Question

If n₁, n₂ and n₃ 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. \(\frac{1}{\sqrt{n}}=\frac{1}{\sqrt{n_1}}+\frac{1}{\sqrt{n_2}}+\frac{1}{\sqrt{n_3}}\)
• B. \(\sqrt{n}=\sqrt{n_1}+\sqrt{n_2}+\sqrt{n_3}\)
• C. n=n₁+n₂+n₃
• D. \(\frac{1}{n}=\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3}\)

Answer 1

The Correct Option is D

To solve the problem of finding the original fundamental frequency of a string that is divided into three segments, we need to use the concept of combined frequencies in the context of wave mechanics. When a string is divided and each segment has a fundamental frequency, the original frequency can be expressed in terms of these segments.

The relationship between the original fundamental frequency \( n \) of the string and the fundamental frequencies of the segments \( n_1 \), \( n_2 \), and \( n_3 \) is given by the equation:

\(\frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3}\)

Explanation:

• The fundamental frequency of a vibrating string is related to its length, mass per unit length, and tension. When a string is divided into segments, each segment acts as an independent vibrating string with its own fundamental frequency.
• This condition resembles the parallel combination of waves where the inverse of the original frequency is equal to the sum of the inverses of the segment frequencies. This is analogous to combining resistances in parallel circuits in electricity.
• Therefore, the correct option is the one where the sum of the reciprocals of the segment frequencies equals the reciprocal of the original frequency.

Conclusion:

The correct expression for the original fundamental frequency \( n \) of the string, given the segment frequencies \( n_1 \), \( n_2 \), and \( n_3 \), is:

\(\frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3}\)