A sonometer wire is vibrating is 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 in resonance with the wire?

Explanation

Wire, of twice the length vibrates in its second harmonic. Thus , if the tuning fork rresonates at L, at 2L. This can be explained as below
The sonometer frequency is given by
`" " v=(n)/(2L) sqrt((T)/(m)) " " ` ( n= number of loops )
Now, as it vibrates wiht 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))/(2xx2L)sqrt((T)/(m))" " ....(ii)`
Dividing Eq.(i) by Eq. (ii) we get,
`" " (v_(1))/(v_(2))=(n_(1))/(n_(2))xx2`
To keep the resonance
`" " (v_(1))/(v_(2))=1=(n_(1))/(n_(2))xx2`
`rArr" " n_(2)=2n_(1)`
Hence when the wire is double the number of loops aslo get doubled to produce the resonance. That is it resonates is second harmonic.