Assume a metal rod with an initial length $L_{0}$ is heated through a temperature increase of $\Delta T$ to a length $L_{1}$ and then cooled to its initial temperature-that is, through a temperature decrease of $-\Delta T$ (same $\Delta T$ increase and decrease). Call the final length of the $\operatorname{rod} L_{2}$ after this thermal cycle.
(a) Show that Eq. $19.3$ implies that $L_{2}=L_{0}\left[1-(\alpha \Delta T)^{2}\right]$, that is, $L_{2} \neq L_{0}$.
(b) What is the implication if the rod were taken through a number of such thermal cycles?
(c) Obviously, something is wrong. Can you explain what it is? (Hint: Think of basis, or reference. For example, if you had an investment that appreciated $100 \%$ in value one day, and you lost $100 \%$ of your investment the next, would you still have any money left?)