How this instrument works
Robert Hooke had the relationship by 1660 and, guarding his priority in the manner of his century, published it in 1676 as an anagram: ceiiinosssttuv. Two years later he unscrambled it — ut tensio, sic vis, as the extension, so the force. Pull any spring twice as far and it pulls back twice as hard. Stiffness k carries units of newtons per metre and describes an object rather than a material: cut one in half and its k doubles, exactly as k = EA/L predicts for a uniform bar of Young's modulus E, cross-section A, and free length L.
Real stiffnesses span six orders of magnitude. A toy slinky sits near 1 N/m, click-pen springs live in the low hundreds, car suspension coils run 20–50 kN/m, and the silicon cantilever inside an atomic force microscope is deliberately floppy — 0.01 to 50 N/m — so that deflections of mere nanometres can register forces measured in piconewtons. Whatever the scale, calibration is identical: hang a known load, record how far it moves, divide.
Linearity is a convenience, not a rule of nature. Every material has its proportional limit, and past it the force-against-displacement graph bends; past the elastic limit, deformation becomes permanent and your spring never returns to its free length. Rubber leaves the straight line almost at once, coil springs stay honest until their turns bind solid, and biological tissue was never on that line to begin with. Written as vectors the sign matters — F = −kx, minus recording that force opposes displacement — while this sheet returns magnitude.
- Type your stiffness into Spring constant (N/m). Take it from a datasheet, or from a load test: known weight divided by the sag it causes.
- Enter how far things have moved from free, unloaded length in Extension or compression — millimetres, centimetres, metres, or inches.
- Read Restoring force, switching its unit to kN or lbf if your bench notes happen to be written that way.
- Remember what that number means: a magnitude, always directed back toward the unstretched position.
Worked example — 100 N/m pulled ten centimetres
A bench spring rated at 100 N/m is clamped at one end and drawn 0.1 m from its free length. Enter 100 into Spring constant (N/m), then 0.1 m into Extension or compression. Arithmetic this bare needs no defence: F = 100 × 0.1 = 10 N. That is tension along the coil and, by Newton's third law, exactly what your fingers feel pulling back.
Ten newtons happens to be roughly what a one-kilogram bag of sugar weighs, which hands you a free calibration check — hang 1 kg on this spring and it should settle about ten centimetres lower. Stored energy comes to ½ × 100 × 0.1² = 0.5 J, released as a sharp snap should you let go, and a fair reason to respect any stretched exercise band.
Questions
Why is Hooke's law usually written with a minus sign?
Because force and displacement point opposite ways. In F = −kx, vector x runs outward from equilibrium, so that negative sign records how springs always act back toward equilibrium. This instrument reports magnitude, F = kx, which is what a force gauge reads off its dial. Keep the sign whenever you feed a result into Newton's second law — that minus is precisely what produces oscillation instead of runaway acceleration, and it is where the period T = 2π√(m/k) comes from.
Do I measure x from the bench or from the spring's rest length?
From rest length — free and unloaded, with nothing hanging on it. This is the error that wrecks most homework: someone measures a coil sitting 25 cm long under load, enters 0.25 m, and reports a force five times too big when natural length was 20 cm and true extension only 0.05 m. Measure twice, loaded and free, then subtract.
What units does spring stiffness use?
Newtons per metre (N/m) in SI, since k is force divided by displacement. Manufacturers frequently quote N/mm or lbf/in instead: 1 N/mm equals 1000 N/m, and 1 lbf/in works out at about 175.1 N/m. Convert any catalogue figure before typing it in, because mistaking N/mm for N/m shifts your answer by three orders of magnitude.
Where does Hooke's law stop working?
At the proportional limit, which arrives long before anything visibly breaks. Past it the force-displacement curve bends away from straight; past the elastic limit, material takes a permanent set. Coil springs also abandon linearity once their turns touch under compression, and most polymers were never linear from the first millimetre. Treat a straight line as a local approximation and check the working range on your datasheet.
Why is stored energy ½kx² rather than kx?
Because force builds as you stretch, starting at zero and finishing at kx. Work equals force times distance only while force holds constant, so here you want area under a straight-line graph — a triangle of height kx and base x, giving ½kx². That factor of one half explains why pulling twice as far stores four times as much energy.
How do springs combine in series and in parallel?
Side by side sharing one load, stiffnesses add: k = k₁ + k₂. End to end in a chain, compliances add: 1/k = 1/k₁ + 1/k₂, so two identical coils in series come out half as stiff as either alone. Suspension designers exploit both arrangements. Work out your combined k first, then bring that single number here.