Calculate spring force, stiffness, or displacement using Hooke's Law F = kx. Supports series and parallel spring configurations with oscillation frequency.
Hooke's Law is the foundational equation of elasticity: the force exerted by a spring is directly proportional to its displacement from equilibrium. Expressed as F = kx, where k is the spring constant (stiffness) and x is the displacement, this deceptively simple relationship governs spring behavior in everything from mechanical watches to vehicle suspensions.
This Hooke's Law Calculator lets you solve for any of the three variables — force, spring constant, or displacement — given the other two. It also handles springs in series (which produce a softer combined stiffness) and in parallel (which produce a stiffer combination). Additionally, it computes the elastic potential energy stored in the spring and the natural oscillation frequency when a mass is attached.
Whether you are designing a spring for a mechanical assembly, analyzing a suspension system, solving a physics homework problem, or studying simple harmonic motion, this tool provides instant results with clear explanations of every output.
While F = kx is simple, real-world problems involve multiple springs in series or parallel, potential energy calculations, and oscillation frequencies — all of which require additional formulas. This calculator handles everything in one place, including automatic effective-stiffness computation for combined spring configurations. Keep these notes focused on your operational context.
Hooke's Law: F = k × x Elastic Potential Energy: PE = ½ × k × x² Series Springs: 1/k_eff = 1/k₁ + 1/k₂ Parallel Springs: k_eff = k₁ + k₂ Natural Frequency (mass–spring): f = (1/2π) × √(k/m) Where: F = restoring force (N) k = spring constant (N/m) x = displacement from equilibrium (m) m = attached mass (kg)
Result: 20 N, PE = 1 J
A spring with k = 200 N/m stretched 0.1 m (10 cm) exerts a restoring force of F = 200 × 0.1 = 20 N. The elastic potential energy stored is PE = ½ × 200 × 0.01 = 1 J.
Robert Hooke first stated his law in 1678 as the Latin anagram "ceiiinosssttuu," which unscrambles to "ut tensio, sic vis" — "as the extension, so the force." This linear relationship is remarkably accurate for most springs within their elastic range. It forms the basis of spring design, vibration analysis, and many mechanical engineering calculations.
Most real-world systems use multiple springs. Vehicle suspensions combine springs (often with dampers) in various parallel and series arrangements. Mattresses use hundreds of springs in parallel. Precision instruments may use series springs for fine adjustment. The mathematical rules for combining springs are analogous to combining resistors in electrical circuits.
A mass-spring system oscillates at its natural frequency. If an external force drives the system at this frequency, resonance occurs — the amplitude grows dramatically. This principle is exploited in mechanical clocks and musical instruments, and must be avoided in structural engineering to prevent catastrophic failure (as in the famous Tacoma Narrows Bridge collapse).
Hooke's Law states that the force needed to extend or compress a spring by some distance x is proportional to that distance: F = kx. It applies in the elastic region where the material returns to its original shape after the force is removed.
The spring constant k (in N/m) measures stiffness — how much force is needed per unit of displacement. A higher k means a stiffer spring.
Hooke's Law is only valid within the elastic limit of the material. Beyond this limit, the spring deforms permanently (plastic deformation) and the force-displacement relationship becomes nonlinear.
Springs in series (end-to-end) have an effective stiffness lower than either spring alone: 1/k_eff = 1/k₁ + 1/k₂. Springs in parallel (side-by-side) add their stiffnesses: k_eff = k₁ + k₂.
When a mass is attached to a spring and displaced, it oscillates back and forth with a natural frequency f = (1/2π)√(k/m). This is simple harmonic motion — sinusoidal oscillation with constant amplitude (in the ideal, frictionless case).
In an ideal mass–spring system, energy continuously converts between elastic PE (at maximum displacement) and kinetic energy (at the equilibrium position). Total mechanical energy is conserved.