Hooke’s Law & the Spring Constant: Definition & Equation

Elasticity & the Spring Constant

After every Thanksgiving, consuming absurd amounts of turkey, stuffing, mashed potatoes and cranberry sauce, I’m extremely thankful for elastic waistbands. And it’s probably a good thing that bungee cords stretch just the right amount. But how exactly do these materials work?

Elasticity is a property of a material which allows it to return to its original shape or length after being distorted. Some materials are not at all elastic — they are brittle and will snap before they bend or stretch. Others, like rubber, for example, will stretch a long way without significant warping or cracking. This is because the materials contain long chain molecules that are wrapped up in a bundle and can straighten out when stretched. Those are the materials we choose for things like waistbands.

But in physics we like numbers. One important number that relates to elasticity is the spring constant. The spring constant is a number that represents how much force it takes to stretch a material — materials with larger spring constants are stiffer.

So these are our properties to describe the stretchiness of materials — elasticity and the spring constant. But what happens when you’re in the process of stretching out a rubber band? Do you need more force at first and less once you’ve stretched part way? Perhaps it’s the other way around? The relationship could be almost anything — linear, quadratic, variable… so what did scientists discover when they started investigating elastic materials?

Hooke’s Law

Robert Hooke investigated how springs and elastic materials stretch. Hooke’s Law states that the force needed to compress or extend a spring is directly proportional to the distance you stretch it. Or, in other words, the more you stretch something, the harder it becomes to keep stretching it. It’s a linear relationship. Or you could think of it this way: As you stretch something out, there is a restoring force that you have to compete with. That restoring force is trying to spring the object back to how it started.

As an equation, Hooke’s Law says that the force applied in newtons (or the restoring force — it amounts to the same thing) is equal to the negative of the spring constant, k, of the material, multiplied by the extension, x, of the material, measured in meters. We use the negative sign when we’re talking about the restoring force because the restoring force is in the opposite direction to the extension. But if F is the force we apply, then the negative sign goes away, and it’s just F = kx.