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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?
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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