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  • A sonometer wire is vibrating in resonance with a tuning fork. Keeping the tension applied same, the length of the wire is doubled. Under what conditions would the tuning fork still be is resonance with the wire?

A sonometer wire is vibrating in resonance with a tuning fork. Keeping the tension applied same, the length of the wire is doubled. Under what conditions would the tuning fork still be is resonance with the wire?

Answers (1)

Let the length of wire be L and tension be T

The frequency of nth harmonic is v=\frac{n}{2L}\sqrt{\frac{T}{m} }where m is mass per unit length

Let us assume two cases

v_{1}=\frac{n_{1}}{2L_{1}}\sqrt{\frac{T_{1}}{m_{1}}}

v_{2}=\frac{n_{1}}{2L_{2}}\sqrt{\frac{T_{2}}{m_{1}}}v

In the given question T_{1} is same as T and the mass is also the same since the same wire is used.

L_{2}=2 L_{1}

\frac{v_{1}}{v_{2}}=\frac{n_{1}}{2L_{1}}\sqrt{\frac{T_{1}}{m_{1}}}\times 2\times \frac{2L_{2}}{n_{1}}\sqrt{\frac{m_{1}}{T_{2}}} = \frac{2n_{1}}{n_{2}}

As tuning fork is same in both harmonics, both the frequencies are equal 

 2n_{1}=n_{2}

Thus, when the length of the wire is doubled, the number of harmonics also get doubled for the same frequency.

Posted by

infoexpert24

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