🧮 Work by a Variable Force

📖 The Physics

Work is easy when the force is constant: W = F × d. But what if the force changes as the object moves, like a stretching spring that gets harder to pull? A single value of F won't do.

The answer is the area under the force–displacement graph — you add up F·Δx over every tiny step, which is exactly the integral ∫F dx.

Set the gradient k = 0 and the force is constant — the area is just a rectangle F₀·d. Set F₀ = 0 and it becomes a triangle ½·k·d² (exactly a stretched spring's energy). Any shape works: the area is the work.