EQ-HOOKE · Classical Mechanics
Hooke's Law (Linear Elastic Restoring Force)
F_spring + k*x = 0
Variables
variable
F_spring
restoring force exerted by the spring on the body
- Object
- rigid_body
- Property
- Force
- Context
- inertial_frame
- Constraint
- net_external
variable
k
spring constant (linear stiffness)
- Object
- spring
- Property
- Stiffness
- Context
- inertial_frame
variable
x
displacement from the spring's natural length (signed)
- Object
- rigid_body
- Property
- Position
- Context
- inertial_frame
- Constraint
- displacement_from_equilibrium
Axioms
algebraic classical commutative_factors conservative_force constant_coefficients deterministic linear non_relativistic
Assumptions
- Linear-elastic regime: k is independent of x (no yielding or hardening)
- No damping: pure elastic, no dissipative term
- Spring has zero mass (massless idealisation)
- One-dimensional motion along the spring axis
- The body is a point mass attached to one end of the spring
Derivation
- Hooke (1676): 'Ut tensio, sic vis' — 'as the extension, so the force'
- Linearisation of a generic potential V(x) near its minimum: V(x) ≈ V(x*) + (1/2) V''(x*) (x − x*)², yielding F = −V'(x) = −V''(x*)(x − x*)
- First term in the Taylor expansion of any restoring force around a stable equilibrium — the universal small-oscillation law
- STRUCTURAL IDENTITY with Newton II via F <-> F_spring: substituting Hooke's F into Newton II gives the simple harmonic oscillator equation m*ẍ + k*x = 0, the archetype of cross-domain composition the coupling sieve must rediscover
References
- Hooke, Lectures De Potentia Restitutiva, or of Spring Explaining the Power of Springing Bodies (1678)
- Goldstein, Classical Mechanics, 3rd ed., §6.1
- Landau & Lifshitz, Theory of Elasticity, 3rd ed., §4