Need help visualizing a conceptual problem; solid mechanics 1

Question

When the instructor posed this problem to the class, no one could answer the question because we all had difficulty 'seeing' what was happening.

I get that because they're different materials, the change in temperature will cause a different change in length in the rod compared to the cylinder; however, apparently this means that one of them will be in tension and one will be in compression and this is where i have troubles visualizing the problem. how exactly does one "see" this? i know this isn't a very specific question, but i'd really like to understand the concepts behind this, so any help is appreciated :)

Answer 1

Suppose the rod expands less than the cylinder, and suppose we unstick the end of the rod. Then after we've heated the system it will look like this:

Because the rod expanded less than the cylinder there will be a gap $x$ between the end of the rod and the plate. So if you want to re-attach the rod to the plate you have stretch the rod by a distance $x_r$ and compress the cylinder by a distance $x_c$ such that $x_r + x_c = x$. You just need to calculate $x_r$ and $x_c$. Use the fact that the force due to extension of the rod must be the same as the force due to compression of the cylinder.