What is the relationship between force and extension in a rubber band?

The relationship between force and extension in a rubber band is non-linear and follows Hooke's law only initially.

When you stretch a rubber band, the force you apply and the extension (how much it stretches) are related, but not in a straightforward way. Initially, for small extensions, the rubber band behaves somewhat like a spring, and Hooke's law can be applied. Hooke's law states that the force needed to extend or compress a spring by some distance is proportional to that distance. In this initial phase, the relationship between force and extension is linear, meaning if you double the force, the extension also doubles.

However, as you continue to stretch the rubber band, this linear relationship breaks down. Rubber bands are made of polymers, which are long, chain-like molecules. When you stretch a rubber band, these polymer chains first uncoil and then align in the direction of the stretch. This initial uncoiling phase is where Hooke's law approximately holds. But as you keep stretching, the chains become fully aligned and start to resist further stretching more strongly. This results in a non-linear relationship where the force required to stretch the rubber band increases more rapidly than the extension.

Additionally, rubber bands exhibit elastic hysteresis. This means that when you stretch and then release a rubber band, the path it follows while returning to its original shape is different from the path it took while being stretched. This is because some of the energy is lost as heat due to internal friction within the material.

In summary, while a rubber band may initially follow Hooke's law for small extensions, its behaviour becomes non-linear as the extension increases, due to the unique properties of the polymer chains that make up the rubber band. This makes the study of rubber bands a bit more complex but also quite fascinating!