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A brass wire 1.8 m long at 27 ^(@) C is ...

A brass wire` 1.8 `m long at `27 ^(@) C` is held taut with little tension between two rigid supports. If the wire is cooled to a temperature of `- 39 ^(@) C`. what is the tension developed in the wire, if its diameter is `2.0` mm ? Co-efficient of tear expansion of brass ` = 2.0 xx 10^(-5) K^(-1)`, Young's modulus of brass = ` 0.91 xx 10^(11)` Pa.

Text Solution

Verified by Experts

Here, L = 1.8 m, `T_(1)=27^(@)C,T_(2)=-39^(@)C,F=?`,
r = 1 mm = `10^(-3)`m,
`alpha=2xx10^(-5)""^(@)C^(-1),Y=0.91xx10^(11)N//m^(2)`
From, `Y=(F*L)/(ADeltaL),DeltaL=(FL)/(AY)`
Also, `DeltaL=alphaLDeltaT" "therefore(FL)/(AY)=alphaLDeltaT` or
`F=alphaDeltaTalphaY=alpha(T_(2)-T_(1))pir^(2)Y`
= `2xx10^(-5)xx(-39-27)xx22/7(10^(-3))^(2)xx0.91xx10^(11)`
= `-3.77xx10^(2)N`
Negative sign indicates that the force is inwards due to contraction of the wire.
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