Question 579: Thermodynamics & KTG

Question

A steel rod of length $$1 \mathrm{~m}$$ and cross sectional area $$10^{-4} \mathrm{~m}^{2}$$ is heated from $$0^{\circ} \mathrm{C}$$ to $$200^{\circ} \mathrm{C}$$ without being allowed to extend or bend. The compressive tension produced in the rod is ___________ $$	imes 10^{4} \mathrm{~N}$$. (Given Young's modulus of steel $$=2 	imes 10^{11} \mathrm{Nm}^{-2}$$, coefficient of linear expansion $$=10^{-5} \mathrm{~K}^{-1}$$ )

Accepted Answer

The change in length of the rod when it is heated is given by the equation:
$\Delta L = L_0 \cdot \alpha \cdot \Delta T$
where
$\Delta L$ is the change in length,
$L_0$ is the original length,
$\alpha$ is the coefficient of linear expansion, and
$\Delta T$ is the change in temperature.
Substituting the given values:
$\Delta L = 1 \, 	ext{m} \cdot 10^{-5} \, 	ext{K}^{-1} \cdot 200 \, 	ext{K} = 0.002 \, 	ext{m}$
The rod is not allowed to extend or bend, so a stress is created in the rod. This stress can be calculated using Young's modulus (Y), which is the ratio of the stress (force per unit area, F/A) to the strain (change in length per unit length, $\Delta L / L_0$):
$Y = \frac{F/A}{\Delta L / L_0}$
Rearranging for F gives:
$F = Y \cdot A \cdot \frac{\Delta L}{L_0}$
Substituting the given values:
$F = 2 	imes 10^{11} \, 	ext{N/m}^2 \cdot 10^{-4} \, 	ext{m}^2 \cdot \frac{0.002 \, 	ext{m}}{1 \, 	ext{m}} = 4 	imes 10^{4} \, 	ext{N}$
So the compressive tension produced in the rod is $4 	imes 10^{4} \, 	ext{N}$.

Thermodynamics & KTG

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