Hooke's Law (Spring Force) Calculator
Hooke's law is the foundational formula for an ideal spring: the restoring force F is proportional to the displacement x — F = k · x, where k is the spring constant. Pick whether to solve for F, k or x, supply the other two values, and instantly get the third quantity together with the elastic potential energy stored in the spring U = ½kx².
Enter finite numbers. When solving for k, x must be non-zero; F must be non-negative (magnitude); k must be positive.
Spring force F
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Calculated
Spring constant k
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Calculated
Displacement x
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Calculated
Elastic potential energy U
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Calculated
Formula
F = k · |x| | U = ½ · k · x²
SI units: F in newtons (N), k in N/m, x in metres (m), U in joules (J). A negative x means compression; force and energy use the magnitude.
Formula
F = k · |x| // restoring-force magnitude (newtons, N) U = ½ · k · x² // elastic potential energy (joules, J)
- · Units: F in newtons (N), k in N/m, x in metres (m), U in joules (J). Keep your inputs consistent in SI.
- · Sign of x: by convention, x is measured from the spring's natural length — positive = stretched, negative = compressed. This tool uses |x| for the force magnitude; direction is your physical convention.
- · A larger spring constant k means a stiffer spring: a stationery clip is roughly 50–200 N/m, while a car suspension spring is typically 10,000–80,000 N/m.
- · Energy identity: since F = kx, U = ½ k x² = ½ F · |x| — i.e. stretching a spring from 0 to x takes an average force of ½ F.
- · Validity: Hooke's law holds only in the elastic region (small strains). Past the proportional limit a real spring deviates non-linearly and eventually yields plastically or fractures.
- · Simple harmonic motion: a mass m on a spring k oscillates with period T = 2π√(m/k). See the separate "Pendulum Period" tool for an analogous case.
- · Sources: Robert Hooke, "De Potentia Restitutiva" (1678); Halliday, Resnick & Walker, "Fundamentals of Physics" §7-7 / §15-2; OpenStax College Physics §5.3 Elasticity.
Frequently asked
What is the spring constant k, and what's a reasonable value?
k has SI units of newtons per metre (N/m) — it tells you how much force is needed to stretch the spring by one metre. Typical values range widely: a ballpoint-pen spring is ~30–80 N/m, a kitchen-scale spring ~100–500 N/m, a car suspension spring 10,000–80,000 N/m, and a lifting cable can be effectively ≥ 10⁶ N/m. If you have a spring but no datasheet, hang a known mass m and measure the stretch x: k = m·g / x. For example, a 1 kg mass stretching the spring by 0.05 m gives k ≈ 9.81 / 0.05 ≈ 196 N/m.
Does Hooke's law apply to every spring?
Only within the elastic limit. In the elastic region, displacement is strictly proportional to force and the spring returns fully to its natural length when unloaded. Beyond the proportional limit the F-x curve bends; deeper in the plastic region the spring deforms permanently and eventually fractures. Practical designs (scales, toys, valves) usually operate at only 50–70 % of the elastic range for linearity and longevity. Rubber bands, biological tissue, and tapered or conical springs are non-linear from the start — you cannot use F = kx directly and must use a more general stress–strain relation.
How do springs combine in series and parallel?
Parallel (both springs share the load and stretch the same distance): k_eq = k₁ + k₂ — adding springs increases stiffness, analogous to resistors in parallel using the reciprocal pattern. Series (one after another, total stretch is the sum): 1/k_eq = 1/k₁ + 1/k₂ — the combination is softer than either spring alone. So two 100 N/m springs in parallel give 200 N/m, and in series give 50 N/m. Engineers sometimes deliberately use series springs to cushion impacts, or parallel stiff springs to support heavy loads.
Why is the elastic energy ½kx² (where does the ½ come from)?
Stretching a spring from 0 to x requires a force that grows linearly from 0 to kx — the average force over the stroke is only ½kx, and the work done is W = ½kx · x = ½kx². From calculus: U = ∫₀ˣ k·s ds = ½kx². This is the elastic potential energy stored in the spring; when released it converts to kinetic energy (a mass m attached reaches v_max = x · √(k/m)) or heat if internal damping is present.
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