Question

The length and diameter of a metal wire used in a sonometer is doubled. The fundamental frequency will change from \(n\) to

• A. \( \dfrac{n}{2} \)
• B. \( \dfrac{n}{16} \)
• C. \( \dfrac{n}{4} \)
• D. \( \dfrac{n}{8} \)

Hint:

Frequency of a sonometer wire varies inversely with length and inversely with diameter.

Answer 1

The Correct Option is C

Step 1: Formula for fundamental frequency of a stretched string.
\[ n = \frac{1}{2L}\sqrt{\frac{T}{\mu}} \] where \(L\) is length and \(\mu\) is mass per unit length.
Step 2: Effect of doubling the length.
If \(L\) is doubled, frequency becomes half:
\[ n \propto \frac{1}{L} \]
Step 3: Effect of doubling the diameter.
Mass per unit length \(\mu \propto d^2\). If diameter is doubled,
\[ \mu' = 4\mu \Rightarrow \sqrt{\mu'} = 2\sqrt{\mu} \]
Step 4: Combined effect on frequency.
\[ n' = \frac{1}{2(2L)} \sqrt{\frac{T}{4\mu}} \] \[ n' = \frac{1}{4} \times \frac{1}{2} \sqrt{\frac{T}{\mu}} = \frac{n}{4} \]
Step 5: Conclusion.
The fundamental frequency becomes \( \dfrac{n}{4} \).