Spring Constant Calculator | Hooke's Law
Calculate spring constant, force, or displacement using Hooke's law (F = kx).
SOLVE FOR
Example springs:
What Is the Spring Constant Calculator | Hooke's Law?
The Spring Constant Calculator solves Hooke's Law (F = kx) in three modes: solve for the spring constant k given force and displacement, calculate force from k and x, or find displacement from k and F. Two additional modes handle springs in series and parallel, and a natural frequency mode computes oscillation frequency from k and mass. A spring diagram updates live to visualise extension or compression.
- ›k (N/m) measures stiffness, higher k = stiffer spring
- ›Series springs: combined k is less than any individual spring
- ›Parallel springs: combined k is the sum of all individual springs
- ›Natural frequency f = (1/2π)√(k/m) is the resonant oscillation frequency
- ›Negative x = compression; positive x = extension
Formula
Hooke's Law & Spring Formulas
Hooke's Law
F = k × x
Spring constant
k = F / x (N/m)
Displacement
x = F / k (m)
Series springs
1/k_eff = 1/k₁ + 1/k₂
Parallel springs
k_eff = k₁ + k₂
Natural frequency
f = (1/2π)√(k/m) (Hz)
How to Use
- 1Select mode: Hooke's Law, Series/Parallel Springs, or Natural Frequency
- 2For Hooke's Law: choose what to solve for (k, F, or x) then enter the other two values
- 3For Series/Parallel: enter the constants of 2 or more springs
- 4For Natural Frequency: enter spring constant k and mass m
- 5Click Calculate, results appear with step-by-step working
- 6Use Quick Presets to load real-world spring examples (car suspension, bungee cord, etc.)
Example Calculation
Car suspension spring (k = 25,000 N/m, driver mass 75 kg):
x = F/k = 735.75 / 25,000 = 0.02943 m = 2.94 cm (compression)
Natural frequency (spring + driver):
f = (1/2π) × √(25,000 / 75) = (1/2π) × √333.3 = 2.90 Hz
Two springs in parallel (k₁ = 200 N/m, k₂ = 300 N/m):
F = 50 N: x = 50/500 = 0.10 m = 10 cm
Hooke's Law and elastic limits
Hooke's Law is only valid within the elastic limit, the point at which permanent deformation begins. Beyond this, the spring does not return to its original length. Steel springs typically obey Hooke's Law up to about 65% of their yield strength. Engineering design usually limits spring deflection to 80% of the elastic limit.
Understanding Spring Constant | Hooke's Law
Spring Constant Reference Values
| Application | k (N/m) | Notes |
|---|---|---|
| Watch hairspring | 0.01–0.1 | Provides restoring torque for balance wheel |
| AFM cantilever | 0.01–100 | Atomic force microscopy probe |
| Pen retract spring | 50–150 | Ballpoint pen click mechanism |
| Door spring | 500–2,000 | Return mechanism, moderate stiffness |
| Mattress coil | 1,000–5,000 | Per coil; bed has hundreds in parallel |
| Human knee cartilage | ~1,000 | Approximate viscoelastic stiffness |
| Car suspension | 15,000–30,000 | Per corner; tuned for ride/handling |
| Valve spring (engine) | 30,000–100,000 | Must resist float at high RPM |
| Railway buffer | 500,000–2,000,000 | Absorbs collision energy |
Frequently Asked Questions
What is the spring constant (k) and what are typical values?
k is determined by spring geometry (wire diameter, coil diameter, number of coils) and material modulus of rigidity (shear modulus G). For a coil spring: k = Gd⁴ / (8D³n) where d = wire diameter, D = coil diameter, n = number of active coils.
- ›Watch hairspring: ~0.01–0.1 N/m
- ›Pen retract spring: ~50–150 N/m
- ›Mattress coil spring: ~1,000–5,000 N/m
- ›Car suspension: 15,000–30,000 N/m per spring
- ›Industrial press spring: 100,000+ N/m
What is the difference between springs in series vs. parallel?
An analogy: resistors in an electrical circuit follow the same rules, series resistors add (like springs in parallel add k), parallel resistors use the reciprocal formula (like springs in series).
- ›Series: same force through each spring, total deflection adds up
- ›Parallel: same deflection at each spring, forces add up
- ›Two equal springs in series: k_eff = k/2 (twice as soft)
- ›Two equal springs in parallel: k_eff = 2k (twice as stiff)
What is the natural frequency of a spring-mass system?
Resonance occurs when an external driving frequency matches the natural frequency. Engineers design systems to avoid resonance, the Tacoma Narrows Bridge (1940) collapsed due to wind-induced resonance.
- ›Angular frequency: ω₀ = √(k/m) rad/s
- ›Period: T = 2π/ω₀ = 2π√(m/k) seconds
- ›Car suspension: ω₀ tuned to ~1.5–2.5 Hz for comfort
- ›Seismic isolation springs: ω₀ ~0.5 Hz, below earthquake frequencies
What is Hooke's Law and when does it break down?
- ›Valid within elastic limit: spring returns to original shape when force removed
- ›Plastic deformation: permanent shape change, spring constant no longer meaningful
- ›Metals obey Hooke's Law more closely than rubber or biological materials
- ›Non-linear springs: progressive-rate springs used in sports car suspensions
- ›Viscoelastic materials (rubber, foam): spring constant depends on rate of deformation
How is spring constant measured experimentally?
The oscillation method avoids friction effects that can affect static measurements, making it more accurate for lightweight springs.
- ›Static: k = mg/x, simple but affected by friction and sag
- ›Dynamic: k = m(2π/T)² from measured oscillation period T
- ›Instron machine: professional tensile testing, measures k across full deflection range
- ›Best practice: measure at 10 different loads, take slope of F-x graph
What are real-world applications of spring constants?
- ›AFM cantilever: k = 0.01–100 N/m, measures forces at atomic scale
- ›Vehicle suspension: k chosen for ride comfort (soft) vs. cornering (stiff)
- ›MEMS accelerometers: tiny etched silicon springs measure acceleration in phones
- ›Watch hairspring: stores and releases energy precisely for timekeeping
- ›Valve springs (engines): must not float at high RPM, high k required
Is this calculator free?
Yes, completely free with no registration required. All calculations run locally in your browser.