Question

When a stretched wire of fundamental frequency f is divided into three segments, the fundamental frequencies of these three segments are \(f_1\), \(f_2\) and \(f_3\) respectively. Then the relation among \(f, f_1, f_2, f_3\) and f is (Assume tension is constant) % "and f is" seems redundant

• A. \( \sqrt{f} = \sqrt{f_1} + \sqrt{f_2} + \sqrt{f_3} \)
• B. \( f = f_1 + f_2 + f_3 \)
• C. \( \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} \)
• D. \( \frac{1}{\sqrt{f}} = \frac{1}{\sqrt{f_1}} + \frac{1}{\sqrt{f_2}} + \frac{1}{\sqrt{f_3}} \)

Hint:

Fundamental frequency of a stretched string: \( f = \frac{v}{2L} = \frac{1}{2L}\sqrt{\frac{T}{\mu}} \). If tension \(T\) and linear mass density \( \mu \) are constant, then \( f \cdot L = \text{constant} \), so \( f \propto 1/L \) or \( L \propto 1/f \). If a wire of length \(L\) is divided into segments of lengths \(L_1, L_2, L_3, \dots\), then \( L = L_1+L_2+L_3+\dots \). Substitute \(L = k/f\) (where \(k\) is a constant) into this relation.

Answer 1

The Correct Option is C

The fundamental frequency \(f\) of a stretched string is given by \( f = \frac{1}{2L}\sqrt{\frac{T}{\mu}} \), where \(L\) is the length of the string, \(T\) is the tension, and \( \mu \) is the linear mass density (mass per unit length).
Since tension \(T\) and linear mass density \( \mu \) are constant for the wire and its segments (assuming the wire is uniform), we can write \( f \propto \frac{1}{L} \), or \( L \propto \frac{1}{f} \).
Let \(L\) be the total length of the original wire, and \(L_1, L_2, L_3\) be the lengths of the three segments.
Then \( L = L_1 + L_2 + L_3 \).
Since \( L \propto 1/f \), let \( L = C/f \), \( L_1 = C/f_1 \), \( L_2 = C/f_2 \), \( L_3 = C/f_3 \), where C is a constant of proportionality \( (\frac{1}{2}\sqrt{T/\mu})^{-1} \).
Substituting these into the length equation: \[ \frac{C}{f} = \frac{C}{f_1} + \frac{C}{f_2} + \frac{C}{f_3} \] Assuming \( C \ne 0 \), we can divide by C: \[ \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} \] This matches option (3).