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A steel rod $\left(Y=2.0 \times 10^{11} \mathrm{~N} / \mathrm{m}^2\right.$ and $\left.\alpha=10^{-50}{ }^{\circ} \mathrm{C}^{-1}\right)$ of length $1 \mathrm{~m}$ and area of cross-section $1 \mathrm{~cm}^2$ is heated from $0^{\circ} \mathrm{C}$ to $200^{\circ} \mathrm{C}$, without being allowed to extend or bend. What is the tension produced in the rod?
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Verified Answer
As given that, Young's modulus of steel $Y=2.0 \times 10^{11} \mathrm{~N} / \mathrm{m}^2$
Coefficient of thermal expansion $\alpha=10^{-5}{ }^{\circ} \mathrm{C}^{-1}$
Length $L_0=1 \mathrm{~m}$
Area of cross-section $A=1 \mathrm{~cm}^2=1 \times 10^{-4} \mathrm{~m}^2$
Increase in temperature $\Delta t=200^{\circ} \mathrm{C}-0^{\circ} \mathrm{C}=200^{\circ} \mathrm{C}$
We know that temperature length of wire
$$
\begin{aligned}
&L_t=L_0[1+\alpha \Delta t] \Rightarrow\left(L_t-L_0\right)=L_0 \alpha \Delta t \\
&\Delta L=1 \times 10^{-5} \times 200 \\
&\therefore Y=\frac{F L_0}{A \Delta L}
\end{aligned}
$$
Tension produced in steel rod $(F)$
$$
\begin{aligned}
&=\left(Y A \Delta L / L_0\right)=\frac{2.0 \times 10^{11} \times 1 \times 10^{-4} \times 10^{-5} \times 200}{1} \\
&=4 \times 10^4 \mathrm{~N}
\end{aligned}
$$
Coefficient of thermal expansion $\alpha=10^{-5}{ }^{\circ} \mathrm{C}^{-1}$
Length $L_0=1 \mathrm{~m}$
Area of cross-section $A=1 \mathrm{~cm}^2=1 \times 10^{-4} \mathrm{~m}^2$
Increase in temperature $\Delta t=200^{\circ} \mathrm{C}-0^{\circ} \mathrm{C}=200^{\circ} \mathrm{C}$
We know that temperature length of wire
$$
\begin{aligned}
&L_t=L_0[1+\alpha \Delta t] \Rightarrow\left(L_t-L_0\right)=L_0 \alpha \Delta t \\
&\Delta L=1 \times 10^{-5} \times 200 \\
&\therefore Y=\frac{F L_0}{A \Delta L}
\end{aligned}
$$
Tension produced in steel rod $(F)$
$$
\begin{aligned}
&=\left(Y A \Delta L / L_0\right)=\frac{2.0 \times 10^{11} \times 1 \times 10^{-4} \times 10^{-5} \times 200}{1} \\
&=4 \times 10^4 \mathrm{~N}
\end{aligned}
$$
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