Spring Force Calculator (Hooke's Law)

Background.

This hooke's law calculator solves the spring-force equation F = -kx for any one unknown, given the other two, and reports the elastic potential energy U = (1/2) k x^2 stored in the spring. It is the workhorse expression of linear elasticity in physics and mechanical engineering, named after Robert Hooke's 1678 essay De Potentia Restitutiva ("On the Power of Springs"), where he summarised his finding as the anagram ceiiinosssttuv — "ut tensio, sic vis", as the extension, so the force.

The relationship is staggeringly useful because it is linear, the restoring force is simply proportional to how far the spring has been pulled or pushed from its natural length, with a constant of proportionality k called the spring constant or stiffness. That linearity is what makes the whole machinery of simple harmonic motion, vibrational normal modes, acoustic waves, and even the small-oscillation limit of every smooth potential well in classical mechanics fall out so cleanly. Drop a mass on the end of a Hookean spring, displace it slightly, let go, and it oscillates sinusoidally forever (in the idealised, frictionless case) with period T = 2*pi*sqrt(m/k). Take any well-behaved potential energy curve, expand it in a Taylor series about a stable minimum, and the leading non-trivial term is automatically (1/2) k x^2 — every smooth oscillator looks Hookean if you zoom in close enough to its equilibrium point.

This calculator handles three solve modes. Solve for force when you know how stiff the spring is (k) and how far it has moved (x). Solve for displacement when you know the spring constant and the force currently acting on (or being delivered by) the spring. Solve for spring constant when you have measured a force at a known displacement, which is exactly the procedure used in undergraduate physics labs to characterise a real coil spring before pushing it into a dynamics experiment. The output also includes elastic potential energy U = (1/2) k x^2, which is the area under the F-vs-x line from 0 to x — energy that is recoverable as kinetic energy when the spring is released.

Real springs are only Hookean over a finite range. Below a material's proportional limit the F-x curve is a straight line and Hooke's law is exact; between the proportional and elastic limits the spring still returns to its original shape when unloaded but the curve bends; beyond the elastic limit the spring is permanently deformed (plastic deformation) and Hooke's law no longer applies. This calculator silently assumes you are inside the elastic, proportional regime — if your displacement is so large that the spring has been visibly stretched out of shape, the numbers it returns are mathematical, not physical.

The applications are everywhere. Car suspensions use coil springs sized so that the static deflection under vehicle weight gives a comfortable ride frequency of about 1 Hz, which fixes k once you know the mass. Mattress springs, watch hairsprings, archery bow limbs, pole-vault poles, and trampoline springs all live or die by their spring constant. In precision instrumentation an atomic-force microscope uses a tiny cantilever whose Hookean k is calibrated to the nano-newton-per-nanometre range so that deflections of the cantilever tip, read out by a laser, translate directly into the inter-atomic force between tip and sample. Vibration analysts model entire buildings, bridges, and engine mounts as networks of lumped masses and Hookean springs to find natural frequencies and avoid resonant catastrophe.

Even chemistry leans on it: the harmonic-oscillator approximation models molecular bond stretches as little springs whose k controls the infrared vibrational frequencies that you see in IR spectra. Get the spring constant right and a vast amount of physics falls into your lap.

What is spring force calculator?

Hooke's law states that the restoring force F exerted by an ideal linear spring is proportional to its displacement x from the spring's natural (unstretched) length, and points opposite to that displacement. In the standard sign convention used in Halliday, Resnick & Walker and most introductory physics texts, F = -k*x. The constant of proportionality k is the spring constant, measured in newtons per metre (N/m), and is always positive — it encodes how stiff the spring is. A small k means a soft, easily stretched spring; a large k means a stiff one. The negative sign is critical: it says the force is a restoring force, always pulling the spring back toward equilibrium. Stretch the spring (x > 0) and the force points back toward x = 0 (F < 0). Compress it (x < 0) and the force pushes outward (F > 0). The energy stored in the deformed spring is U = (1/2) k x^2, a non-negative quantity equal to the work you had to do against the spring to displace it. Hooke's law is the defining equation of a linear elastic system and is the small-displacement limit of essentially every stable mechanical system in nature.

How to use this calculator.

1. 01Decide which quantity you want to compute: the spring force F, the displacement x, or the spring constant k. Select it from the Solve For dropdown.
2. 02Enter the spring constant k in N/m. This is always a positive number — a steel coil spring used in a physics lab is typically 10 to a few thousand N/m; a car suspension spring is roughly 20,000 to 40,000 N/m per corner.
3. 03Enter the displacement x in metres, with a sign. Positive x means the spring has been stretched past its natural length; negative x means it has been compressed. A 5 cm stretch is x = 0.05 m.
4. 04If you are solving for displacement or spring constant, enter the force F in newtons instead, also signed. A force you apply to stretch the spring is positive; the spring's restoring force pulling back on your hand is negative.
5. 05Read the calculated value alongside the elastic potential energy U = (1/2) k x^2. Confirm the sign of F is opposite to the sign of x — if it isn't, you have either swapped a sign or pushed the spring past its elastic limit and Hooke's law no longer applies.
6. 06Sanity-check the magnitude. For a typical pen-sized lab spring (k around 50 N/m) stretched 10 cm, you should expect a few newtons of force and tens of millijoules of stored energy. If your result is wildly larger or smaller, recheck the units of k (N/m, not N/cm or N/mm).

The formula.

Hooke's law in its standard form is F = -k*x, where F is the restoring force the spring exerts (in newtons), k is the spring constant (in N/m, always positive), and x is the signed displacement from the spring's natural length (in metres). Rearranged: x = -F/k for the displacement form, and k = |F/x| for the spring-constant form, with the absolute value reflecting that k itself is always positive. The negative sign in F = -k*x is not optional cosmetic notation, it carries the physics — it tells you the force always opposes the displacement, which is what makes a Hookean spring a restoring system rather than a runaway one. The elastic potential energy stored in the spring is U = (1/2) * k * x^2, which is the integral of the magnitude of the restoring force from 0 to x: U = integral from 0 to x of k*x' dx' = (1/2) k x^2. Because x is squared, U is always non-negative; a stretched and a compressed spring at equal |x| store identical energy. From Hooke's law plus Newton's second law, m * (d^2 x / dt^2) = -k*x, you get the differential equation of simple harmonic motion (SHM). Its solution is x(t) = A * cos(omega*t + phi) with angular frequency omega = sqrt(k/m), so the period is T = 2*pi*sqrt(m/k) and the frequency is f = 1/(2*pi) * sqrt(k/m). Notice that the period depends only on the mass and the spring constant, not on the amplitude — that amplitude-independence is the hallmark of a linear (Hookean) oscillator and is why pendulum clocks and watch balance wheels can keep time. For springs combined in series, the effective spring constant is 1/k_eff = 1/k_1 + 1/k_2 + ..., which is always less than the smallest k_i (series springs are softer than the softest individual spring). For springs in parallel — both attached between the same two endpoints, sharing the load — the effective constant adds: k_eff = k_1 + k_2 + ..., so parallel springs are stiffer than any one of them. These rules let you decompose complex elastic systems into networks of equivalent Hookean springs.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Hooke, R. (1678). De Potentia Restitutiva, or Of Spring. Cutlerian Lecture, London. Original statement of the law: 'ut tensio, sic vis' — as the extension, so the force.
2. [2]Halliday, D., Resnick, R., & Walker, J. (2018). Fundamentals of Physics, 11th edition, Chapter 15 (Oscillations), Section 15-1: Simple Harmonic Motion. Wiley.
3. [3]Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Volume I, Chapter 21: The Harmonic Oscillator. Addison-Wesley.
4. [4]ASTM International. ASTM E8/E8M-22, Standard Test Methods for Tension Testing of Metallic Materials. West Conshohocken, PA: ASTM International, 2022.
5. [5]Goldstein, H., Poole, C., & Safko, J. (2002). Classical Mechanics, 3rd edition, Chapter 2: Variational Principles and Lagrange's Equations. Addison-Wesley.
6. [6]Sader, J. E., Chon, J. W. M., & Mulvaney, P. (1999). Calibration of rectangular atomic force microscope cantilevers. Review of Scientific Instruments, 70(10), 3967–3969. Standard AFM spring-constant calibration reference.