Question

A brass wire $2m$ long at $27{}^{\circ }C$ is held taut with negligible tension between two rigid supports. If the wire is cooled to a temperature of $-33{}^{\circ }C$ , then the tension developed in the wire, its diameter being $2mm$ , will be (coefficient of linear expansion of brass = $2.0\times 10-5{}^{\circ }{C}^{-1}$ and Young's modulus of brass = $0.91\times 10^{11}Pa$ )

• A. $3400N$
• B. $34kN$
• C. $0.34kN$
• D. $6800N$

Answer 1

Answer: (C)

Thermal stress that is produced in an elastic wire is Y $\alpha \theta$ per unit area, wher Y is Young's modulus, $\alpha$ coefficient of linear expansion and $\theta$ change of temperature.
Thus, tension developed i the wire is
T = Thermal stress x area of cross section
$=\text{Y}\alpha \times \left(\Delta \text{T}\right)\left[\pi {\left(\text{r}\right)}^2\right]\text{N}$
$=\text{0.91}\times 10^{11}\times 2\times 10^{-5}\times \left[27-\left(-33\right)\right]\left[\pi \times 1\times 10^{-6}\right]\text{N}$
$=\text{0.91}\times 2\times 6\times \text{3.14}\times 10N$
$=\text{34.4}\times 10N=\text{0.34}kN$