Overview

When an object is heated, its particles expand and move faster, making the object itself bigger. Similarly, when an object is cooled, its particles move closer together and move more slowly, making the object smaller. The differential equation for this process is

$\frac{\ⅆ L}{\ⅆ T}\= a L$,

where $L$ is the length of the object, $T$ is its temperature, and $a$ is the thermal expansion coefficient of the material. For a given initial length and initial and final temperatures, the final value of the length can be expressed as

 $L_{\mathrm{final}}\=L_{\mathrm{initial}} \cdot {\ⅇ}^{a\left(T_{\mathrm{final}}-T_{\mathrm{initial}}\right)}$.

For small changes in temperature, you can use a linear approximation instead:

 $L_{\mathrm{final}}-L_{\mathrm{initial}}\=a \cdot L_{\mathrm{initial}} \cdot \left(T_{\mathrm{final}}-T_{\mathrm{initial}}\right)$.

$L_{\mathrm{initial}}$ : m

$L_{\mathrm{final}}$ :  m

 $T_{\mathrm{initial}}$ : ${}^{\circ }$C

  $T_{\mathrm{final}}$ :  ${}^{\circ }$C$$

$a$ : $\times {10}^{-6}$ K^-1

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