A spring pulls back in proportion to its stretch: pull it twice as far, it pulls twice as hard, and always toward home. That single straight line — F = -k·x — is elasticity's first law, and it is runnable: the restoring force is linear, the stored energy is the parabola ½k·x² underneath it, and the same law in a bar reads stress = E·strain. Down the center, data flows: a stiffness and a stretch go in, the engine computes force and energy, the verdict comes out. The blue team builds and defends it; the red team tries to break it.
source Hooke, Lectures de Potentia Restitutiva, or Of Spring — "ut tensio, sic vis" (London: John Martyn, 1678) — archive.org · bim_early-english-books · de-potentia-res · Hooke 1678. Facsimile, not a critical edition — AMBER. Rendered, not quoted.
Correctness is not memorized; it falls out of proportionality. Four facts define the law:
L1 the force is linear in stretch — double x, double F. L2 it is restoring — the minus sign; F always points back to rest. L3 the stored energy is the area under the line, U = ½k·x² = ∫k·x dx. L4 in a bar it is the same law with stiffness E: σ = E·ε, E the elastic slope.
For the current setting, the law checked at x and 2x (linearity is exact, not fitted):
| stretch | force F = -k·x | energy ½k·x² |
|---|
Hooke's straight line is the elastic beginning of a longer curve. Follow the stretch past the proportional limit and the material stops obeying — it yields, then necks, then fractures. That whole shape is the neighbouring sphere: the stress–strain curve.
Hooke 1678 is its first segment — σ = E·ε, the initial slope every real specimen traces before it leaves the line. One deletion — the line is not eternal — is the door from elasticity to plasticity and to Griffith's fracture. Each sphere is the next one's premise.
The blue team's live check: sweep the law across a range of stretches and confirm it stays linear and restoring, and that energy equals the integral of force. If red tampers, this badge is where it shows.
The engine takes two numbers and a reading. In the spring reading: a stiffness k (N/m) and a displacement x (m). In the bar reading: a Young's modulus E (GPa) and a strain ε (dimensionless). The two readings are the same law:
| reading | input | stiffness | law |
|---|---|---|---|
| spring | stretch x | k | F = -k·x |
| bar | strain ε | E | σ = E·ε |
| energy | either | — | U = ½k·x² |
"Restoring" = the force opposes the stretch (the minus sign). "Linear" = the response to a sum of loads is the sum of responses. That is the whole game — and it is what you feed the panel below.
Spring: pull the mass by x and the force pulls back, straight and linear, toward rest.
Move any control — force and energy are computed from the law on the spot, never looked up.
What the machine produces, proven: force is linear in stretch (double x doubles F, to 1e-12) and restoring (the minus sign); energy U = ½k·x² equals the integral of k·x to a finite-difference tolerance of 1e-9; the bar form σ = E·ε has Young's modulus as its exact slope; superposition holds while elastic. The current force and energy are above; the guarantees are the output.
The blue team's witness (left) confirms these live; the red team (right) tries to make the line curve.
"Constant" k and E are themselves idealizations — they drift with temperature, rate and fatigue. So "elastic" is already a range, which is why the panel lets you push toward the limit. The straight line is not the mechanics of solids; it is the first proof that a material's response has a computable shape.
"Hooke's law holds for any deformation." Cut. Only below the proportional/elastic limit; beyond it the material yields, then necks, then fractures — the line ends.
"Stress equals strain." Cut. Stress = E × strain; the modulus E carries the units and the stiffness. Drop it and the equation is dimensionally false.
"Hooke published the equation F = -k·x in 1678." Kept, corrected. He published the anagram ceiiinosssttuu in 1676 and revealed "ut tensio, sic vis" in 1678; Young's modulus (1807) and the modern F = -k·x came later.
The red team's move: bend the law — make the force quadratic in x (F = -k·x²) instead of linear. Then "double x" no longer doubles F. The blue team's witness (window 7) is watching.
Curve the law and the doubling test fails: at 2x a quadratic force is four times as large, not twice. The witness recomputes, disagrees with linearity, and turns red. Nothing is faked; the attack is real and it is caught.