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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

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Here, L = 1.8m , `T_(1) = 27^(0) C , T_(2) = - 39^(0)C, F = ? `
r =1 mm = `10^(-3)` m,
`alpha = 2 xx 10^(-5)""^(0) C^(-1), Y = 0.91 xx 10^(11) N//m^(2)`
From, Y = `(F.L)/(A Delta L) " " Delta L = (FL)/(AY)`
Also, `Delta L = alpha L Delta T " " therefore (FL)/(AY) = alpha L Delta T ` or
F = `alpha Delta T alpha Y = alpha (T_(2) - T_(1)) pi r^(2) Y`
`= 2 xx 10^(-5) xx (- 39 - 27 ) xx (22)/(7) (10^(-3))^(2) xx 0.91 xx 10^(11)`
= `-3.77 xx 10^(2)` N
Negative sign indicates that the force is inwards due to contraction of the wire.
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