Hooke's law: F = -k·x

Hooke's law (Robert Hooke, 1676) is the foundation of elasticity theory: the restoring force of an elastic spring is proportional and opposite to the displacement from rest position. F = -k·x, where F is restoring force (N), k is spring constant (N/m), and x is displacement from equilibrium (m). Minus sign means force always points toward equilibrium. This calculator solves F, k, x, elastic energy U, or oscillator period T of a mass-spring system.

Fundamental formulas

• F = -k·x — restoring force (magnitude F = k·|x|)
• k = F/x — spring constant from force and displacement
• x = F/k — displacement from force and k
• U = ½·k·x² — elastic potential energy (J)
• T = 2π·√(m/k) — mass-spring oscillator period
• ω = √(k/m) — angular frequency (rad/s)
• f = (1/(2π))·√(k/m) — frequency (Hz)

The spring constant k

The spring constant (or stiffness) measures resistance to deformation. Unit: N/m. Interpretation:

• High k → stiff spring (much force for little displacement)
• Low k → soft spring
• Measured experimentally: apply F, measure x, compute k = F/x

Typical k values

Elastic vs plastic zone

Hooke's law is valid only in elastic zone. Past elastic limit:

• Material enters plastic zone: permanent deformation
• Past yield limit: material fails
• Past ultimate strength: spring breaks

Quality steel springs: elastic zone typically to 10-20% of rest length. Beyond, spring deforms permanently.

Elastic potential energy

Stretching or compressing a spring does work against the restoring force, stored as elastic potential energy:

U = ½·k·x²

Always positive (quadratic in x). Measured in joules. Doubling x quadruples U.

Energy examples

Practical numerical examples

Example 1: weight on spring

Spring k = 100 N/m, 1 kg object hung (F = 9.81 N):

• x = F/k = 9.81/100 = 9.81 cm
• Stored energy U = ½·100·0.0981² = 0.481 J

Example 2: dynamometer

Stretches 5 cm under 2 N:

• k = F/x = 2/0.05 = 40 N/m
• Classic school lab instrument

Example 3: car shock absorber

1,000 kg car, 4 springs k = 25,000 N/m each:

• Weight per spring: 2,452 N
• Compression: x = 2,452/25,000 = 9.8 cm
• Energy per spring: 120 J (480 J total)

Example 4: mass-spring oscillator

2 kg mass on k = 200 N/m spring:

• ω = √(200/2) = 10 rad/s
• T = 2π/10 = 0.628 s
• f = 1.59 Hz

Example 5: crossbow

k = 3,000 N/m, drawn 30 cm:

• F = 3,000 × 0.3 = 900 N (need strong arms!)
• U = ½·3,000·0.3² = 135 J
• 30 g arrow: v = √(2U/m) = 94.9 m/s (100% efficiency; real ~70%)

The harmonic oscillator

Mass on spring oscillates with simple harmonic motion (SHM):

x(t) = A·cos(ω·t + φ)

with A amplitude, ω = √(k/m), φ initial phase. Features:

• Period T = 2π·√(m/k): NOT amplitude-dependent (isochronism)
• Period grows with m, shrinks with k
• Energy conservation: ½·m·v² + ½·k·x² = constant

SHM is the base model for many systems: pendulum (small oscillations), LC circuits, vibrating strings, atoms in crystals, mechanical clocks.

Springs in series and parallel

Series (end-to-end)

Same force, displacements add:

1/k_tot = 1/k₁ + 1/k₂

Total softer than individuals.

Parallel (side by side)

Same displacement, forces add:

k_tot = k₁ + k₂

Total stiffer. Car: 4 parallel 25,000 N/m → 100,000 N/m total.

Beyond Hooke: real elasticity

Young's modulus (solids)

For a material cable:

k = E·A / L

E = Young's modulus (Pa), A = cross-section, L = length. Typical values:

• Rubber: 0.01-0.1 GPa
• Wood: 10-15 GPa
• Aluminum: 70 GPa
• Steel: 200 GPa
• Diamond: 1,000 GPa

Fatigue

After many load cycles, materials can fail below elastic limit — fatigue. Critical in aerospace, automotive, bridges.

Special materials

• Shape memory alloys (nitinol): return to original shape
• Rubber: 500-800% stretch, non-linear
• Elastomers: silicones, high elastic strain
• Leaf springs: composite laminates

Applications

Measurement

• Dynamometers (force)
• Spring scales (weight)
• Strain gauges (structural deformations)
• Accelerometers (spring + mass)
• Pressure transducers

Suspensions

• Cars, motorcycles, trains
• Seismic building dampers
• Bicycle forks
• Spring toys

Musical instruments

• Guitars, pianos, violins (strings)
• Tuning forks
• Drum membranes
• Wind instrument air columns

Mechanical clocks

• Mainspring (energy reservoir)
• Balance wheel (spiral spring oscillator)
• Rolex, Patek Philippe

Weapons and sports

• Bows, crossbows (elastic energy)
• Ancient catapults
• Slingshots
• Trampolines

Atomic physics

• Crystal lattice vibrations (phonons)
• Diatomic molecules (harmonic approximation)
• Classical atom model

Structural engineering

• Bridge and skyscraper design
• Settlement calculations
• Materials strength

History

Robert Hooke (1676): publishes law as anagram "ceiiinosssttuu", rearranged to "ut tensio sic vis" (as extension, so force). Hooke was Royal Society secretary, Newton rival.

Thomas Young (1807): introduces longitudinal elastic modulus, generalizing Hooke's law to extended materials.

Cauchy (1820s): formalizes elasticity theory with stress and strain tensors.

Navier, Saint-Venant, Lamé (19th c.): modern isotropic elasticity theory.

Harmonic oscillator: most studied physics system

• Exactly solvable mathematically
• Good model for small oscillations around any energy minimum
• Basis for many systems: pendulum, LC circuit, molecular vibrations
• Quantum harmonic oscillator has quantized energy levels (phonons, photons)

Damping

In practice, friction dissipates energy and amplitude decays. Equation:

m·ẍ + b·ẋ + k·x = 0

Three regimes:

• Underdamped: decreasing oscillations
• Critically damped: fastest return without oscillating
• Overdamped: slow return, no oscillation

Car shocks are critically damped: fast return without bouncing.

Resonance

If we force oscillator at external frequency ω_ext, response amplitude is maximum when ω_ext = ω_natural = √(k/m). This is resonance:

• Swing push: push in phase to maximize height
• Glass shattered by sound at its natural frequency
• Tacoma Bridge (1940): collapsed by wind resonance
• MRI: hydrogen atoms resonate at specific frequency

FAQ

Hooke's law formula?

F = -k·x, F restoring force (N), k spring constant (N/m), x displacement (m). Minus sign: force opposite to displacement.

What is k?

Spring stiffness. Unit N/m. From experiment k = F/x. Depends on material, geometry, dimensions.

When does Hooke's law fail?

Past elastic limit: material enters plastic zone or breaks. Typically > 10-20% stretch for metals. Rubber non-linear even at 5-10%.

Spring energy?

U = ½·k·x². Always positive. Doubling x quadruples U.

Mass-spring oscillator?

SHM: x(t) = A·cos(ω·t + φ), ω = √(k/m). T = 2π·√(m/k) independent of amplitude.

Combined springs?

Series: 1/k_tot = 1/k₁ + 1/k₂. Parallel: k_tot = k₁ + k₂.

Does elastic band follow Hooke?

Only approximately for small stretches. Non-linear behavior and hysteresis at larger stretches.

Common errors

• Forgetting the minus sign
• Using angle instead of linear displacement
• Applying past elastic limit
• Confusing k with m
• Wrong units (k in N/m)
• Applying to non-springs
• Forgetting gravity in vertical springs

Updated July 2026