Question

A steel wire is heated to `170^(@)C` and held between two rigid supports which are `20cm` apart. The wire is allowed to cool to a temperature of `29.6^(@)C`. If the frequency of the note produced when the wire is plucked at the middle is `100 k`, then find the value of `k`. (The density `D` of steel is `7.8xx10^(3)kg//m^(3)` and for steel `alpha=16xx10^(-6)K^(-1)` and `Y` of steel `=20xx10^(10)Pa`.).

Answer 1

Thermal stress in the wire `=alphaYDeltaT`
Tension in the wire `=alphaYDeltaTpir^(2)`
`n=(1)/(2L)sqrt((T)/(pir^(2)D))=(1)/(2L)sqrt((alphaYDeltaT.pir^(2))/(pir^(2)D))=(1)/(2L)sqrt((alphaYDeltaT)/(D))`
Putting values `n=600Hz`.
In the fundamental mode as the wire is plucked in the middle.