Question

A steel rod of length 1 m and area of cross-section $1\, cm^2$ 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 $= 2 × 10^{11}\, N\, m^{-2}$ and coefficient of linear expansion of steel $= 10^{-5} \,{}^{\circ}C^{-1})$

• A. $4 × 10^3\, N$
• B. $4 × 10^4\, N$
• C. $4 × 10^5\, N$
• D. $4 × 10^6\, N$

Answer 1

Answer: (B)

Let $ΔL$ be increase in the length of the rod due to increase in temperature of the rod. Then
$ΔL = LαΔT$
where, $α$ is the coefficient of the linear expansion, $ΔT$ 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, $Y = 2 × 10^{11}\, N\,m^{-2}$
$α = 10^{-5} \,{}^{\circ}C^{-1}$
$ΔT = 200^{\circ}C - 0^{\circ}C = 200^{\circ}C$
$L = 1\,m, A = 1\, cm^{2} = 1 × 10^{-4}\, m^{2}$
$\therefore $ $T = 2 × 10^{11}\, N \,m^{-2} × 10^{-5}\,{}^{\circ}C^{-1} × 200^{\circ}C × 1 × 10^{-4} \,m^{2}$
$= 4 × 10^{4}\, N$