Recently in my physics class we conducted a lab where we had to calculate the k value of a rubber band and spring. In the lab we are supposed to discover that the rubber band does not obey Hooke's Law as nicely as the spring, and that the k value of the spring remains constant with more applied force while the k value of the rubber band changes as more force is applied. The motion of the weight attached to the elastic object demonstrates simple harmonic motion and I observed that the spring caused more uniform motion while the rubber band caused (pretty extreme) dampening. This lab made me wonder: Why do springs like to conform to Hooke's Law more than a segment of rubber band? Beyond material science, does the circular nature of the spring vs the linear nature of the rubber band effect how precise the movement of the object is over time?
The main reason for the difference is that you are deforming the rubber band more than the spring. In fact you are deforming the rubber band much, much more than the spring.
To understand this you need to appreciate how a helical spring works. When you stretch the spring you are not stretching the metal wire that it is made from. In fact you are twisting the wire and you aren't twisting it very much.
To see this let's zoom in on the spring:
As you extend the spring you rotate the top segment right and the bottom segment left so you are twisting the bit of the wire I've shaded light blue. So you aren't stretching the metal at all - just twisting it slightly. Exactly how much the wire is twisted depends on the spring geometry and the total spring extension but as a general rule springs are designed to keep the overal twisting quite small.
So suppose the extension in your experiment is a factor of two. With the rubber band you are stretching the rubber by a factor of two and that's a huge deformation. Rubber is only capable of doing this because the polymers that make it up can adjust thir conformation, but even so it's hardly any surprise that with such a large deformation the response is not linear. By contrast, when you stretch a spring by a factor of two you are twisting the wire by a relatively small amount. Because the strain of the wire remains small the response remains linear.