Hooke's law describes linear elasticity: within a small-deformation (linear) range, the restoring force of an elastic element is proportional to its displacement from equilibrium. It is a model, not a universal law, and it breaks down when deformations are large or materials yield.
Spring form (force–displacement)
For a spring displaced by x:
F = -k x
k is the spring constant. The negative sign indicates the force points back toward equilibrium. The corresponding potential energy is
U(x) = 1/2 k x^2
Equivalent spring constants
Common combinations:
Series: 1/k_eq = 1/k1 + 1/k2 + ...
Parallel: k_eq = k1 + k2 + ...
Material form (stress–strain)
For a linearly elastic material in 1D:
σ = E ε
where σ is stress, ε is strain, and E is Young's modulus. For a uniform rod of length L and cross-sectional area A, the effective spring constant is roughly k = EA/L.
Range of validity
Hooke's law is accurate only in the elastic, small-strain regime. Beyond that, materials can show nonlinear elasticity, viscoelasticity, hysteresis, plastic deformation, and fracture.
Link to oscillations
For a mass–spring oscillator, Hooke's law leads to simple harmonic motion with
ω = sqrt(k/m)
T = 2π sqrt(m/k)
Common pitfalls
- k is a property of the spring element (material + geometry), not just the material.
- The sign reflects direction; |F| = k|x|.