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Hooke's Law Calculator

Force equals negative spring constant times displacement from equilibrium

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Hooke’s Law (Force)

Hooke’s Law states that the restoring force of a spring is proportional to its displacement from equilibrium. The spring constant k measures stiffness — a higher k means a stiffer spring. You can solve for force, distance, equilibrium position, or spring constant.

F = −k(x − x₀)

Spring Potential Energy

The elastic potential energy stored in a spring equals one-half times the spring constant times the displacement squared. This energy is released as kinetic energy when the spring returns to its natural length.

U = ½kx²

How It Works

Hooke’s Law states that the force needed to stretch or compress a spring is proportional to the displacement: F = −k(x − x₀). The spring constant k measures stiffness — a higher k means a stiffer spring. The stored potential energy is U = ½kx². This law applies to any elastic material within its proportional limit, including rubber bands, metal beams, and biological tissues.

Example Problem

A spring with spring constant k = 200 N/m is stretched 0.15 m from its natural length. What restoring force does the spring exert, and how much elastic energy is stored?

  1. Identify the knowns. Spring constant k = 200 N/m, displacement x = 0.15 m, equilibrium x₀ = 0 m (the natural length is the reference point).
  2. Identify what we're solving for. We want the magnitude of the restoring force Fₓ and the stored elastic potential energy U.
  3. Write Hooke's law for the force: Fₓ = −k × (x − x₀). The minus sign indicates the force points back toward equilibrium; for the magnitude we use |Fₓ| = k × |x − x₀|.
  4. Substitute the values: |Fₓ| = 200 × |0.15 − 0| = 200 × 0.15.
  5. Compute the force: |Fₓ| = 30 N pulling the mass back toward x₀.
  6. Apply the elastic energy formula U = ½ × k × x² = ½ × 200 × 0.15² = 2.25 J — this is the work the spring will do as it returns to its natural length.

When to Use Each Variable

  • Solve for Force — when you know the spring constant and displacement, e.g., calculating the restoring force of a compressed suspension spring.
  • Solve for Distance — when you know the force and spring constant, e.g., finding how far a spring stretches under a known load.
  • Solve for Spring Constant — when you know the force and displacement, e.g., determining stiffness from a load-deflection test.
  • Solve for Potential Energy — when you know the spring constant and displacement, e.g., calculating the energy stored in a drawn bowstring.

Key Concepts

Hooke's Law states that the restoring force of an elastic material is proportional to its displacement from equilibrium: F = -k(x - x₀). The spring constant k (in N/m) measures stiffness — higher k means more force is needed for the same displacement. The elastic potential energy stored in a spring is U = ½kx². Hooke's Law applies only within the elastic limit; beyond that point, permanent deformation occurs.

Applications

  • Automotive engineering: designing suspension springs for ride comfort and load-bearing capacity
  • Mechanical watches: calibrating mainsprings and balance springs for accurate timekeeping
  • Civil engineering: analyzing deflection in beams and structural members within the elastic range
  • Medical devices: designing spring-loaded syringes, retractors, and prosthetic joints

Common Mistakes

  • Applying Hooke's Law beyond the elastic limit — once a material yields, the linear relationship breaks down
  • Forgetting the sign convention — the restoring force opposes the displacement direction (negative sign)
  • Confusing displacement from equilibrium with total length — x in the formula is the change from natural length, not the stretched length
  • Using the wrong units for k — spring constant must be in force per length (e.g., N/m), not force alone

Frequently Asked Questions

What equation relates spring force, stiffness, and displacement?

Hooke's law: Fₓ = −k × (x − x₀). Force scales linearly with how far the spring is displaced from its natural length, with stiffness k as the slope and the minus sign indicating the restoring direction.

How do I solve for the spring constant k?

Rearrange Hooke's law: k = |Fₓ| / |x − x₀|. Apply a known force, measure the resulting deflection from the spring's free length, and divide. The standard load-deflection test in mechanical QA uses this exact relationship.

What is a spring constant, in plain terms?

It's the stiffness of the spring, measured in newtons per meter (N/m). A k of 500 N/m means the spring exerts 500 N of restoring force for every meter it is stretched or compressed from its natural length.

Why does Hooke's law have a negative sign?

The sign indicates direction: the restoring force always points opposite to the displacement, pulling the spring back toward equilibrium. When you stretch the spring (positive x), the force points in the negative direction, and vice versa.

When does Hooke's law stop working?

Beyond the material's elastic limit. Once a spring or any elastic material yields plastically, deformation becomes permanent and force no longer scales linearly with displacement. For steel springs the limit is the proportional yield stress; for coil springs it's often the solid-height point where coils touch.

How is elastic potential energy related to spring force?

The energy U = ½kx² equals the area under the force-vs-displacement line. It represents the work done to stretch or compress the spring, and it's released back as kinetic energy when the spring returns to its natural length — the principle behind catapults, suspension bounce, and clockwork mechanisms.

Is Hooke's law the same as the spring force equation?

Yes — Hooke's law and "the spring force equation" refer to the same relationship Fₓ = −k(x − x₀). Some textbooks drop the equilibrium term and write F = −kx when x₀ is taken as the origin.

Reference: Tipler, Paul A. 1995. Physics For Scientists and Engineers. Worth Publishers. 3rd ed.

Worked Examples

Automotive Engineering

How much force does a car suspension spring exert at maximum compression?

A coil spring in a vehicle's suspension has a spring rate k = 35,000 N/m. During a hard bump it compresses by 8 cm from its preload position.

  • Knowns: k = 35,000 N/m, x − x₀ = 0.08 m
  • F = k × |x − x₀|
  • F = 35,000 × 0.08

F = 2,800 N restoring force

Real automotive springs become progressively stiffer near full compression — Hooke's law applies only inside the linear (elastic) range.

Sports Equipment

How much elastic energy is stored in a fully drawn archery bow?

A recurve bow modelled as a linear spring has k = 600 N/m. The archer draws the string 0.65 m back from rest.

  • Knowns: k = 600 N/m, x = 0.65 m
  • U = ½ × k × x²
  • U = ½ × 600 × 0.65²

U ≈ 126.75 J of elastic potential energy

When released, most of this energy transfers into arrow kinetic energy — heavier draw weights launch arrows faster for the same draw length.

Manufacturing QA

What's the spring constant from a load-deflection test?

A QA fixture applies a measured force of 250 N to a coil spring and records 15 mm of deflection from its free length.

  • Knowns: Fₓ = 250 N, x = 0.015 m, x₀ = 0 m
  • k = |Fₓ| / |x − x₀|
  • k = 250 / 0.015

k ≈ 16,667 N/m (≈ 16.7 kN/m)

Industry spec sheets often list spring rates as N/mm or lb/in — divide N/m by 1000 for N/mm, or by 175.13 for lb/in.

Hooke's Law Formula

Hooke's law describes the restoring force of a linear spring as proportional to its displacement from equilibrium:

Fₓ = −k × (x − x₀)

Where:

  • Fₓ — restoring force along the spring's axis (newtons). The minus sign indicates the force points back toward equilibrium.
  • k — spring constant or stiffness (N/m). Larger k means a stiffer spring.
  • x — instantaneous position of the free end (meters).
  • x₀ — equilibrium (natural, unstretched) position (meters). Often taken as 0.

The companion equation gives the elastic potential energy stored in the spring when displaced by x from equilibrium:

U = ½ × k × x²

Hooke's law is exact only within a material's elastic (proportional) limit. Beyond that point the spring or material yields plastically and the linear relationship breaks down. For coil springs, that limit is typically the point where coils touch (solid height) or where the wire begins to permanently deform.

Spring Stretch Diagram

A linear spring obeys Hooke's law within its elastic limit: the restoring force is proportional to the displacement from equilibrium. The diagram below shows the spring in three states — natural length at equilibrium x₀, stretched by Δx, and the resulting restoring force F pulling the mass back.

mx₀F = −kxx

k — spring constant (N/m, slope of the F-vs-x line) · x₀ — equilibrium (natural) position · x — displacement from equilibrium (positive when stretched, negative when compressed) · F — restoring force, always pointing toward equilibrium (the negative sign in F = −kx) · U = ½kx² — elastic potential energy stored in the stretched/compressed spring.

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