Question
A metallic bar of Young's modulus, $0.5 \times 10^{11}\text{ }N{\text{ }m}^{- 2}$ and coefficient of linear thermal expansion $10^{- 5}\ ^{\circ}C^{- 1}$, length 1 m and area of cross-section $10^{- 3}{\text{ }m}^{2}$ is heated from $0^{\circ}C$ to $100^{\circ}C$ without expansion or bending. The compressive force developed in it is :
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Solution
Sol. [Key Concept: Thermal stress in a clamped bar (σ = Y·α·ΔT)]
Thermal strain $=$ Longitudinal strain $= \alpha\Delta T$
⇒ Longitudinal strain, $\delta = 10^{- 5} \times 10^{2} = 10^{- 3}$
⇒ Compressive stress $= \delta \times$ Young's Modulus
$$\begin{matrix} & \ = 10^{- 3} \times 0.5 \times 10^{11} \\ & \ = 0.5 \times 10^{8} \end{matrix}$$
⇒ Compressive force $= 0.5 \times 10^{8} \times 10^{- 3} = 0.5 \times 10^{5}$
$$\begin{matrix} & \ = 5 \times 10^{4} \times \frac{10}{10} \\ & \ = 50 \times 10^{3}\text{ }N \end{matrix}$$
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