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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 HzThe natural frequency of nth mode isfn=n2L√Tμwhere, 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=12L√Tμ......................................(1)Frequency of the first harmonic after :f′=12L′√T′μ.........................................(2)Given L′=1.5L T′=0.81TDividing (1) by (2)ff′=L′L√TT′=1.5LL√T0.81T=1.50.9∴160f′=1.50.9⇒f′=160×0.91.5=96HzFrequency of the 1st overtone is 2f′=2×96=192Hz

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