Hooke's Law — Calculate Spring Force & Constant
Hooke's Law states that the force a spring exerts is directly proportional to its deformation: F = k · x, where F is force in Newtons (N), k is the spring constant in N/m, and x is the displacement in meters (m). This linear relationship holds within the elastic limit of the material — once that limit is exceeded, the spring undergoes permanent deformation and Hooke's Law no longer applies. Applications include mechanical engineering, vehicle suspension design, spring balances, vibration analysis, and biomechanics.
When to use this calculator
- Calculate the force exerted by a hook balance spring (k = 500 N/m) stretched 8 cm: F = 500 × 0.08 = 40 N.
- Determine the spring constant of a car suspension by measuring how much it compresses under a known weight (k = F/x).
- Verify whether a spring's deformation stays within the elastic limit before permanent deformation occurs.
- Design a vibration isolation system by selecting k for a desired natural frequency: f = (1/2π)·√(k/m).
Example: force of a laboratory spring
- Given: spring with k = 200 N/m, stretched x = 0.05 m (5 cm).
- Apply F = k × x = 200 × 0.05 = 10 N.
- Elastic potential energy stored: E = ½ × 200 × (0.05)² = 0.25 J.
How it works
2 min readHow It Works
Hooke's Law relates three quantities — force (F), spring constant (k), and displacement (x) — in a simple linear equation. Depending on which quantity you need:
F = k × x → Spring force [N]
k = F / x → Spring constant [N/m]
x = F / k → Deformation / displacement [m]Variables:
> ⚠️ Hooke's Law is valid only within the elastic region of the material. Beyond the yield point, the spring deforms permanently and the law no longer applies.
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Elastic Potential Energy
The energy stored in a compressed or stretched spring:
U = ½ · k · x²Example: k = 400 N/m, x = 0.10 m → U = ½ × 400 × 0.01 = 2 J.
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Typical Spring Constant Values
| Spring Type | Typical k | Application |
|---|---|---|
| Ballpoint pen spring | 1–5 N/m | Everyday use |
| Hook balance spring | 100–800 N/m | Lab / retail scale |
| Motorcycle suspension | 10,000–25,000 N/m | Motorcycles |
| Car suspension spring | 15,000–40,000 N/m | Passenger vehicles |
| Industrial press spring | 50,000–500,000 N/m | Heavy machinery |
| Achilles tendon (biomechanics) | ~3,000 N/m | Medicine / biomechanics |
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Springs in Series and Parallel
Example: k₁ = 200 N/m, k₂ = 300 N/m.
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Common Mistakes
1. Confusing deformation with total length. x = Δl is the change in length from equilibrium, not the final spring length.
2. Inconsistent units. If k is in N/m, x must be in meters. k = 500 N/m, x = 5 cm: F = 500 × 0.05 = 25 N (not 2,500 N).
3. Exceeding the elastic limit. For spring steel, maximum elastic strain is about 0.1–0.2 %. Beyond that, the spring doesn't return to its original shape.
4. Ignoring the vector sign. The restoring force is F = −k·x; the negative sign means it opposes displacement. This calculator works with magnitudes.
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Frequently asked questions
What is Hooke's Law and where does it apply?
Hooke's Law states that the force exerted by a spring is directly proportional to its deformation: F = k·x. It was formulated by Robert Hooke in 1676 and applies to any elastic material within its linear (elastic) region — metal springs, rubber bands, tendons, structural beams under low loads, and even atomic bonds.
What does the spring constant k mean physically?
The spring constant k (N/m) measures stiffness: how many newtons of force are needed to stretch or compress the spring by one meter. A higher k means a stiffer spring. A ballpoint pen spring has k ≈ 1–5 N/m; a car suspension spring has k ≈ 15,000–40,000 N/m.
How do I measure the spring constant experimentally?
Apply known forces (e.g., known weights: 100 g ≈ 0.981 N) to the spring and measure the resulting displacement with a ruler. Calculate k = F/x for each load and average the values. Plot F vs. x — a straight line confirms Hooke's Law is valid; the slope equals k.
What is the elastic limit and how do I know if I've exceeded it?
The elastic limit (or yield point) is the maximum stress a spring can handle before it deforms permanently. Beyond this point, Hooke's Law breaks down. Signs: the spring doesn't return to its original length after unloading. For spring steel (ASTM A228), the yield strength is about 1,600–1,900 MPa. In practice, visible deformation exceeding 30–40% of the free length is a warning sign.
How do springs in series and parallel differ?
Parallel springs (supporting load together): k_total = k₁ + k₂ — the system is stiffer. Series springs (force passes through each in sequence): 1/k_total = 1/k₁ + 1/k₂ — the system is more flexible. Example: k₁ = 200 N/m, k₂ = 300 N/m → parallel gives 500 N/m; series gives 120 N/m.
What is the connection between Hooke's Law and Simple Harmonic Motion?
The restoring force F = −k·x is exactly the condition that produces Simple Harmonic Motion (SHM). A mass m attached to a spring k oscillates with period T = 2π·√(m/k) and frequency f = (1/2π)·√(k/m). Note: the frequency depends only on m and k, not on the amplitude of oscillation.
How do I calculate the elastic potential energy stored in a spring?
Elastic potential energy is U = ½ · k · x². Example: k = 400 N/m, x = 0.10 m → U = ½ × 400 × 0.01 = 2 J. This energy is released when the spring returns to equilibrium, converting to kinetic energy. In a frictionless system, total mechanical energy (U + KE) is conserved.
Can this calculator be used for torsional springs?
Torsional springs follow an analogous law: M = κ · θ, where M is the torque (N·m), κ is the torsional stiffness (N·m/rad), and θ is the angle of twist (rad). The math is identical to Hooke's Law for linear springs. You can use this calculator by substituting k → κ and x → θ, as long as you use consistent units.
Is Hooke's Law used in biomedical applications?
Yes. Biological tissues — tendons, ligaments, cartilage, and skin — exhibit spring-like behavior within their elastic range. The Achilles tendon, for example, has an effective spring constant of roughly 3,000 N/m. In orthopedics, Hooke's Law underpins prosthetic limb design and bone fracture analysis. Note that biological tissues often show nonlinear behavior, so Hooke's Law is an approximation valid at low strains.