Hooke's Law Calculator

**Hooke's Law:** F = k × x Where: - F = restoring force (N), opposing displacement - k = spring constant (N/m) - x = displacement from equilibrium (m) Sign convention: the minus sign (F = −kx) indicates the force opposes displacement. **Elastic potential energy:** PE = ½ × k × x² This is the work done to stretch/compress, stored in the spring. **Worked example: stretching a spring** A spring with k = 200 N/m is stretched by 5 cm (0.05 m). F = 200 × 0.05 = 10 N restoring force PE = ½ × 200 × 0.0025 = 0.25 J stored To stretch from 0 to 5 cm requires 0.25 J of work. **Springs in series and parallel:** **Parallel** (springs stacked side-by-side, sharing load): k_total = k₁ + k₂ + ... Stiffer than individual. **Series** (springs end-to-end, same force): 1/k_total = 1/k₁ + 1/k₂ + ... Softer than weakest individual. **Simple harmonic motion (SHM):** Mass m on spring k oscillates with: ω = √(k/m) T = 2π / ω = 2π × √(m/k) f = 1/T = (1/2π) × √(k/m) Worked example: 1 kg mass on 100 N/m spring. ω = √(100/1) = 10 rad/s T = 2π/10 ≈ 0.628 s f = 1/T ≈ 1.59 Hz **Common spring constants:** | Spring/structure | k (N/m) | |---|---| | Pen spring | 100-500 | | Mousetrap | ~1,000 | | Bicycle brake spring | ~1,000 | | Car valve spring | 30,000-100,000 | | Car coil suspension | 20,000-50,000 | | Truck/SUV suspension | 40,000-100,000 | | Mattress (typical) | 5,000-15,000 | | Trampoline spring | 2,000-5,000 | | Door return spring | 100-1,000 | | Slinky | ~1 | | AFM cantilever | 0.01-50 | **Young's modulus form:** For a rod or wire under axial load: F = (EA/L) × x Where E = Young's modulus, A = cross-section area, L = original length. Effective spring constant: k = EA/L. | Material | E (GPa) | |---|---| | Rubber | 0.01-0.1 | | Wood | 10-20 | | Concrete | 30 | | Glass | 70 | | Aluminum | 70 | | Brass | 100 | | Copper | 110 | | Steel | 200 | | Diamond | 1,200 | **Worked example: steel bar deflection** 10 cm long steel bar, 1 cm² cross-section, load 1,000 N. k = (200e9 × 1e-4) / 0.1 = 200e6 N/m x = F/k = 1000 / 200e6 = 5 × 10⁻⁶ m = 5 μm Steel barely stretches under typical loads. **Bow energy (archery):** Drawing a bow stores PE = ½kx². Compound bows have k ~ 200-500 N/m at draw, peak force 200-400 N over 0.6 m draw → 60-120 J energy. Released arrow gains ~50-80% of that as kinetic energy (efficiency loss). **Forces beyond Hooke's law:** Materials transition through: 1. **Linear elastic** (Hooke's Law): F ∝ x. 2. **Nonlinear elastic**: still recoverable but F not linear. 3. **Yield**: permanent deformation begins. 4. **Ultimate strength**: maximum stress before necking. 5. **Fracture**: material breaks. Elastic limit varies hugely by material: rubber stretches several times its length elastically; steel only ~0.5% before yielding. **Hooke's law in different contexts:** | System | Restoring force | |---|---| | Spring | F = −kx | | Pendulum (small angle) | F = −mg sin(θ) ≈ −mgθ | | Torsion | τ = −κθ | | Atomic bond (near equilibrium) | F = −kΔr | | Electrical (LC) | V = −Q/C | All produce SHM with frequency determined by k/m equivalent ratio. **Damped oscillation:** Real springs have damping (friction/air resistance): F = −kx − bv Where b = damping coefficient. Three regimes: - Underdamped: oscillates with decreasing amplitude. - Critically damped: fastest return to equilibrium without overshoot. - Overdamped: slow return without oscillation. Critical damping condition: b = 2√(km). Used in car shock absorbers (critically/slightly under-damped for good ride). **Hookean springs in everyday life:** - Trampolines, retractable pens, mattress springs, car suspensions, watches, clocks, bows, slingshots, pogo sticks, climbing dynamic ropes. Most are designed to operate well within elastic limit for repeatability and longevity.