Hooke's law says that the force a spring exerts is proportional to how far it has been stretched or squashed from its natural length. Written as an equation it is F = −kx, and it is the single most useful approximation in the mechanics of deformable objects — not because it is exactly true, but because it is very nearly true for small deformations of almost every solid material. This Hooke's law calculator rearranges it for whichever of the three quantities you are missing and adds the two figures people usually need next: the elastic potential energy stored in the spring and the mass that would produce the same force if you hung it from the spring on Earth.
Arb Digital builds free calculators that are explicit about their assumptions, and Hooke's law has an important one: it holds only within the material's elastic region. Push past that and the relationship stops being linear, the spring may not return to its original length, and any number this page produces stops meaning anything. That boundary gets a whole section below, because it is where the physics actually lives rather than being a footnote to it.
What This Hooke's Law Calculator Does
Pick which quantity to solve for, fill in the other two, and read the answer. Each of the three inputs has its own unit selector, because these quantities are quoted in wildly inconsistent units in practice — a spring catalogue may list stiffness in pounds-force per inch, a physics problem in newtons per metre, and a mechanical drawing in newtons per millimetre. Everything is converted to SI internally, solved, and converted back.
The supporting grid does real work rather than restating the input. Elastic potential energy is calculated as ½kx², which is the area under the force-displacement line and the energy you would recover if the spring were released. The equivalent hanging mass divides the force by the standard acceleration of gravity, defined by CODATA as exactly 9.806 65 m s⁻², which converts an abstract force into something you can picture holding. Stiffness in newtons per millimetre is the engineering shorthand most people actually use, and extension in millimetres is given regardless of which unit you entered so that two runs of the tool can be compared at a glance.
Two presets show the range this law covers. A retractable pen spring sits at a few hundred newtons per metre and a few millimetres of travel. A car coil spring is three orders of magnitude stiffer and carries a substantial fraction of the vehicle's weight. The same two-symbol equation describes both.
How to Use It
- Choose what you are solving for. Force if you know the spring and how far it moved, stiffness if you measured a force and a displacement, extension if you know the spring and the load.
- Enter the two known values with their units. Mixing systems is fine — pounds-force with millimetres works, because the conversion happens before the arithmetic.
- Measure displacement from the free length. This is the single most common source of error. The x in Hooke's law is displacement from the unloaded, natural length of the spring, not from wherever it sits under a preload.
- Check the stored energy figure. Because energy goes as x², doubling the extension quadruples the energy. That is why a fully compressed spring releasing unexpectedly is far more dangerous than intuition suggests.
- Sanity-check against the elastic limit. If your extension is a large fraction of the spring's free length, the linear assumption is probably no longer safe and the number is an estimate at best.
The Formula: How Hooke's Law Is Calculated
The relationship is F = −k·x, where F is the restoring force the spring exerts in newtons, k is the spring constant in newtons per metre, and x is the displacement from the natural length in metres. The negative sign encodes direction: stretch the spring in one direction and it pulls back the other way. When you only want the magnitude of the force — which is what a calculator displays — the sign is dropped and the equation reads F = kx.
Rearranged, k = F ÷ x and x = F ÷ k. Both rearrangements guard against division by zero here, because a spring constant of zero describes something that is not a spring, and a displacement of zero gives you no information about stiffness at all.
Work an example. A spring with k = 200 N/m stretched by 150 mm: convert the displacement to 0.15 m, then F = 200 × 0.15 = 30 N. The elastic energy is ½ × 200 × 0.15² = ½ × 200 × 0.0225 = 2.25 J. Dividing the force by 9.806 65 gives 3.06 kg, so hanging just over three kilograms from this spring would produce that extension. Those are exactly the numbers the calculator shows with its default values, and you can reproduce every step on paper.
The energy formula deserves its own line of reasoning. Force is not constant during the stretch — it rises linearly from zero to kx — so the work done is not force times distance. It is the average force, kx/2, times the distance x, which gives ½kx². The HyperPhysics elasticity and periodic motion pages at Georgia State University set out the same derivation and the connection between spring stiffness and oscillation.
Where Hooke's Law Stops Being True
Every real spring has a proportional limit, an elastic limit and a yield point, and they are three different things. Below the proportional limit, force and displacement follow a straight line and Hooke's law is accurate. Between the proportional limit and the elastic limit the relationship curves, but the spring still returns to its original length when the load is removed. Past the elastic limit the material deforms permanently — release the load and the spring is now longer than it started, with a different constant.
This matters because nothing in the equation warns you. Enter an enormous extension and the calculator will faithfully return an enormous force, because F = kx has no concept of a material failing. A coil spring compressed until its coils touch, a condition engineers call solid height, has effectively infinite stiffness and Hooke's law is meaningless there. As a rough working guide, if the displacement you are entering exceeds a modest fraction of the spring's free length, treat the result as indicative rather than exact and check the manufacturer's rated travel.
The same caveat applies to materials generally. Steel obeys Hooke's law beautifully over a wide range. Rubber does not — its force-extension curve is noticeably nonlinear almost immediately, which is why rubber bands are a poor choice for a physics demonstration of this law despite being the object most often reached for.
Springs in Series and in Parallel
Combining springs behaves in a way that surprises people who have learned the electrical resistor rules, because it is exactly backwards. Springs in parallel — side by side, sharing a load, each stretching by the same amount — add their constants directly: ktotal = k₁ + k₂. Two identical springs side by side are twice as stiff as one.
Springs in series — end to end, each carrying the full load and each contributing its own extension — combine reciprocally: 1/ktotal = 1/k₁ + 1/k₂. Two identical springs end to end are half as stiff as one, because the same force produces twice the total extension. The intuition is that in series each spring feels the whole force, while in parallel each feels only a share of it.
To use this calculator on a combination, work out the effective constant first with those rules, then enter it as k. The results grid will then describe the combination as a single equivalent spring, which is exactly how it behaves from the outside.
Why Stored Energy Is the Number That Bites
Force scales linearly with displacement; energy scales with its square. Compress a spring twice as far and you have doubled the force but quadrupled the energy waiting to come out. That asymmetry explains a great deal of practical mechanical behaviour, and most of the accidents.
It also explains why spring stiffness and safety are not the same question. A stiff spring compressed a little may store less energy than a soft spring compressed a lot, even though the stiff one exerts more force at full compression. If you are trying to reason about what happens when something lets go, the energy figure in the results grid is the more relevant number, and the kinetic energy calculator converts that energy into the speed a released mass would reach. The energy converter is there if you need it in calories, foot-pounds or watt-hours instead of joules.
From Stiffness to Oscillation
Hooke's law is the reason springs oscillate rather than simply moving and stopping. A restoring force proportional to displacement produces simple harmonic motion, with angular frequency ω = √(k/m) and period T = 2π√(m/k). Every mass-on-a-spring problem in introductory mechanics follows from those two lines.
Two consequences are worth carrying away. The period does not depend on amplitude, so a spring pulled far and a spring pulled a little oscillate at the same rate — which is precisely what makes spring-regulated clocks possible. And stiffer springs oscillate faster while heavier masses oscillate slower, in a square-root relationship, so quadrupling the mass only halves the frequency. The identical mathematical form turns up in electrical circuits, where inductance plays the role of mass and the reciprocal of capacitance plays the role of stiffness; the LC resonant frequency calculator works that analogue.
Getting an Accurate Spring Constant by Measurement
If you have a spring and no data sheet, measuring k is straightforward and more reliable than any single calculation. Hang a known mass, measure the extension from free length, and divide the weight in newtons by the extension in metres. Do it with several different masses and plot the results — the slope of the straight-line region is your spring constant, and the point where the plot starts curving away tells you where the proportional limit is.
Two practical notes improve the measurement. Measure the free length with the spring hanging under its own weight in the orientation you will use it, since a long spring sags measurably. And always approach each load from the same direction, because many real springs show a small hysteresis between loading and unloading. If you are converting measured masses to weights, the weight converter handles the mass units and the force converter handles the force units, while the gravitational force calculator covers the case where local gravity itself is the question. The full free tools hub has the rest of the physics set, including the friction force calculator for the other common resisting force in these problems.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Measuring displacement from the loaded position — x is always measured from the spring's natural free length, never from wherever it currently sits.
- Using the equation past the elastic limit — beyond that point the material deforms permanently and the linear relationship no longer describes it, though the arithmetic will happily continue.
- Swapping the series and parallel rules — springs add directly in parallel and reciprocally in series, which is the opposite of how resistors combine.
- Assuming energy scales with force — force is linear in displacement but stored energy goes as the square, so doubling the stretch quadruples the energy.
- Mixing millimetres into a newtons-per-metre constant — a factor of a thousand hides easily here. Use the unit selectors rather than converting mentally.
Related Free Tools From Arb Digital
Stored spring energy converts into motion, which the kinetic energy calculator handles, and into other energy units through the energy converter. For the oscillation analogue in electronics, see the LC resonant frequency calculator. Related mechanics tools include the friction force calculator and the gravitational force calculator. Unit work belongs in the force converter or the weight converter, and the free online tools hub lists everything else.
Frequently Asked Questions
Because the spring's restoring force acts in the opposite direction to the displacement. The sign describes direction, not size, so a calculator that reports the magnitude of the force displays a positive number and states the direction separately.
It is the force needed per unit of displacement, measured in newtons per metre. A constant of 200 N/m means every additional metre of stretch requires 200 more newtons, so it is a direct measure of stiffness.
From the spring's natural free length with no load applied. Measuring from a preloaded position is the most common source of error in these calculations and will make the spring appear stiffer or softer than it is.
In parallel the constants add directly, so two identical springs side by side are twice as stiff. In series the reciprocals add, so two identical springs end to end are half as stiff. This is the opposite of the rule for electrical resistors.
Because the force is not constant through the stretch. It rises linearly from zero to kx, so the work done is the average force of kx divided by two, multiplied by the distance x, which gives one half k x squared.
Only within their elastic region, and the size of that region varies enormously. Steel follows it closely over a wide range. Rubber departs from linearity almost immediately, which makes it a poor demonstration material despite being the obvious one to reach for.
The calculator will return the arithmetic answer, because the equation has no concept of material failure. If your displacement is a large fraction of the spring's free length, treat the result as indicative and check the manufacturer's rated travel.
This tool is provided for educational and study use. It applies an idealised linear model within the elastic region and is not a substitute for engineering analysis, manufacturer specifications or any safety assessment of a real mechanism.
