Hooke's law says a spring's extension is proportional to the force stretching it: F = k·x. Drag the sliders below to change the applied force and the spring constant, and watch the extension, force, stored energy and the force-vs-extension graph respond in real time.
Hooke's Law
Hang a load and watch the extension grow in proportion to the force — until it passes the elastic limit. F = kx.
Within the elastic limit — extension is proportional to force (F = kx).
Force vs extension
What Is the Hooke's Law Simulator?
The Hooke's law simulator is a free interactive physics lab that runs in your browser — nothing to install and no sign-up. Change the applied force and spring constant and watch the extension, force and stored energy respond through F = k·x.
| Control | Range | Step |
|---|---|---|
| Applied force | 0 – 20 N | 0.5 |
| Spring constant per metre | 20 – 200 N/m | 5 |
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.
Frequently asked questions
What is Hooke's law?
Hooke's law states that the extension of a spring is proportional to the force stretching it: F = k·x, where k is the spring constant. It holds up to the limit of proportionality; the stored elastic energy is E = ½·k·x².
Does a stiffer spring stretch more or less?
Less. Since x = F/k, a larger spring constant k means a smaller extension for the same force. A stiffer spring resists more, so it stretches less — halve the stiffness and the same pull stretches it twice as far.
What is the elastic limit?
The elastic limit is the point beyond which a spring no longer returns to its natural length — stretch it past that and it stays permanently deformed. It sits at or just beyond the limit of proportionality, the point where the force-extension line stops being straight and F = k·x no longer holds. Between the two the spring is non-linear but still springs back.
What does the slope of a force-extension graph represent?
For a graph of force (vertical) against extension (horizontal), the straight-line slope is the spring constant k. A steeper line means a stiffer spring. The area under the line equals the elastic potential energy stored.
References & formula source
- Halliday, Resnick & Walker — Fundamentals of Physics, Chapter 7 (Kinetic Energy and Work), Hooke's-law spring.
- Young & Freedman — University Physics with Modern Physics, §6.3 (Work Done by a Varying Force: Spring Forces).
- R. Nave — HyperPhysics, Georgia State University, "Hooke's Law" / elasticity section.
- Further reading: Hooke's law — Wikipedia