Question

A steel wire of cross-sectional area 0.5${\text{m}}{{\text{m}}^{\text{2}}}$ is held between two fixed supports. If tension in the wire is negligible and it is just taut at 20 ${}^{\text{0}}{\text{C}}$, determine tension when temperature falls 0 ${}^{\text{0}}{\text{C}}$. Young's modulus of elasticity is $21 \times {10^{11}}$ dyne/${\text{c}}{{\text{m}}^{{\text{ - 2}}}}$and coefficient of linear expansion is $12 \times {10^{ - 6}}$per degree centigrade. Assume that the distance between the supports remains the same.

Answer 1

Hint- To solve this question, we use the basic theory of Thermal expansion. As we know Thermal expansion occurs when an object expands and becomes larger due to a change in the object's temperature. Similarly, in this case it will affect the operation of steel wire when temperature changes to ${\text{20}}{}^{\text{0}}{\text{C}}$. Some basic formulas are used to get our desired result in this problem.

Formula used- $ \Rightarrow Strain = \dfrac{{\Delta L}}{L} = \alpha \Delta T$
                                         Stress = Y × Strain
Given: Y=$21 \times {10^{11}}dyn/c{m^2}$
$\alpha = 12 \times {10^{ - 6}}/{}^0C$
Change in temperature $\Delta T = 20{}^0C$
Change in temperature $A = 0.5m{m^2} = 0.005c{m^2}$

Complete answer:
Let the initial length of steel wire be L and let L be the steel wire's length when the temperature is reduced to$0{}^0C$.
Decrease in length due to compression,$\Delta L = {L^I} - L$
Using$\Delta L = L\alpha \Delta T$
$ \Rightarrow Strain = \dfrac{{\Delta L}}{L} = \alpha \Delta T = 12 \times {10^{ - 6}} \times 20 = 2.4 \times {10^{ - 4}}$
From Hooke's law, Stress=Y×Strain
⟹ Tension T=Y×Strain × A
$\therefore T = 21 \times {10^{11}} \times 2.4 \times {10^{ - 4}} \times 0.005 = 252 \times {10^4}dyn$
⟹ T=25.2N (1N=${10^4}dyn$)
Therefore, tension of the steel wire is 25.2 N.

Note- The expansion can occur in length of iron pendulum in which case it is called Linear Expansion. And If we take a square tile and then after heat it, the expansion will be on two fronts that is length and breadth, and it is called Area Expansion. Similarly, if we take a cube shape structure and heat it, all its sides expand and now the body experiences an increase in the overall volume of the structure and it is called Volume Expansion.