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Question
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?
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Solution
Wire of twice the length vibrates in its second harmonic. Thus, if the tuning fork resonates at L it will resonate at 2 L This can be explained below The sonometer frequency is given by
`v = n/(2L) sqrt(T/m)` .....(n = number of loops)
Now, as it vibrates with length L, we assume `v = v_1`
`n = n_1`
∴ `v_1 = n_1/(2L) sqrt(T/m)` ......(i)
When length is doubled, then `v_2 = n_2/(2 xx 2L) sqrt(T/m)` ......(ii)
Dividing equation (i) by equation (ii), we get
`v_1/v_2 = n_1/n_2 xx 2`
To keep the resonance `v_1/v_2 = 1 = n_1/n_2 xx 2`
⇒ `n_2 = 2n_1`
Hence, when the wire is doubled the number of loops also gets doubled to produce the resonance. That is it resonates in the second harmonic.
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