Hooke's Law Simulator

What Is the Hooke's Law Simulator?

The Spring That Fights Back Harder the Further You Pull

A car's suspension coil can carry well over a tonne, yet the same spring barely resists your fingers over the first centimetre — because a spring's push is not fixed, it builds. Drag the Applied force, F slider and watch the extension grow with it: the readout for Extension x tracks x = F/k, so doubling the force doubles the stretch. The restoring force starts at zero at the natural length and climbs in step with the pull, which is exactly what F = k·x encodes on the Force-vs-extension graph — a straight line rising from the origin. The stored elastic potential energy follows E = 1/2·k·x², so its readout climbs even faster than the force, growing with the extension squared.

Now hold the force steady and raise the Spring constant, k. Here is where intuition trips: a stiffer spring does not stretch more, it stretches less. Because x = F/k, a bigger k shrinks the Extension x for the same load, and the Force-vs-extension graph tilts steeper — its slope simply is k. Halve a spring's stiffness and the same pull stretches it twice as far.

Push F far enough and the elastic-limit indicator trips. Up to the limit of proportionality the graph stays a straight line and F = k·x holds; push a spring past its elastic limit, though, and it no longer springs back — it stays permanently stretched once the load comes off. Put numbers to your own coil with the Hooke's law calculator, or take the controls for a spin across our collection of hands-on physics simulations.

Where the ideal Hooke's law breaks down

The straight line lasts only while the steel stays elastic, and this simulator shows the exact moment it stops. Set k to 20 N/m and raise the force to 5.0 N: the extension reads 0.250 m and the status is still Elastic. Nudge the force to 5.5 N and the extension leaps to 0.313 m — far more than an extra half-newton deserves — and the status flips to Past limit. Underneath, the sim has switched to a second straight line at 40 per cent of the original stiffness. A real spring is both gentler and nastier: the curve bends smoothly rather than kinking, and the stiffness keeps falling as dislocations glide through the wire, until the coil necks and breaks.

Even inside the elastic region, E = ½kx² is the energy you put in, not the energy you get back. Load a real spring and unload it and the two force-extension curves do not lie on top of one another; the loop between them left as heat in the wire. For steel that gap is small, which is why steel makes good springs. For a rubber band it is enormous, and rubber is not linear to begin with anyway.

The spring constant is also a property of the whole spring, not of the metal in it. Cut a coil spring in half and each half has double the k; join two end to end and the pair has half. That is why a catalogue quotes a rate in newtons per metre rather than naming a material.

Two more walls the graph does not draw. A real spring runs out of travel: coils straighten in tension and touch one another in compression, at which point the stiffness effectively becomes infinite. And F = k·x describes equilibrium only. Lower a 6 N load gently onto a 50 N/m spring and it settles at the 0.120 m shown; release the same load suddenly from the natural length and it swings down to about twice that, near 0.24 m, before oscillating back. Static and maximum extension are different numbers, and lifting gear is designed around the larger one.

Where this is actually used

A hanging spring balance is Hooke's law with a printed ruler. Set k to 200 N/m and the force to 20 N: the extension reads 0.100 m and the stored energy 1.000 J. Since 20 N is roughly a 2 kg load, that spring gives 10 cm of pointer travel for 2 kg. Weigh a 20 kg suitcase across the same 10 cm and you need a spring nearer 2000 N/m, ten times stiffer. Every mechanical scale is that compromise: stiff enough for the range, soft enough to stay readable.

A car's coil spring is the same object two hundred times stiffer. Take one corner carrying about 400 kg, close to 4000 N, sitting 100 mm down on its spring: that is a rate near 40,000 N/m, against the 200 N/m maximum on this slider. Feed those into E = ½kx² and the corner is holding roughly 200 J, handed straight back when the wheel drops into a pothole.

At the far end of the range, an atomic force microscope images individual atoms with a cantilever whose spring constant is of order 0.1 N/m, some two hundred times softer than the softest spring on this slider. Because x = F/k, a force of one nanonewton bends it by about 10 nanometres, which a laser bounced off its back can measure. The instrument never measures a force directly: it measures a deflection and multiplies by k.