Question

A string of length \( l \) is divided into three segments of lengths \( l_1, l_2 \) and \( l_3 \) with the fundamental frequencies \( n_1, n_2 \) and \( n_3 \) respectively. The original fundamental frequency 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. \( \sqrt{n} = \sqrt{n_1} + \sqrt{n_2} + \sqrt{n_3} \)
• D. \( \frac{1}{\sqrt{n}} = \frac{1}{\sqrt{n_1}} + \frac{1}{\sqrt{n_2}} + \frac{1}{\sqrt{n_3}} \)
• E. \( n = n_1 n_2 n_3 \)

Hint:

Frequency inversely proportional to length in strings.

Answer 1

The Correct Option is B

Concept: \[ n \propto \frac{1}{l} \]

Step 1: Relation.
\[ n_1 = \frac{k}{l_1}, \quad n_2 = \frac{k}{l_2}, \quad n_3 = \frac{k}{l_3} \]

Step 2: Total length.
\[ l = l_1 + l_2 + l_3 \Rightarrow \frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3} \]