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Question

The frequency of the 1st harmonic of a sonometer wire is 160Hz. If the length of the wire is increased by 50% and the tension in the wire is decreased by 19%, the frequency of its first overtone is :

( Assume linear mass density to be constant )

A
180 Hz
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B
192 Hz
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C
220 Hz
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D
232 Hz
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Solution

The correct option is B 192 Hz
The natural frequency of nth mode is
fn=n2LTμ
where, L is the length of the sonometer wire,
μ is the mass per unit length of the wire and T is the tension in the string.

Frequency of the first harmonic before :f=12LTμ......................................(1)
Frequency of the first harmonic after :f=12LTμ.........................................(2)
Given L=1.5L
T=0.81T
Dividing (1) by (2)
ff=LLTT=1.5LLT0.81T=1.50.9
160f=1.50.9
f=160×0.91.5=96Hz
Frequency of the 1st overtone is 2f=2×96=192Hz

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