What is the purpose of the linear approximation ΔL=αL0ΔT?
When using this, we run in all kinds of problems. For example, when a material heates up twice by 1K we get L=L1+αL1=L0+αL0+α(L0+αL0)=L0+2αL0+α2L0.
However when it heats up by 2K we get L=L0+2αL0. A similiar thing has been discussed in Bug in linear thermal expansion, L0 must be 0. This is not a duplicate.
So is the correct formula L=eαΔTL0? This looks the most logical when looking to a material heating up increasingly many times, since it then resembles the Taylor series of ex in αΔT. 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 L0 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.
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