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A bungee cord barely resists at first, then yanks hard at the bottom. A trampoline gives little from a small dip but launches you from a deep one. In every case, the more you deform it, the harder it pushes back. Is there one rule behind all of this?
A bungee cord barely resists at first, then yanks hard at the bottom. A trampoline gives little from a small dip but launches you from a deep one. In every case, the more you deform it, the harder it pushes back. Is there one rule behind all of this?
Hooke found it in 1676: double the stretch, double the force. But the ENERGY stored grows with the SQUARE of the stretch — which is why twice the speed does four times the crash damage.
Hooke's Law: a spring's force is proportional to how far it is stretched or compressed from its natural length, and always points back toward that length (a restoring force). The constant k measures stiffness.
Written fully as F = −kx, the minus sign says the force OPPOSES the displacement — stretch right, it pulls left. That permanent tug toward equilibrium is what makes a spring oscillate (the SHM tab) instead of sitting deformed. The stored elastic energy is PE = ½kx² — note the SQUARE, the source of the '4× energy for 2× stretch' result. That energy is handed back as the spring relaxes, which is how bows and catapults work. Push past the elastic limit, though, and the material deforms permanently — Hooke's Law no longer applies. **All forms:** , with stored energy . **Limiting case:** the law holds only up to the limit of proportionality — past it the – line bends and the spring keeps a permanent stretch; the straight line is the whole law. **Connect it:** feed into and you get — Hooke's law is the engine of simple harmonic motion.
Forgetting that x is the extension FROM the natural length (not the total length), and that Hooke's law only holds up to the elastic limit.