Playground
Interactive spring: drag the mass to stretch/compress and see restoring force update. Spring coils animate.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Restoring Forceoutput Force exerted by the spring (magnitude) | N | MLT^-2 | 0 – 500 | |
| Spring Constant Stiffness of the spring | N/m | MT^-2 | 1 – 500 | |
| Displacement Displacement from the equilibrium position | m | L | 0 – 2 |
Deep dive
Derivation
Empirical law: for small deformations, the restoring force is proportional to displacement. Derived from Taylor expansion of any potential energy minimum: U(x) ≈ U₀ + ½kx².
Experimental verification
Verified with calibrated springs, atomic force microscopes, and crystal lattice vibration measurements.
Common misconceptions
- Hooke's law is not a fundamental law — it's a linear approximation
- The negative sign indicates direction (restoring), not that force is negative in magnitude
- Not all springs obey Hooke's law — only within the elastic limit
Real-world applications
- Vehicle suspension systems
- Seismograph design
- Atomic bond modeling
- Mattress and cushion engineering
Worked examples
Bungee cord spring constant
Given:
- m_person:
- 70
- x:
- 4
- g:
- 9.8
Find: k
Solution
At equilibrium: kx = mg → k = mg/x = 70 × 9.8 / 4 = 171.5 N/m
Car suspension compression
Given:
- k:
- 25000
- x:
- 0.02
Find: F
Solution
F = kx = 25000 × 0.02 = 500 N per spring
Scenarios
What if…
- scenario:
- What if you stretch beyond the elastic limit?
- answer:
- The spring deforms permanently. F is no longer proportional to x — the material enters plastic deformation.
- scenario:
- What if k → 0 (infinitely soft)?
- answer:
- The spring offers no resistance. Any displacement produces zero restoring force — like pushing through air.
- scenario:
- What if you connect two springs in series?
- answer:
- Effective k = (k₁·k₂)/(k₁+k₂). Two 100 N/m springs in series act like one 50 N/m spring — softer, not stiffer.
Limiting cases
- condition:
- x → 0
- result:
- F → 0
- explanation:
- No displacement means no restoring force.
- condition:
- k → 0
- result:
- F → 0
- explanation:
- An infinitely soft spring exerts no force.
- condition:
- x → large
- result:
- Law breaks down
- explanation:
- Beyond the elastic limit, permanent deformation occurs and linearity fails.
Context
Robert Hooke · 1676
First published as a Latin anagram 'ceiiinosssttuv' (ut tensio, sic vis — as the extension, so the force).
Hook
A bungee cord stretches 4 m under a 70 kg jumper. What's the cord's spring constant?
Find the spring constant k given that a 70 kg person stretches the cord 4 m at equilibrium, where the restoring force equals weight.
Dimensions:
- lhs:
- F → [MLT⁻²]
- rhs:
- k·x → [MT⁻²]·[L] = [MLT⁻²]
- check:
- Both sides are [MLT⁻²] = Newton. ✓
Validity: Valid within the elastic limit of the material. Breaks down for large deformations where stress-strain is nonlinear.