Question

A steel wire of cross section area $0.5m{m}^2$ is held between two fixed supports. If the tension in the wire is negligible, and it is just taut at a temperature of ${20}^{\circ }C$ determine the tension when the temperature falls to $0^{\circ }C$. Assume the distance between supports to be the same,Young's modulus of steel $Y=2.1\times {10}^{12}$ dynes and coefficient of linear expansion for steel $\alpha =12\times {10}^{-6}{}^{\circ }{C}^{-1}$

• A. $2.52\times {10}^6\text{dynes}$
• B. $3.52\times {10}^6\text{dynes}$
• C. $3.52\times {10}^5\text{dynes}$
• D. $2.52\times {10}^4\text{dynes}$

Answer 1

The correct option is A $2.52\times {10}^6\text{dynes}$

Cross sectional area of wire
$=A=0.5\times {10}^{-2}{m}^2$
Initial temperature of wire $={20}^{\circ }C$
Final temperature of wire $=0^{\circ }C$
Young's modulus $=Y=2.1\times {10}^{12}$ dynes
Coefficient of linear expansion $=\alpha =12\times {10}^{-6}{}^{\circ }{C}^{-1}$
Let, tension is string when temperature falls to $0^{\circ }C$ be F, so,
$Y=\frac{stress}{Strain}=\frac{\frac{F}{A}}{\frac{\Delta l}{l}}=\frac{\frac{\frac{F}{A}}{l\alpha \Delta \theta }}{l}\Rightarrow F=AY\Delta \theta \alpha$
$F=0.5\times {10}^{-2}\times 2.1\times {10}^{12}\times (20-0)\times 12\times {10}^{-6}$
$=2.1\times 0.5\times 12\times 20\times {10}^{12-2-6}$
$=2.52\times {10}^6\text{dynes}$