Episode 2: the true nature of pushing is electromagnetism → Episode 3: friction, tension, and normal force were all the same single force
Textbooks line up a whole row of names for forces ── friction, tension, normal force, elastic force, viscous drag… There are so many that they seem hard to memorize. But just as peeling back "pushing" in Episode 2 brought out electromagnetism, these differently named forces are in fact almost all different faces of the same single electromagnetism. The abundance of names is an abundance of situations, not an abundance of kinds of force. The sister series' refrain of "content over names" applies straight to everyday forces. And the deep reason Hooke's law \(F=kx\) holds for any material also drops right out of Episode 2's "valley."
In Episode 2 we saw that between two atoms there is a "valley in the potential energy." Closer than the valley, repulsion; further than the valley, attraction. The everyday forces are merely differences in which direction and which arrangement this one valley is used in.
| Name of the everyday force | True nature (which use of Episode 2's valley?) |
|---|---|
| Normal force / contact force | The repulsion on the near side of the valley. The pressed-in electron cloud pushes back (Episode 2 itself). |
| Tension (string / rope) | The attraction on the far side of the valley. Atoms within the string hold onto neighboring atoms electromagnetically. |
| Elastic force (spring) | A displacement from the bottom of the valley. Compress it and it repels, stretch it and it attracts, trying to return to its original state. |
| Friction | The electromagnetic catching and sticking between the atoms of touching surfaces (roughness and adhesion). |
None of these is a "new force." The carrier of them all is the electromagnetic force between electrons and nuclei. One of the four forces, changing its arrangement, just looks like many different things. A different name makes it seem like a different thing, but it was only a different situation.
Hooke's law \(F=kx\) (a restoring force proportional to the stretch \(x\)) holds ── for a small deformation ── for almost anything: metal springs, rubber, atomic bonds, strings. Why is it so universal? The answer is in Episode 2's valley.
Expand the potential energy around the stable spacing (the bottom of the valley) \(r_0\), and
$$V(r_0+x)\approx V_0+\tfrac12\,k\,x^2\qquad(k=V''(r_0))$$
The first-order term vanishes (at a minimum the slope is zero). What remains as the star is the \(x^2\) term = a parabola. Force is its slope, so \(F=-dV/dx=-kx\).
The point is that no matter what shape a valley has, up close near its bottom it looks like a parabola. So "a restoring force under small deformation is proportional to the displacement" ── \(F=kx\) ── appears regardless of the details of the material. The universality of the spring came not from the properties of the material, but from the geometry of "near any stable equilibrium, everything is a parabola." In the figure below, watch how the actual force (the force from the valley) and the parabolic approximation \(F=-kx\) overlap perfectly for small displacements and drift apart as you displace things by a large amount.
The conclusion of Episode 3. The names of forces that flood daily life ── friction, tension, normal force, elasticity ── were just the same single electromagnetism, called by different names for different situations. It's not that there are many kinds of force; there are many situations. The sister series' refrain of "look not at the name (the appearance) but at the content (the definition)" happens, unchanged, right at the introduction to mechanics.
"Friction, tension, normal force, elastic force" (many nouns) → the reality is "the electromagnetic interaction between atoms (= Episode 2's valley), used in whichever direction: pushing / pulling / displacing."
The names are names for the "scenes" of a relationship. The content is one relationship.
"Friction = the catching of surface roughness" is an entry-level picture; actual friction is more complicated. The area really in contact (the true contact area) is quite small, and adhesion (sticking) and plastic deformation there matter; a complete explanation of "why friction is roughly proportional to the surface pressure (Amontons' law)" is a topic still under active research. That said, this episode's conclusion ── that the carrier is electromagnetism ── is unshaken.
Also, \(F=kx\) is an approximation that holds only for small deformations (the parabolic approximation at the bottom of the valley). Stretch it far and it deviates from proportionality, and eventually crosses the valley and breaks (the bond snaps). That "drifting apart" in the figure is exactly this.
Friction, tension, normal force, elastic force ── the everyday names for forces are many, but their true nature is the one "interatomic electromagnetic valley" of Episode 2. The pushing side (repulsion) is the normal force, the pulling side is tension, a displacement from the bottom is elasticity, and the catching of surfaces is friction. The names are names for situations, not kinds of force. In particular, the reason \(F=kx\) holds for any material is the geometry that near any stable equilibrium, every valley is a parabola.
What the peeling revealed is the sister series' "content over names." Force = a relationship, and that relationship is a single electromagnetism ── that was the conclusion at the everyday scale. ── So far we've seen "everyday forces = electromagnetism." Next time the flavor changes: we step into "forces that aren't forces at all" ── the centrifugal force and the Coriolis force. It's the story of phantom forces that well up the moment you rotate your coordinates.
Print / PDF: ⌘+P (Ctrl+P on Windows). On screen, the slider lets you see that for small displacements the actual force and the spring approximation overlap. Click "See the answer" to open a solution.