# Springs -- Hooke's law

#### Prerequisites

The spring was our starting example for thinking about how to quantify impulse and force in the context of Newton's laws because it displayed how much force was being exerted in a way we knew how to measure — by changing it's length. Let's look more closely at a spring and develop a model of the forces it exerts.

As we hold a spring and pull it from both ends, we can feel that as it stretches, the force it is exerting on us increases. (And correspondingly, by Newton's 3^{rd} law, the force we have to exert on it to stretch it increases as well.) Let's consider two hands pulling a spring. Though there are a lot of forces, nothing is moving so the forces on each object — spring, left hand, right hand — are all balanced.

Here are the free-body diagrams of each of the hands and the spring. (We are only showing the horizontal forces in order to focus on the stretch of the spring. There have to be weights pulling down and friction forces pushing up to keep everything from falling.)

We have labelled each of the forces felt by each object with our full-blown labeling convention — type of force, causing object, and feeling object.

For the spring the left hand pulls to the left with a tension force and the right hand pulls to the right. Each of the hands feels the spring pulling it toward the center. Since the hands aren't moving, there must be another force pulling back on them — the force of the arms pulling on the hands. (In our Newtonian framework we can designate anything as an object, even dividing connected things into parts.) Each of these forces is pulling out on the object feeling it, so they are all tension forces (T).

Since none of these objects are accelerating (all are at rest at a constant velocity = 0), each pair of forces felt by each object must be equal and opposite (by N2). Note that the neighboring arrows in the middle — the force of a hand on the spring and the spring on that hand — are Newton's 3rd law pairs: they are of the same type and have their "cause-feel" labels reversed. By N3 they must therefore be equal. As a result, all 6 forces are equal. We'll write their magnitude as *T* — the tension.

If we do some experiments with good springs and small amounts of stretch, we will find that two identical springs hooked one to the other pull the same as a third spring with twice the stretch. After a bunch of experiments like this, Robert Hooke (1635-1703) proposed a law describing springs ("Hooke's law") that the stretch was proportional to the pull. We write it like this:

$$T = k \Delta L$$

In words:

If a spring is pulled (or squeezed) from opposite ends with an equal and opposite tension, the amount the length of the spring changes is proportional to the tension. The proportionality constant is a property of the particular spring.

Now it turns out that this is not really a great law for springs. A lot of springs won't support a compression (squeeze) at all and will just collapse. Hooke's law is a *toy model* of a spring — a simplification of a system that lets us understand what's going on in complete detail.

Many springs only have stretch proportional to how hard they are being pulled over a small range of stretches. But what is really important is that the Newton's law equations for a mass being pulled by a spring described an oscillation. (We'll get into this in great detail later.) Think of a mass hanging from a spring and oscillating up and down. These equations turn out to be the basic equations that lets you build mathematical models of almost anything that oscillates — from electric circuits generating radio waves, to vibrations of molecules, to muscular tremors.

"Springs" seems like a rather obscure and uninteresting topic. But because the same equations that describe them describe so many other phenomena, it's really useful to get comfortable with springs. They are a simple system that is easy to understand completely, and they can be used as analogies and ways into mathematical modeling more complex and interesting phenomena.

## Scales

Let's try to get a semi-quantitative idea of how big the forces are in typical springs (an Estimation). In the figure at the right we show a real spring that is about 4 inches long (10 cm). I've held and pulled such springs while doing physics demonstrations. I can stretch to about 1/2-again its length by pulling pretty hard on both ends. That's about a 5 cm stretch. How hard do I pull when I do that? Well, I can think about lifting various of the set of weights that we put on a balance. I suspect the force I'm exerting on the spring is about the same as I exert when I lift a 1/2-kg mass.

That weighs mg = (0.5 kg)(9.8 N/kg) ~ 5 N. So the spring constant k is on the order of

$k = T/ΔL$ ~ (5 N)/(5 cm) = (5 N)/(5 cm) x (100 cm)/(1 m) = 100 N/m

so a fairly stiff hand held spring might have a spring constant of 100 N/m (or 1 N/cm).

Julia Gouvea, Joe Redish, and Mark Eichenlaub 9/26/17

#### Follow-ons

Last Modified: August 24, 2019