About this Hooke's law calculator
Springs are everywhere, and they all obey one beautifully simple rule. The spring in a pen, the suspension under a car, the coil in a mattress, and the balance spring in a mechanical watch all push back with a force proportional to how far they are stretched or compressed. A spring calculator applies Hooke's law to that everyday situation: enter the spring constant and the displacement, and it returns the restoring force, the energy stored in the spring, and, if you add a mass, the period of oscillation. It also handles the combinations that trip people up, because two springs joined end to end behave very differently from two springs side by side.
Hooke's law states that the restoring force equals the spring constant multiplied by the displacement from the spring's natural length, F = kx, with the force pointing back towards equilibrium, which is why the full form is often written with a minus sign, F = -kx. The spring constant k, measured in newtons per metre, describes stiffness: a car suspension spring might have a k of tens of thousands, while a pen spring is a few hundred. The energy stored in a stretched or compressed spring is one half k times x squared, PE = (1/2)kx^2, which grows with the square of the displacement, so doubling the stretch quadruples the stored energy. And a mass on a spring oscillates with a period of two pi times the square root of mass over k, a result that depends on the mass and the spring but, remarkably, not on the amplitude, for small oscillations.
Using the tool starts with the two numbers that define any spring problem: k and the displacement in metres, taking care to convert centimetres to metres first, since a factor of a hundred in displacement becomes a factor of a hundred in force and ten thousand in energy. The optional mass field adds the oscillation period, which turns a static force problem into a dynamics one, and the presets cover a stiff spring stretched a little, a soft spring stretched a lot, and a car-suspension-like case for comparison. Below the main calculator, the combination panel takes two spring constants and returns both the series and parallel equivalents, which is the part of spring physics most students get backwards, and the history keeps a record of each stretch you tried.
The headline example is worth doing by hand once. A spring with k equal to 200 newtons per metre, stretched by 0.1 metres, exerts a restoring force of 200 times 0.1, which is 20 newtons, roughly the weight of a two-kilogram bag. Its stored energy is one half times 200 times 0.01, which is exactly 1 joule. Add a one-kilogram mass and the oscillation period is two pi times the square root of one divided by 200, which is about 0.444 seconds, a brisk jiggle of just over two cycles per second. Notice how the energy depends on the square: stretch the same spring twice as far, to 0.2 metres, and the force doubles to 40 newtons while the energy quadruples to 4 joules, which is why over-stretching a spring is punished so disproportionately.
The combination rules are the part most worth memorising, because they run opposite to the rules for electrical resistors. Springs in series, joined end to end, share the same force but split the displacement, so the combination is softer than either spring alone: two 100 newton-per-metre springs in series give an equivalent of 50, because each stretches half as much for the same force. Springs in parallel, side by side sharing the displacement, share the load, so the combination is stiffer: the same two springs in parallel give 200. Car suspensions exploit the parallel idea with multiple springs and dampers, while the series idea appears wherever a long, soft travel is needed from stiff elements. Getting this backwards is the classic exam error, and this tool's combination panel exists to make the right answer the easy one.
Real springs, of course, eventually break Hooke's law, and knowing where is part of using the tool wisely. Every spring has an elastic limit: stretch it gently and it returns exactly to its original length, stretch it too far and it stays permanently deformed, and stretch it further still and it breaks. The law also assumes the spring itself is light compared with the attached mass, and that the motion is slow enough that the spring moves as a whole rather than rippling like a slinky. Damping, the friction that makes real oscillations die away, is ignored here, so the period is exact for an ideal spring but real oscillations will gradually shrink. Within those limits, though, Hooke's law is astonishingly accurate, which is why it has survived three and a half centuries of use, from clockmaking to vehicle design to the atomic-force microscopes that image individual atoms by feeling the forces on a microscopic spring.
The applications repay a moment's reflection, because springs are load-bearing members of modern life. Vehicle suspensions use springs to absorb bumps and keep tyres on the road, and their stiffness is tuned as a compromise between comfort and handling. Mechanical watches keep time with a balance spring whose period is nearly independent of amplitude, which is why Hooke's law and the oscillation formula together made portable timekeeping possible. Pogo sticks, trampolines, diving boards, and prosthetic running blades all store energy in elastic deformation and return it, and the energy formula tells you exactly how much. Even at the molecular scale, chemical bonds behave like tiny springs, and the same mathematics describes how molecules vibrate, which is how infrared spectroscopy identifies substances. A simple law, a simple tool, and a very long reach.
A second worked example, the car suspension case, shows the numbers at realistic scale. A suspension spring might have a constant of 30,000 newtons per metre and compress by 0.05 metres under load, giving a force of 1,500 newtons and a stored energy of 37.5 joules. Support a 300-kilogram corner of the car on that spring and the oscillation period is two pi times the square root of 300 divided by 30,000, which is about 0.628 seconds, or roughly 1.6 cycles per second, a frequency chosen deliberately because it feels comfortable to passengers. Engineers tune exactly these numbers, balancing softness for comfort against stiffness for handling, and the trade-off is visible directly in the formulas this calculator uses.
The law is named for Robert Hooke, who published it in 1678 as a Latin anagram, ceasit ut tensio, sic vis, meaning as the extension, so the force, a secrecy trick he used to claim priority while he worked out the details. Hooke's contemporaries included Newton, with whom he quarrelled famously, but the spring law outlived their disputes and became a foundation of elasticity theory. The same mathematics reappears wherever a restoring force is proportional to displacement, from the vibrations of molecules to the oscillations of buildings in earthquakes, and the simple spring remains the first model every physics student meets for any system that wobbles, bounces, or rings.
A final practical note: always convert centimetres to metres before entering a displacement, and check that your spring constant is in newtons per metre rather than newtons per centimetre, a unit error that inflates forces a hundredfold and has embarrassed many a careful student.