A steel wire of length \(20 \mathrm{~cm}\) and area of crosssection \(1 \mathrm{~mm}^2\) is tied rigidly at…
A steel wire of length \(20 \mathrm{~cm}\) and area of crosssection \(1 \mathrm{~mm}^2\) is tied rigidly at both the ends. When the temperature of the wire is changed from \(40^{\circ} \mathrm{C}\) to \(20^{\circ} \mathrm{C}\), find the change in its tension. Given, the coefficient of linear expansion for steel is \(1.1 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}\) and Young's modulus of steel is \(2.0 \times 10^{11} \mathrm{Nm}^{-2}\).
\(22 \mathrm{~N}\)
\(44 \mathrm{~N}\)
\(16 \mathrm{~N}\)
\(8 \mathrm{~N}\)
Solution
Given, Young's modulus of steel wire,
\(Y=2 \times 10^{11} \mathrm{Nm}^{-2}\)
Length of wire, \(l=20 \mathrm{~cm}=0.2 \mathrm{~m}\)
Area of cross-section, \(A=1 \mathrm{~mm}^2=10^{-6} \mathrm{~m}^2\)
Change in temperature of wire,
\(\begin{aligned}
\Delta t & =(40-20)^{\circ} \mathrm{C} \\
& =20^{\circ} \mathrm{C} \\
\alpha & =1.1 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}
\end{aligned}\)
\(\therefore\) Change in tension in the steel wire,
\(\begin{aligned}
T & =Y A \alpha \Delta t \\
& =2 \times 10^{11} \times 10^{-6} \times 1.1 \times 10^{-5} \times 20 \\
& =44 \mathrm{~N}
\end{aligned}\)