The Hooke's Law Calculator solves F = −kx for any one of spring constant, displacement and restoring force when the other two are known. Enter the pair you have; the tool returns the third, preserves the minus sign that marks a restoring force, and reminds that the law holds only up to the elastic limit.
Hooke's law says the restoring force of a spring is proportional to how far it is stretched or compressed, and opposite in direction. The constant of proportionality is k, the spring constant, measured in newtons per metre.
Calculate the restoring force of a spring
Restoring force equals minus the spring constant times displacement from the unstretched length. The magnitude is kx; the minus sign points the force back toward equilibrium. Pull a spring to the right and the force points left; compress it leftward and the force points right.
| Symbol | Quantity | SI unit |
|---|---|---|
| F | restoring force | newtons (N) |
| k | spring constant | newtons per metre (N/m) |
| x | displacement from equilibrium | metres (m) |
F = −kxDisplacement x is measured from the unloaded equilibrium position, positive in the direction you choose as positive. Force then comes out negative when x is positive, which is the restoring behaviour the law encodes.
Enter k and x to receive F. Unit selectors cover common length and force units; scientific notation in the 3.45e9 form is accepted on every numeric field.
Calculate the spring constant
Spring constant k measures stiffness: how many newtons of restoring force appear per metre of displacement. Rearrangement of Hooke's law gives k = −F/x. A larger k means a stiffer spring; small k means a soft spring that stretches farther under the same load.
k = −F / xBoth F and x must be signed consistently. If a positive displacement produces a negative measured force, −F/x is positive, as a physical spring constant must be. Entering magnitudes without signs and omitting the minus in the formula also yields a positive k; mixing the two conventions is what produces a nonsense negative stiffness.
Typical lab springs sit in the tens to hundreds of N/m. Vehicle suspension springs are much stiffer. The calculator accepts any positive k the inputs imply and warns if a negative k would be required for consistency.
Understand the minus sign
The minus sign in F = −kx means the force opposes the displacement. It is a direction marker, not a claim that the force is "less than" something. Drop the minus and you still get the correct magnitude, but you lose the automatic direction that keeps free-body diagrams consistent.
Choose a positive axis along the spring. If x is positive when the free end moves to the right, F is negative (to the left) for a stretch. If you compress the spring so x is negative, F comes out positive (to the right). In both cases the force points toward x = 0.
Exam answers that ask only for magnitude often want kx without the sign. Answers that ask for the force as a vector on a line want the signed value. The calculator returns the signed restoring force and also reports k and x so either form is one glance away.
Calculate force for k = 200 and x = 0.05
A spring constant of 200 N/m and a displacement of five centimetres is the worked fixture for this page. Substituting into F = −kx produces a restoring force of ten newtons toward equilibrium, with the sign recording direction on the chosen axis.
1. List what is known. k = 200 N/m, x = 0.05 m.
2. Apply F = −kx. F = −(200) × (0.05)
3. Solve. F = −10 N
The magnitude is 10 N toward equilibrium. If the same spring is compressed by 0.05 m (x = −0.05 m), F = −(200)×(−0.05) = +10 N, again toward equilibrium.
Solving for k from F = −10 N and x = 0.05 m:
k = −F / x = −(−10) / 0.05 = 200 N/m
Solving for x from F = −10 N and k = 200 N/m:
x = −F / k = −(−10) / 200 = 0.05 m
Any two values determine the third.
Recognise the elastic limit
Hooke's law is linear only while the spring stays within its elastic limit. Beyond that limit the material yields, the force-extension graph bends, and unloading leaves a permanent set. The formula F = −kx then ceases to describe the spring.
Every metal spring has a region of linear response near equilibrium. Inside that region, doubling the displacement doubles the force. Past the elastic limit, permanent deformation begins; past the ultimate strength, the spring may break.
The calculator always evaluates F = −kx from the numbers entered and attaches a warning that the law holds only up to the elastic limit. It cannot know the limit for a particular spring; that comes from the manufacturer or from a measured force-extension curve.
Rubber bands and some polymers are nonlinear even at modest extensions. They can still be useful springs, but a single k does not describe them across their full range.
A quick lab check: hang successive known masses, record extension, and plot F against x. A straight segment through the origin means Hooke's law applies there and k is the slope. Curvature or a permanent offset after unloading means the elastic limit has been passed or the material was never linear.
Apply Hooke's law to real springs
Spring balances, kitchen scales that use a spring, and vehicle suspensions all rely on a known k in the linear region. Series and parallel combinations change the effective constant: springs in parallel add their k values; springs in series combine like resistors in parallel.
For two springs:
Parallel: k_eff = k₁ + k₂
Series: 1/k_eff = 1/k₁ + 1/k₂Lab work usually measures extension under known weights and fits a straight line through the origin to extract k. A spring balance reads weight by measuring extension and converting through a calibrated k. A suspension spring is chosen so that typical loads keep x inside the linear range and the ride frequency falls in a comfortable band. If two identical springs support a load side by side, each takes half the force and the effective stiffness doubles, which halves the sag for the same mass.
Elastic potential energy stored in a spring is U = ½kΔx². That energy relation lives on the Potential Energy Calculator; this page owns the force law F = −kx and the determination of k. Using both pages on the same spring is fine: force from displacement here, stored energy there, with the same k and Δx.
When a problem gives the force needed to stretch a spring a stated distance, solve for k first, then keep that k for later force or energy questions on the same spring. Changing the spring (or linking springs) changes k_eff; do not reuse an old k after the hardware changes.
Frequently asked questions
What is Hooke's law?
Hooke's law states that the restoring force of a spring is proportional to displacement from equilibrium and opposite in direction: F = −kx. The constant k is the spring stiffness in N/m. The law applies inside the elastic limit of the material.
Why is there a minus sign in F = −kx?
The minus sign makes the force restoring: it always points opposite to the displacement. A positive stretch produces a negative force on the same axis. Without the sign you can still compute magnitude, but direction must be assigned by hand.
How do I find the spring constant?
Measure force and displacement in the linear region and use k = −F/x with consistent signs, or k = |F|/|x| if you work in magnitudes only. Enter any two of F, k and x into the calculator to solve for the third.
What are the units of k?
Newtons per metre (N/m). A spring with k = 200 N/m produces 200 N of restoring force magnitude per metre of displacement, or 10 N for a 0.05 m stretch.
What is the elastic limit?
The elastic limit is the maximum displacement (or stress) from which the spring returns to its original length when unloaded. Beyond it, permanent deformation remains and Hooke's law no longer holds with the original k.
Does Hooke's law work for compression as well as extension?
Yes, within the elastic limit, for springs designed to take compression. Displacement is negative under the usual sign convention, and the restoring force comes out positive, still toward equilibrium. Coil springs used in compression must be prevented from buckling sideways.
How do springs in series and parallel combine?
Parallel springs add: k_eff = k₁ + k₂. Series springs soften: 1/k_eff = 1/k₁ + 1/k₂. Two equal springs in parallel double k; two equal springs in series halve k.
Is the force always exactly proportional to x?
Only inside the linear elastic region. Real springs eventually deviate; rubber and some polymers deviate early. Treat F = −kx as an accurate model near equilibrium and verify with a force-extension plot when precision matters.
What does a negative force result mean?
On the chosen axis, a negative force points opposite the positive direction. For a positive stretch, F = −kx is negative, meaning the spring pulls back toward equilibrium. Report the magnitude alone only when the question asks for how large the force is, not which way it points.
Can k be found from a graph?
Yes. On a force-extension graph in the linear region, k is the gradient (slope) of F versus x when magnitudes are plotted, or the negative of the gradient when signed restoring force is plotted against positive x. A straight line through the origin confirms Hooke's law holds for that range.
Summary
The Hooke's Law Calculator solves F = −kx for force, spring constant or displacement from any other pair, keeping the minus sign so the force is restoring. Stiffness k is reported in N/m. The linear model holds only up to the elastic limit; beyond that point the material yields and a single k no longer describes the spring.
Stored elastic energy uses a different formula and is covered on the potential energy page.