Hooke's law

Hooke's Law provides a mathematical model for the behaviour of elastic materials. It states that the force required to extend or compress a spring by some distance is directly proportional to that distance. For A-Level Further Mathematics, we use a specific form of this law that involves the material's properties.

$T = \frac{\lambda x}{l_0}$

It's crucial to distinguish between elastic strings and elastic springs. An elastic string, like a rubber band, can only be in tension. If you try to compress it, it goes 'slack', and the tension instantly drops to zero. An elastic spring, like a coil spring, can be both stretched (in tension) and compressed (in thrust). The same formula applies for thrust, where $x$ represents the compression distance.

When a particle attached to an elastic string or spring is moving, we can no longer assume forces are balanced. Instead, we use Newton's Second Law, $F_{net} = ma$. The tension $T$ (or thrust) from Hooke's Law becomes one of the forces contributing to the net force. Remember that the tension is not constant; it changes as the extension $x$ changes.