A brass wire `1.8 m` long at `27^(@)C` is held taut with little tension between two rigid supports. If the wire cooled to a temperature of `-39^(@)C`, what is the tension developed in the wire, if its diameter is `2.0 mm`? Coefficient of linear expansion of brass `= 2.0 xx 10^(-5)//^(@)C`, Young's modulus of brass `= 0.91 xx 10^(11) Pa`.

Explanation

Here `I=1.8m,t_(1)=27^(@)C,t_(2)=-39^(@)C`
`r=(2.0)/(2)=1.0 mm =1.0xx10^(-3)m`
`r=2.0xx10^(-5)""^(@)C^(-1),Y=0.91xx10^(11)Pa`
As `" " DeltaI=Ialpha(t_(2)-t_(1))`
`:.` Strain `(DeltaI)/(I)=alpha(t_(2)-t_(1))`
`"Stress" = "Strain" xx "Young’s modulus" =alpha(t_(2)-t_(1))xx`Y
`=2.0xx10^(-5)xx(-39-27)xx0.91xx10^(11)=1.2xx10^(8)Nm^(-2)` [Numerically]
Tension developed in the wire = Stress xx Area of cross-section
`= Strees xxPr^(2)=1.2xx10^(8)xx3.14xx(1.0xx10^(-3))^(2)=3.77xx10^(2)N`.