Spring Forces

A spring resists being moved from its relaxed length. Stretch it and it pulls back in; compress it and it pushes back out. That is the whole idea of a restoring force: it always points back toward the spring's natural length. Hooke's law writes it as F = −k·Δx, where Δx is the displacement from the natural length and k is the spring constant. The minus sign is not decoration. It is the part that says “back toward equilibrium,” and dropping it is where most points go.

Points leak in three places. Writing Hooke's law as F = k·Δx and losing the minus sign, so the force ends up pointing the wrong way. Swapping the rules for combining springs: in series the springs carry the same force and their stretches add, so 1/k_eq = 1/k₁ + 1/k₂, softer than either spring, while in parallel they share one stretch and their forces add, so k_eq = k₁ + k₂, stiffer than both. And forgetting, when a spring acts alongside gravity or another force, that the spring force is still −k·Δx while the equilibrium and the stretch are set by the full force balance, not by the spring alone.