A wire of cross-sectional area A at temperature T is held taut with negligible tension between two rigid supports. If the wire is cooled to a temperature $(T - ΔT)$ , what tension is developed in the wire ? The coefficient of linear expansion is $α$ and Young's modulus of the wire is Y

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$YA / α ΔT$

$Y A α ΔT$

$Yα ΔT / A$

$A α ΔT / Y$

answer is A.

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Detailed Solution

Let I be the length of the wire at temperature T. If $∆ l$ be the contraction in length when the temperature falls by $∆ T$ , then $Δl = αlΔT$
Strain in wire $= (Δl / l) = αΔT$
Now $Y = \frac{stress}{strain} = \frac{(F / A)}{Δl / l} = \frac{Fl}{AΔl}$
So, tension in the wire is given by $F = \frac{Y A Δl}{l} = Y A α ΔT$