string is divided into three parts having lengths ℓ1, ℓ2...

A string is divided into three parts having lengths $\ell_{1}, \ell_{2}$ and $\ell_{3}$ each. If the fundamental frequency of these parts are $\mathrm{f}_{1}, \mathrm{f}_{2}$ and $\mathrm{f}_{3}$ respectively, then the fundamental frequency of the original string $\mathrm{f}=\ldots \ldots \ldots$ (A) $\sqrt{f}=\sqrt{f}_{1}+\sqrt{f}_{2}+\sqrt{f}_{3}$ (B) $\mathrm{f}=\mathrm{f}_{1}+\mathrm{f}_{2}+\mathrm{f}_{3}$ (C) $(1 / \mathrm{f})=\left(1 / \mathrm{f}_{1}\right)+\left(1 / \mathrm{f}_{2}\right)+\left(1 / \mathrm{f}_{3}\right)$ (D) $(1 / \sqrt{f})=\left(1 / \sqrt{f}_{1}\right)+\left(1 / \sqrt{f}_{2}\right)+\left(1 / \sqrt{f}_{2}\right)$

A string is divided into three parts having lengths $\ell_{1}, \ell_{2}$ and $\ell_{3}$ each. If the fundamental frequency of these parts are $\mathrm{f}_{1}, \mathrm{f}_{2}$ and $\mathrm{f}_{3}$ respectively, then the fundamental frequency of the original string $\mathrm{f}=\ldots \ldots \ldots$
(A) $\sqrt{f}=\sqrt{f}_{1}+\sqrt{f}_{2}+\sqrt{f}_{3}$
(B) $\mathrm{f}=\mathrm{f}_{1}+\mathrm{f}_{2}+\mathrm{f}_{3}$
(C) $(1 / \mathrm{f})=\left(1 / \mathrm{f}_{1}\right)+\left(1 / \mathrm{f}_{2}\right)+\left(1 / \mathrm{f}_{3}\right)$
(D) $(1 / \sqrt{f})=\left(1 / \sqrt{f}_{1}\right)+\left(1 / \sqrt{f}_{2}\right)+\left(1 / \sqrt{f}_{2}\right)$

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Key Concepts

Fundamental Frequency of a Vibrating String

The fundamental frequency of a vibrating string is determined by the relationship f = (1/2L)?(T/?), where T is the tension in the string, ? is its linear mass density, and L is the length of the string. This indicates that the frequency is inversely proportional to the length of the string, assuming constant tension and mass density.

Relationship Between Frequency and Length

Since the frequency is inversely proportional to the length, dividing a string into segments means that each segment’s fundamental frequency will be inversely related to its individual length. Thus, the longer the segment, the lower its fundamental frequency, and vice versa.

Reciprocal Frequency Summation

Given the inverse proportionality between frequency and length, if the original string is divided into segments with lengths corresponding to frequencies f1, f2, and f3, then the total length being the sum of the individual lengths implies that the reciprocal of the overall string’s frequency is the sum of the reciprocals of the segment frequencies. This is why the correct relation is expressed as (1/f) = (1/f1) + (1/f2) + (1/f3).

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