A steel wire is rigidly fixed at both ends its length mass and cross sectional area are 1m, 0.1 kg and `10^(6)m^(2)` respectively then the temperature of the wire is lowered by `20^(@)C` if the transverse waves are setup by plucking the wire at 0.25m from one end and assuming that wire vibrates with minimum number of loops possible for such a case. Find the frequency of vibration (in Hz). [coefficient of linear expansion of steel `=1.21xx10^(-6)//.^(@)C` and young's modulus `=2xx10^(11)//N//m^(2)`]

Explanation

The mechanical strain `=(trianglel)/(l)=alphatriangleT=1.21xx10^(-5)xx20=2.42xx10^(-5)`
The tension in wire `=T=Y(trianglel)/(l)A=2xx10^(11)xx2.42xx10^(-5)xx10^(-6)=48.4N`
`therefore` speed of wave in wire
`V=sqrt((T)/(mu))=sqrt((48.4)/(0.1))=22m//s`
Since the wire is plucked at `(l)/(4)` from one end the wire shall oscillate in `1^(st)` overtone (for minimum number of loops)
`lamda=l=1m`
Now `V=flamda` or `f=(V)/(lamda)=22Hz`