Intuition The one core ideaA spring pushes back harder the more you deform it, so the work you spend stretching it piles up unevenly — small at first, large at the end. To store that idea in symbols we need just three ingredients: a displacement x stiffness k add up tiny bits of work — and everything on the parent page is built from those three.
This page assumes you have seen nothing . We meet each symbol, draw the picture it stands for, and say why the derivation cannot proceed without it. Read top to bottom — every item leans on the one above it.
Definition Natural length and equilibriumA spring left completely alone — nobody pulling, nobody pushing — sits at its natural length . The single point where the spring exerts no force is called the equilibrium position . We mark it as the origin, the place where our measuring ruler reads zero.
Picture a coil lying on a table with a marker painted on its free end. Untouched, that marker sits at a spot we will call "0". Everything else on the parent page is measured from this spot .
Intuition Why start here?The parent note keeps warning "measure x
x x how far the free end has moved from equilibrium , measured in metres (m x positive . Squash it left → x negative . At rest, x = 0
The picture: an arrow drawn from the "0" mark to where the end currently sits. The length of that arrow is how much x direction gives its sign.
Intuition Why we need a signed number, not just "how much"The spring's force flips direction depending on which side you are on. A plain distance ("0.1 m") can't tell left from right. A signed displacement can, and that sign is exactly what carries the minus into Hooke's law next.
Definition Force and the restoring directionA force is a push or a pull, measured in newtons (N restoring because it always points back toward equilibrium — opposite to the displacement you imposed.
See Hooke's Law for the full story of why real springs obey this to good approximation.
Intuition Why the topic can't skip the signThe parent page splits force into two versions: the spring's F spring = − k x your applied force F applied = + k x slowly you must exactly cancel the spring, so your push equals + k x
Definition Spring constantk k how stiff the spring is : newtons of force produced per metre of stretch. Units: N/m k k
The picture: the steepness of the straight line when you plot force against displacement. A stiff spring gives a steep line; a soft spring gives a shallow one. k is that slope.
Intuition Why a constant, not a variableThe word ideal in "ideal spring" means k triangle rather than some lumpy blob. If k x 2 1 k x 2
Work is energy transferred when a force pushes something along a distance. When the force is constant , work is simply
W = F × d J
The picture: the area of a rectangle — height F d
Common mistake The trap the parent page fightsWhy W = F × d it's the first work formula everyone learns.
Why it fails for a spring: it demands a single, constant F k x F The fix (previewed): slice the motion so thin that F
See Work done by a variable force for the general machinery.
Definition The tiny sliced x ′ d x ′ extremely small piece of displacement " — so short that across it the force doesn't visibly change. The prime (′ x ′ running position as we sweep from 0 x
Definition The integral sign∫ ∫ 0 x ( something ) d x ′ d x ′ x ′ 0 x ∫ Sum .
Intuition Why an integral is exactly the right toolWe chose the integral because the force varies. On each hair-thin slice the force is effectively constant, so on that slice W = F × d d W = k x ′ d x ′ triangle under the force line, and a triangle's area is 2 1 × base × height = 2 1 x ⋅ k x = 2 1 k x 2
Definition Potential energyU Stored energy that can be released later, measured in joules (J U = 2 1 k x 2
Intuition Why "work in = energy stored" is allowedOnly for a conservative force does every joule of work go into recoverable storage — none leaks to heat. A spring is (idealised as) conservative, so work done against it equals PE gained exactly. That is the promise of Conservative forces and potential energy . When the spring later releases, that U Conservation of mechanical energy .
Notice x squared : compressing by 0.04 m 0.04 m same positive energy, because ( − 0.04 ) 2 = ( 0.04 ) 2
Work = F times d for constant F
Slice dx' force nearly constant
Triangle area one half base height
Test yourself — can you answer each before revealing?
Where is x = 0 At the free end's resting position when nobody touches it (natural length / equilibrium).
What does a negative x The spring is compressed — the free end moved to the opposite side of equilibrium.
What does the minus sign in F = − k x The force always points back toward equilibrium, opposite to the displacement.
What are the units of k k N/m
Why can't you use W = F × d That formula needs a single constant force, but the spring's force grows from 0 k x
What does ∫ 0 x k x ′ d x ′ Add up k x ′ d x ′ d x ′ x ′ 0 x
Why is the stored energy a triangle's area, not a rectangle's? Because force rises linearly from 0 k x 2 1 ×
Why does work-in equal energy-stored for a spring? The spring force is conservative — no energy leaks to heat, so all work is recoverable as U
Why do stretch and compress by the same amount store equal energy? U = 2 1 k x 2 x
