Question

Calculate the compressional force required to prevent the metallic rod of length $l$ $cm$ and cross-sectional area $A \, cm^{2} \, $ when heated through $t \,{}^\circ C$ , from expanding lengthwise. Young's modulus of elasticity of the metal is $E$ and mean coefficient of linear expansion is $\alpha $ per degree celsius.

• A. $\textit{EA}\alpha \textit{t}$
• B. $\frac{\textit{EA} \alpha \textit{t}}{\left(1 + \alpha \textit{t}\right)}$
• C. $\frac{\textit{EA}\alpha \textit{t}}{\left(1 - \alpha \textit{t}\right)}$
• D. $\textit{E}l \alpha \textit{t}$

Answer 1

Answer: (B)

The change in natural length $= \Delta l _{\text{t}} = l ⁡ \alpha \text{t}$
The natural length of rod at temperature $\text{t}^{^\circ } \text{C} = l + l ⁡ \alpha \text{t}$
The decrease in natural length due to developed stress $= \Delta l $
But the length of rod remains constant.
$∴ \, \, \Delta l _{\text{t}} - \Delta l ⁡ = 0$
$∴ \, \, \Delta l = \Delta l ⁡_{\text{t}} = l ⁡ \alpha \text{t}$
$∴ \, \, \text{E} = \frac{\text{stress}}{\text{strain}} = \frac{\text{F/A}}{\frac{- \Delta l }{l ⁡ + \Delta l ⁡_{\text{t}}}}$
$∴ \, \, \text{E} = - \frac{\text{EA} \Delta l }{l ⁡ + \Delta l ⁡_{\text{t}}} = - \frac{\text{EA} \Delta l ⁡_{\text{t}}}{l ⁡ + \Delta l ⁡_{\text{t}}}$
$= - \frac{\text{EA} l \alpha \text{t}}{l ⁡ + l ⁡ \alpha \text{t}} = - \frac{\text{EA} \alpha \text{t}}{\left(1 + \alpha \text{t}\right)}$