N/m
Enter a valid spring constant greater than zero.
m
Enter a valid displacement greater than zero.
💡 F = k × x
F in Newtons, k in N/m, x in m
Positive x = extension or compression
F is the restoring force (opposes displacement)
F in Newtons, k in N/m, x in m
Positive x = extension or compression
F is the restoring force (opposes displacement)
N
Enter a valid force greater than zero.
m
Enter a valid displacement greater than zero.
💡 k = F / x
Measure force with a known weight (F = mg) and displacement with a ruler. Average multiple measurements.
Measure force with a known weight (F = mg) and displacement with a ruler. Average multiple measurements.
N
Enter a valid force greater than zero.
N/m
Enter a valid spring constant greater than zero.
💡 x = F / k
Displacement is in the direction of applied force. Both extension and compression follow the same formula.
Displacement is in the direction of applied force. Both extension and compression follow the same formula.
N/m
Enter a valid spring constant greater than zero.
m
Enter a valid displacement greater than zero.
💡 E = ½ × k × x²
Doubling x quadruples energy. Doubling k doubles energy.
Also computes spring force F at this displacement.
Doubling x quadruples energy. Doubling k doubles energy.
Also computes spring force F at this displacement.
Spring Force
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Sources & Methodology
Hooke’s Law formulas verified against Shigley’s Mechanical Engineering Design and Engineering Toolbox spring stiffness data.
Engineering Toolbox — Hooke’s Law and Spring Stiffness
Spring constant calculation, elastic potential energy, series and parallel spring combinations, and unit conversions.
Engineering Toolbox — Typical Spring Constants
Reference values for spring constants across common applications from precision instruments to automotive suspension systems.
Hooke’s Law: F = k × x (valid within elastic limit)
Find k: k = F / x (N/m) · Find x: x = F / k (m)
Elastic potential energy: E = ½ × k × x² (J)
Oscillation period: T = 2π√(m/k) · Frequency: f = (1/2π)√(k/m)
Unit conversions: 1 N/mm = 1000 N/m · 1 lbf/in = 175.127 N/m · 1 lbf = 4.44822 N
Find k: k = F / x (N/m) · Find x: x = F / k (m)
Elastic potential energy: E = ½ × k × x² (J)
Oscillation period: T = 2π√(m/k) · Frequency: f = (1/2π)√(k/m)
Unit conversions: 1 N/mm = 1000 N/m · 1 lbf/in = 175.127 N/m · 1 lbf = 4.44822 N
⏱ Last reviewed: April 2026
How to Apply Hooke’s Law in 2026
Hooke’s Law describes the linear elastic behaviour of springs and many other materials: the restoring force is directly proportional to displacement from equilibrium. Robert Hooke published this relationship in 1678 (originally as the Latin anagram “ceiiinosssttuv” — “ut tensio sic vis” — “as the extension, so the force”). It underpins spring mechanics, structural elasticity, vibration analysis, and energy storage calculations.
The Hooke’s Law Formulas
F = k × x k = F/x x = F/k E = ½kx²
F (N) · k (N/m) · x (m) · E (J)
Example 1 — Force: k = 2000 N/m, x = 0.08 m: F = 2000 × 0.08 = 160 N
Example 2 — Spring constant: 50 N causes 5 cm extension: k = 50/0.05 = 1000 N/m
Example 3 — Energy: k = 1000 N/m, x = 0.05 m: E = 0.5 × 1000 × 0.0025 = 1.25 J
Example 4 — Period (1 kg mass): T = 2π√(1/1000) = 0.199 s (f = 5.03 Hz)
Example 1 — Force: k = 2000 N/m, x = 0.08 m: F = 2000 × 0.08 = 160 N
Example 2 — Spring constant: 50 N causes 5 cm extension: k = 50/0.05 = 1000 N/m
Example 3 — Energy: k = 1000 N/m, x = 0.05 m: E = 0.5 × 1000 × 0.0025 = 1.25 J
Example 4 — Period (1 kg mass): T = 2π√(1/1000) = 0.199 s (f = 5.03 Hz)
Typical Spring Constants
| Application | k (N/m) | k (N/mm) | Typical Deflection |
|---|---|---|---|
| Watch/clock spring | 10–50 | 0.01–0.05 | <5 mm |
| Pen click mechanism | 100–500 | 0.1–0.5 | 2–5 mm |
| Kitchen scale spring | 1,000–5,000 | 1–5 | 10–50 mm |
| Trampoline mat spring | 5,000–10,000 | 5–10 | 100–300 mm |
| Car suspension spring | 15,000–30,000 | 15–30 | 50–150 mm |
| Engine valve spring | 20,000–50,000 | 20–50 | 10–15 mm |
| Seismic isolator | 1,000,000+ | 1,000+ | 200–500 mm |
💡 Springs in series vs parallel: Two equal springs k in series: k_total = k/2 (softer). Two equal springs in parallel: k_total = 2k (stiffer). For different springs in series: 1/k_total = 1/k1 + 1/k2. In parallel: k_total = k1 + k2. Car suspension springs are effectively in parallel when both wheels on an axle compress together under a central load, giving double the stiffness of one spring alone.
Frequently Asked Questions
Hooke's Law states the spring restoring force is proportional to displacement: F = k times x. F is force in Newtons, k is the spring constant in N/m, and x is displacement in meters. Published by Robert Hooke in 1678, it applies within the elastic limit — the point beyond which the spring permanently deforms and the linear relationship breaks down.
F = k times x. Rearranged: k = F divided by x (spring constant); x = F divided by k (displacement). Elastic energy stored: E = 0.5 times k times x squared in joules. Spring-mass oscillation: T = 2 pi times sqrt(m/k) seconds. All four forms are available in this calculator's four tabs.
k = F divided by x. Apply a known force and measure the resulting displacement. For example, hanging a 5 kg mass (F = 5 times 9.81 = 49.05 N) causing 5 cm (0.05 m) extension: k = 49.05 divided by 0.05 = 981 N/m. Repeat at several different loads and average the results for the most accurate k value.
E = 0.5 times k times x squared in joules. For k = 1000 N/m and x = 0.05 m: E = 0.5 times 1000 times 0.0025 = 1.25 J. Doubling displacement quadruples stored energy (x squared relationship). When the spring returns to natural length, this energy converts entirely to kinetic energy (in the absence of friction), the basis of spring-powered devices.
The elastic limit (or yield point) is the maximum displacement at which the spring returns to its natural length when the force is removed. Beyond this, permanent plastic deformation occurs and Hooke's Law no longer applies. Engineering springs are designed to operate at 60 to 70 percent of the yield stress to ensure reliable performance over millions of load cycles.
Springs in series (end to end): 1 divided by k_total = 1/k1 plus 1/k2. Result is softer than either alone — extends more for the same force. Springs in parallel (side by side, same deflection): k_total = k1 plus k2. Result is stiffer — needs more force for the same deflection. Two identical springs in series: k/2. In parallel: 2k.
Vehicle suspension springs, shock absorbers, engine valve springs, clock and watch mechanisms, weighing scales, force gauges, seismic isolation bearings, atomic force microscope cantilevers, and any elastic element that stores and releases energy predictably. Beam bending and structural deflection calculations also follow Hooke's Law at the material level through Young's modulus E.
Young's modulus E is the material-level spring constant: stress = E times strain, or sigma = E times epsilon. For a rod of length L, cross-section A, and modulus E: spring constant k = E times A divided by L. Hooke's Law for springs (F = kx) and for elastic materials (sigma = E epsilon) describe the same linear elastic behaviour at different physical scales.
Period T = 2 pi times sqrt(m divided by k). For m = 1 kg and k = 1000 N/m: T = 2 pi times 0.0316 = 0.199 s, frequency f = 5.03 Hz. Heavier masses oscillate slower (T proportional to sqrt of m). Stiffer springs oscillate faster (T inversely proportional to sqrt of k). This relationship governs everything from watch balance wheels to building earthquake resonance.
Compression springs have open coils and resist compression — they push back when shortened. Tension springs have tightly wound coils with hooked ends and resist extension — they pull back when stretched. Both follow F = kx within their elastic range. Torsion springs resist rotation and follow the rotational equivalent: torque = k_torsion times angle, where k_torsion has units N m per radian.
1 N/mm = 1000 N/m. 1 lbf/in = 175.127 N/m. 1 kgf/mm = 9806.65 N/m. Engineering springs are commonly specified in N/mm for automotive and industrial applications. Physics uses N/m. This calculator converts all inputs to SI base units internally, then displays results in both N and additional requested units.
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