How the normal force keeps you from falling through the ground

Brian Wong
Jul 9 · 4 min read

In physics, the normal force is invoked to explain why objects do not fall through the ground despite the force of gravity. By some miraculous coincidence, the ground exerts an upward force that exactly cancels out gravity. Let’s demystify this.

Unicorn on a Spring

A unicorn’s considerable weight is supported by a spring

The normal force appears mystical only because the relevant effects occur on a microscopic scale. Fortunately, the same interaction is at work when a unicorn sits on a tense spring, and this is much easier to visualize.

Here we see immediately why the spring musters exactly the upward force required to support the unicorn against gravity. The spring is able to provide a variable upward force; the more it is compressed, the more opposing force it provides. The equilibrium is a level of compression where the upward force from the spring is in precise balance with the downward force of gravity. If at some point the spring produced less force than the gravity of the unicorn, the unicorn would sink further down and compress the spring it reached the balance point.

It is also clear in this case that the spring provides no upward force as soon as the unicorn is removed and all compression is lost.

What’s so special about a spring?

In order to extend this analogy, we must identify the relevant property of a spring that allows it to exactly counterbalance the weight of any object.

Consider the distance between the bottom of the spring and the bottom of the unicorn. We know that as this distance decreases, the force that the spring applies upward against the unicorn increases.

We can make a stronger statement than that, though. It turns out that as this distance goes to zero, the upward force does not cap out at some max value. Instead, it goes to infinity.

Because of this characteristic, no matter how heavy the unicorn, the spring can always sink down sufficiently far to support it.

Atoms Behave Like a Spring

It turns out that every atom shares this key property of springs.

Consider the force that one atom exerts on another as they are brought closer and closer together. Because the nucleus of each atom is positively charged, there is a strong repulsive force when the nuclei are brought sufficiently close.¹

In particular, the strength of this electric repulsion is proportional to the inverse square of the distance between them. If we plot a graph of 1/x², we see that as x approaches 0, y goes to infinity:

This explains how the atoms of the ground function like a spring on a microscopic scale. If you zoomed in enough, you could see that as you step onto the ground, the atoms on the bottom of your shoes sink down into the atoms of the ground until the nuclei are so close together that the repulsive force balances out the force of gravity acting on your body.

Thus, the atoms of the ground behave like microscopic springs. When weight is applied, they “compress” (shortening the distance between their nuclei and the nuclei of the pressing object) to an equilibrium distance that supports the object’s weight. This “normal” force disappears as soon as the weight is removed because because the electrostatic force decays so quickly with distance (equivalent to a very tiny spring).

Addendum: Gravity vs. Electromagnetism

The astute reader will point out that gravity has the same inverse square scaling with respect to distance as the electric force. Yet in the context of a shoe pressing into the earth, the force of gravity is approximately constant while the electric repulsion between atoms varies from negligible to infinite over a microscopic distance!

There are two reasons for this inconsistency:

  1. Gravity is a universal attraction between any two particles, whereas the electric force can either be repulsive or attractive depending on the particles involved.
  2. Although the asymptotic scaling is the same (inverse square of distance), the constant multiplier is much smaller for gravity.

Each individual atom of Earth exerts a negligible gravitational pull. But all of the forces combine additively, even the ones from atoms on the other side of the planet. The net effect is a substantial force toward the center of the Earth. Because we are 4,000 miles from the Earth’s core, the force of gravity is effectively constant.

On the other hand, only the atoms right next to our shoe exert non-negligible electrical force. This is because most atoms have the same number of protons and electrons, so they look like uncharged particles from sufficient distance. It is only at very close distances — like when our shoe is shoved up against the positively charged nuclei of other atoms — that the subatomic protons and electrons behave mathematically as separate particles with substantial charge.

[1] There is, of course, some attraction between the nucleus of one atom and the electrons of the other. The nucleus-nucleus repulsion dominates at short separation because the light and energetic electrons are able to move out of the way.