Hooke's Law Calculator
Calculate the restoring force and elastic potential energy of a spring using Hooke's Law: F = -kx and PE = ½kx². Enter the spring constant and displacement to find the force, stored energy, and oscillation properties. See also our Spring Constant Calculator and Potential Energy Calculator.
How to Calculate Using Hooke's Law
Hooke's Law, formulated by Robert Hooke in 1660, describes the behavior of elastic materials: the force needed to extend or compress a spring is proportional to the displacement from its natural length. This linear relationship (F = -kx) is one of the most fundamental laws in physics and engineering, applying not just to springs but to any elastic deformation within the material's elastic limit.
To calculate the spring force, multiply the spring constant (k) by the displacement (x): F = -kx. The negative sign indicates that the force is a restoring force — it always acts in the direction opposite to the displacement, pulling the spring back toward its equilibrium position. The spring constant k (measured in N/m) characterizes the stiffness of the spring: a larger k means a stiffer spring that requires more force for the same displacement.
The elastic potential energy stored in a stretched or compressed spring is PE = ½kx². This energy is available to do work when the spring is released. Unlike the force (which is linear in x), the energy depends on x² — doubling the displacement quadruples the stored energy. This stored energy is what powers mechanical watches, enables pogo sticks, and allows shock absorbers to smooth out bumps.
Hooke's Law is valid only within the elastic limit of the material. Beyond this limit, permanent deformation occurs and the relationship becomes nonlinear. For metals, the elastic limit is typically at strains of 0.1-0.5%. For rubber, it can be much larger. The elastic limit is a critical design parameter — engineers must ensure that springs and structural members operate well within their elastic range to avoid permanent damage.
Hooke's Law Formula
Hooke's Law (force):
F = -kx
Elastic Potential Energy:
PE = ½kx²
Spring constant from force:
k = F/x (magnitude)
Period of oscillation (mass-spring):
T = 2π√(m/k)
Natural frequency:
f = (1/2π)√(k/m)
Springs in series:
1/k_total = 1/k₁ + 1/k₂ + ...
Springs in parallel:
k_total = k₁ + k₂ + ...
Example Calculation
A spring with k = 100 N/m is compressed by 0.1 m. Calculate the force and stored energy:
Given: k = 100 N/m, x = 0.1 m
Force: F = -kx = -100 × 0.1 = -10 N (restoring)
Magnitude: |F| = 10 N
Potential energy: PE = ½kx² = ½ × 100 × 0.1²
PE = ½ × 100 × 0.01 = 0.5 J
If a 0.5 kg mass is attached and released:
Max velocity: v = x√(k/m) = 0.1×√(100/0.5) = 1.414 m/s
Verify: ½mv² = ½×0.5×1.414² = 0.5 J = PE ✓
Period: T = 2π√(m/k) = 2π√(0.5/100) = 0.444 s
Frequency: f = 1/T = 2.25 Hz
Hooke's Law Reference Table
| k (N/m) | x (m) | Force (N) | PE (J) |
|---|---|---|---|
| 10 | 0.01 | 0.1 N | 0.0005 J |
| 10 | 0.1 | 1.0 N | 0.05 J |
| 50 | 0.05 | 2.5 N | 0.0625 J |
| 100 | 0.1 | 10 N | 0.5 J |
| 100 | 0.2 | 20 N | 2.0 J |
| 200 | 0.05 | 10 N | 0.25 J |
| 500 | 0.1 | 50 N | 2.5 J |
| 1000 | 0.01 | 10 N | 0.05 J |
| 1000 | 0.05 | 50 N | 1.25 J |
| 5000 | 0.02 | 100 N | 1.0 J |
Frequently Asked Questions
What is Hooke's Law?
Hooke's Law states that the force exerted by a spring is proportional to its displacement from equilibrium: F = -kx. The constant k is the spring constant (stiffness), measured in N/m. The negative sign means the force is always directed back toward equilibrium (restoring force). This law applies to any elastic deformation within the material's elastic limit — not just coil springs, but also bending beams, stretching wires, and compressed gases (approximately). It was first stated by Robert Hooke in 1660 as the Latin anagram "ceiiinosssttuv" (ut tensio, sic vis — as the extension, so the force).
What is the spring constant?
The spring constant (k) measures a spring's stiffness — how much force is needed per unit of displacement. A spring with k = 100 N/m requires 100 newtons to stretch or compress it by 1 meter (or 1 N per centimeter). Stiffer springs have larger k values. The spring constant depends on the material (Young's modulus), wire diameter, coil diameter, number of coils, and whether the spring is in tension or compression. Typical values range from ~1 N/m (soft toy springs) to ~100,000 N/m (car suspension springs).
When does Hooke's Law break down?
Hooke's Law is only valid within the elastic limit — the maximum stress a material can withstand without permanent deformation. Beyond this point, the material yields (deforms plastically) and the force-displacement relationship becomes nonlinear. For steel springs, this typically occurs at strains of 0.1-0.3%. For rubber, the elastic range is much larger but the relationship is nonlinear even within it. Temperature, fatigue, and corrosion can also reduce the elastic limit over time.
How do springs in series and parallel differ?
Springs in series (end-to-end) are softer: 1/k_total = 1/k₁ + 1/k₂. Each spring extends independently, so total extension is the sum. Two identical springs in series have half the stiffness of one spring. Springs in parallel (side-by-side) are stiffer: k_total = k₁ + k₂. They share the load, so each extends less. Two identical springs in parallel have double the stiffness. Car suspensions use parallel springs for higher stiffness; series arrangements are used when greater flexibility is needed.
What is simple harmonic motion?
When a mass attached to a spring is displaced and released, it oscillates back and forth in simple harmonic motion (SHM). The restoring force (F = -kx) is proportional to displacement, producing sinusoidal oscillation: x(t) = A×cos(ωt + φ), where A is amplitude, ω = √(k/m) is angular frequency, and φ is phase. The period T = 2π√(m/k) is independent of amplitude — a remarkable property that makes springs useful for timekeeping. SHM is the foundation for understanding waves, vibrations, and oscillations throughout physics.
How is Hooke's Law used in engineering?
Hooke's Law is fundamental to structural engineering, mechanical design, and materials science. It governs the design of springs (suspension systems, mattresses, mechanical watches), the deflection of beams and structures under load, the vibration analysis of machines and buildings, and the calibration of force-measuring instruments (spring scales, load cells). Finite element analysis — the primary tool for structural simulation — is based on Hooke's Law applied to small elements. Understanding elastic behavior is essential for designing anything that must support loads without permanent deformation.
Springs in Everyday Life
Springs and elastic elements are everywhere in daily life. Mattresses use hundreds of coil springs for comfort. Vehicle suspensions use springs (coil, leaf, or torsion) to absorb road bumps. Mechanical watches store energy in mainsprings and regulate time with balance springs. Pens use small compression springs. Trampolines store and release elastic energy. Even buildings are designed with elastic principles — skyscrapers sway in the wind within their elastic range, and earthquake-resistant structures use base isolation springs.
At the atomic level, Hooke's Law describes the bonds between atoms in a crystal lattice. The "spring constant" of atomic bonds determines material properties like Young's modulus, speed of sound, and thermal expansion. This connection between microscopic atomic springs and macroscopic material behavior is one of the beautiful unifying themes in physics, linking quantum mechanics to everyday engineering.