First slide

Thermal expansion

Question

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

Easy

A

$Y A α ΔT$
B

$Yα ΔT / A$
C

$A α ΔT / Y$
D

$YA / α ΔT$

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 $F = \frac{Y A Δl}{l} = Y A α ΔT$

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