Question

The fundamental frequency of a sonometer wire is 50 Hz for some length and tension. If the length is increased by 25% by keeping tension same then frequency change of second harmonic is

• A. increased by 20%
• B. decreased by 20%
• C. increased by 10%
• D. decreased by 10%

Hint:

The percentage change in frequency is the same for all harmonics if only length is changed.

Answer 1

The Correct Option is B

Step 1: Fundamental Formula
Fundamental frequency $n = \frac{1}{2L}\sqrt{\frac{T}{m}}$. Since $T$ and $m$ are constant, $n \propto \frac{1}{L}$.
Step 2: Change in Length
New length $L' = L + 0.25L = 1.25L$.
New fundamental frequency $n' = \frac{n}{1.25} = \frac{50}{1.25} = 40 \text{ Hz}$.
Step 3: Second Harmonic
Second harmonic $n_2 = 2n$.
Initial second harmonic $= 2 \times 50 = 100 \text{ Hz}$.
Final second harmonic $= 2 \times 40 = 80 \text{ Hz}$.
Percentage change $= \frac{80 - 100}{100} \times 100 = -20%$.
Final Answer: (B)