Spring constant, stiffness and stored energy
What the spring constant really measures, why stored energy grows with the square of the stretch, and where Hooke's law breaks down.
A straight line between force and stretch
Hooke's law describes a spring whose force rises in direct proportion to how far it is displaced. Plot force against stretch and you get a straight line whose steepness is the spring constant. A high spring constant means a steep line and a stiff spring that fights back hard for a small stretch, while a low constant means a soft, easily stretched spring. This linear behaviour is what makes springs so predictable and useful in scales, suspensions and clocks.
Reading the spring constant physically
The spring constant k is the force needed per meter of stretch, measured in newtons per meter. A 200 N/m spring takes 200 N to stretch one meter, or a proportional 20 N to stretch 0.1 m. Because k is a property of the spring itself, it stays the same across the elastic range no matter how far you stretch within it. Measuring the force at a known stretch is the standard way to determine k for a real spring.
Why stored energy is quadratic
Elastic potential energy is the work done to stretch the spring, which is the area under the force-versus-stretch line. Since the force itself grows with the stretch, that area is a triangle, giving energy equal to one half times k times the stretch squared. Doubling the stretch therefore quadruples the stored energy, not just doubles it. That squared relationship is why a slightly longer draw on a bow or catapult stores dramatically more energy.
The elastic limit
Every real spring obeys Hooke's law only up to its elastic limit. Below that limit it returns exactly to its original length when released, and the calculations here hold. Push past it and the material yields, taking a permanent set or snapping, and the neat proportionality disappears. Design work keeps working stresses comfortably inside the elastic region so the spring behaves linearly and does not fatigue prematurely.