Question

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?

Answer 1

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.