Question

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

• A. $YA\alpha \Delta t$
• B. $\frac{Y\alpha \Delta t}{A}$
• C. $\frac{A\alpha \Delta t}{Y}$
• D. $\frac{YA}{\alpha \Delta T}$

Answer 1

The correct option is A

$YA\alpha \Delta t$

Let l be the length of the wire at temperature t and $\Delta l$ be the contraction in the length as the temperature falls by $\Delta t$. Then, $\Delta l=\alpha l\Delta t$
$\therefore$ Strain in the wire $\left(\frac{\Delta l}{l}\right)=\alpha \Delta t$
Now, young's modulus, $Y=\frac{Stress}{Strain}=\frac{\frac{F}{A}}{\frac{\Delta l}{l}}$
$\therefore$ Tension in the wire is $F=\frac{YA\Delta l}{l}=YA\alpha \Delta t$ and the correct choice is (a).