homework and exercises - What is the purpose of the linear approximation $Delta L = alpha L_0 Delta T$?

What is the purpose of the linear approximation $\Delta L = \alpha L_0 \Delta T$?

When using this, we run in all kinds of problems. For example, when a material heates up twice by 1K we get $L=L_1+\alpha L_1=L_0+\alpha L_0+\alpha (L_0+\alpha L_0) = L_0+2\alpha L_0+\alpha^2L_0$.

However when it heats up by 2K we get $L=L_0+2\alpha L_0$. A similiar thing has been discussed in Bug in linear thermal expansion, $L_0$ must be $0$. This is not a duplicate.

So is the correct formula $L = e^{\alpha\Delta T} L_0$? This looks the most logical when looking to a material heating up increasingly many times, since it then resembles the Taylor series of $e^x$ in $\alpha\Delta T$. If so, what is the purpose of the linear approximation?

Answer

The formula will be a good approximation for a "reasonable" temperature range. However, note in your original problem description, that

$$L_0$$ is the length of the object at a standard temperature. If necessary, you will have to calculate this length, and base all of your temperature differences on the standard temperature that corresponds to this length. Once that is done, the linear relationship that you noted in your question should work well.