Question

When a string is divided into three segments of length \(l_1, l_2,\) and \(l_3\), the fundamental frequencies of these three segments are \(v_1, v_2,\) and \(v_3\) respectively. The original fundamental frequency of the string is:

• A. \(\sqrt{v} = \sqrt{v_1} + \sqrt{v_2} + \sqrt{v_3}\)
• B. \(v = v_1 + v_2 + v_3\)
• C. \(\dfrac{1}{v} = \dfrac{1}{v_1} + \dfrac{1}{v_2} + \dfrac{1}{v_3}\)
• D. \(\dfrac{1}{\sqrt{v}} = \dfrac{1}{\sqrt{v_1}} + \dfrac{1}{\sqrt{v_2}} + \dfrac{1}{\sqrt{v_3}}\)

Hint:

For strings under same tension, frequency is inversely proportional to length.

Answer 1

The Correct Option is C

Step 1: For a stretched string, fundamental frequency: \[ v \propto \frac{1}{l} \]
Step 2: Hence, \[ v_1 \propto \frac{1}{l_1},\quad v_2 \propto \frac{1}{l_2},\quad v_3 \propto \frac{1}{l_3} \]
Step 3: Total length \(l = l_1 + l_2 + l_3\): \[ \frac{1}{v} \propto l_1 + l_2 + l_3 \] \[ \Rightarrow \frac{1}{v} = \frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3} \]