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

`L = 1.8 m , T_1 = 27^@C , T_2 = -39^@C`
`D = 2mm = 2 xx 10^(-3) m , alpha = 2 xx 10^(-5) ""^@C^(-1)`
As `Y = (F//A)/(Delta l // l ) = (Fl)/(F Delta l)`
`Delta l = (F)/(A) l/Y`
` Delta l = alpha l Delta T`
` therefore l alpha Delta T = (F)/(A) l/Y `
` rArr F = alpha Delta TAY`
` = | 2 xx 10^(-5) (-39 - 27) xx 3.14 xx ( (2 xx 10^(-3) )^2 )/(4) xx 0.91 xx 10^11 | `
` = | - 3.77 xx 10^2 N | = 3.77 xx 10^(-2) N`