Question

A steel rod of length 1 m and area of cross-section $1c{m}^2$ is heated from $0^{°}C$ to $200^{°}C$, without being allowed to extend or bend. The tension produced in the rod is (Given : Young’s modulus of steel $=2\times 10^{11}N{m}^{-2}$ and coefficient of linear expansion of steel $=10^{-5}{}^{\circ }{C}^{-1})$

• A. $4\times 10^{3}N$
• B. $4\times 10^{4}N$
• C. $4\times 10^{5}N$
• D. $4\times 10^{6}N$

Answer 1

Answer: (B)

Let $\Delta L$ be increase in the length of the rod due to increase in temperature of the rod. Then
$\Delta L=L\alpha \Delta T$
where, $\alpha$ is the coefficient of the linear expansion, $\Delta T$ is the rise in temperature and L is the length of the rod.
$\therefore \frac{\Delta L}{L}=\alpha \Delta T...\left(i\right)$
Let the compressive tension of the rod be T and A becross-section area. Then
$Y=\frac{T/A}{\Delta L/L}$
$\therefore T=Y\frac{\Delta L}{L}A=Y\times \alpha \Delta T\times A$ (Using (i))
Here, $Y=2\times 10^{11}N{m}^{-2}$
$\alpha =10^{-5}{}^{\circ }{C}^{-1}$
$\Delta T=200^{\circ }C-0^{\circ }C=200^{\circ }C$
$L=1m,A=1c{m}^2=1\times 10^{-4}{m}^2$
$\therefore$ $T=2\times 10^{11}N{m}^{-2}\times 10^{-5}{}^{\circ }{C}^{-1}\times 200^{\circ }C\times 1\times 10^{-4}{m}^2$
$=4\times 10^{4}N$