What is Hooke’s Law?

Hooke’s Law states that the force needed to extend or compress a spring is directly proportional to the displacement from its natural length, expressed as F = −kx, where F is the restoring force, k is the spring constant (stiffness), and x is the displacement. This law applies within the elastic limit of the material.

What is the spring constant?

The spring constant (k) measures a spring’s stiffness in N/m. It is the force required to stretch or compress the spring by one unit of length. A higher k value means a stiffer spring. It is determined from the slope of a Force vs. Displacement graph: k = F/x.

What happens when Hooke’s Law limit is exceeded?

When the elastic limit is exceeded, the material enters plastic deformation — it will not return to its original shape when unloaded. Beyond this point, Hooke’s Law no longer applies. Continued loading leads to yielding, necking, and eventually fracture. The stress-strain curve becomes non-linear past the proportional limit.

Hooke’s Law Simulator

F = kx • Spring Constant • Elastic PE • Series & Parallel — Simulate • Explore • Practice • Quiz

Mode

📖 User Guide

Spring k 80 N/m

Mass 0.0 kg

Presets

Config

Show Elastic Limit Show Energy Area

Force

0.00 N

Extension

0.0000 m

Energy

0.000 J

k effective

80.0 N/m

User Guide — Hooke’s Law Simulator

1 Overview

This free Hooke’s law spring constant simulator lets you stretch virtual springs, plot real-time F = kx graphs, and explore elastic behaviour interactively. Adjust the spring constant k (10–200 N/m) and add mass to see the force-extension relationship plotted live on the canvas. The simulator also demonstrates series and parallel springs, the elastic limit, and elastic potential energy (E_p = ½kx²).

Built for physics and engineering students, this tool provides a complete virtual spring experiment — equivalent to a classroom Hooke’s Law investigation — with instant numerical readouts, visual energy area shading, and no equipment required.

2 Getting Started

The simulator opens in Simulate mode showing a single spring with k = 80 N/m and no load. The canvas displays the spring animation on the left and a force-extension graph on the right. Four readout cards show Force (N), Extension (m), Energy (J), and Effective k (N/m).

Use the Mode pills to switch between Simulate, Explore, Practice, and Quiz. Mass preset buttons (1, 2, 5, 10 kg) let you quickly add common loads. The Config tabs switch between Single, Series, and Parallel spring arrangements.

3 Simulate Mode

Drag the Mass slider (0–15 kg) to load the spring. As mass increases, the spring stretches and a data point appears on the F-x graph. The Force readout shows F = mg, the Extension shows x = F/k, and the Energy card displays ½kx².

Adjust the Spring k slider to change stiffness. A stiffer spring (higher k) stretches less for the same load, and the graph line has a steeper slope. Toggle the Show Elastic Limit checkbox to see a red zone on the graph marking the region beyond which permanent deformation would occur. Toggle Show Energy Area to see the shaded triangle under the F-x curve representing stored elastic potential energy.

Switch the Config to Series or Parallel to see how spring combinations change the effective stiffness: series springs are softer (1/k_eff = 1/k₁ + 1/k₂), while parallel springs are stiffer (k_eff = k₁ + k₂).

4 Explore Mode

Explore mode provides concept cards across four categories: Fundamentals (Hooke’s Law equation, proportionality, elastic limit), Springs (series, parallel, effective k), Energy (elastic PE, work done, area under graph), and Applications (spring scales, shock absorbers, spring design). Each card includes a formula, diagram, and worked example.

This mode is ideal for understanding the theory behind the simulation — study the cards, then switch back to Simulate to verify each concept visually.

5 Practice & Quiz

Practice mode generates unlimited random problems: calculate spring force given k and x, find the extension for a given load, compute elastic PE, or determine the effective k of series/parallel combinations. Step-by-step solutions are revealed after each attempt. Your running accuracy score is displayed.

Quiz mode presents 5 randomised questions per session, mixing conceptual and numerical items about Hooke’s Law, spring constant, energy storage, and spring combinations. A detailed score breakdown is shown at the end.

6 Tips & Best Practices

Use mass preset buttons for quick loading — they simulate the standard laboratory procedure of adding known masses.
Compare Single vs Series vs Parallel: Apply the same load to all three configurations to see how effective stiffness changes the extension.
Watch the energy shading grow as you add mass — it provides a visual proof that E_p = ½kx² is the area of a triangle.
Push past the elastic limit (visible as the red zone) to understand why springs must be designed to operate within their proportional range.
Vary k while keeping mass constant to see how spring stiffness affects extension — stiffer springs stretch less for the same force.
The simulator works on mobile devices in landscape mode — ideal for quick revision anywhere.

Understanding Hooke’s Law — Free Interactive Simulator

Hooke’s Law is one of the foundational principles in physics and mechanical engineering. It states that the force needed to extend or compress a spring is directly proportional to the displacement from its natural length: F = kx. The constant k is the spring constant (or stiffness), measured in N/m. This law holds within the elastic limit — beyond that, permanent deformation occurs and the relationship becomes non-linear.

Force-Extension Graphs & Energy

A force-extension graph for a spring obeying Hooke’s Law is a straight line through the origin with gradient k. The area under the graph equals the elastic potential energy stored: E_p = ½kx². This energy can be fully recovered when the spring returns to its natural length (within the elastic limit). The simulator above lets you see this energy area grow in real time as you increase the load.

Series & Parallel Spring Combinations

When springs are placed in series (end to end), the effective spring constant is lower than either individual spring: 1/k_eff = 1/k₁ + 1/k₂. In parallel (side by side), the stiffness adds up: k_eff = k₁ + k₂. Toggle between configurations in the simulator to see the difference in extension for the same load.

How to Use This Tool

In Simulate mode, drag the weight or use the sliders to apply force to the spring. Watch the force-extension graph plot in real time, and see the energy stored. Switch to Explore mode to study 12 key concepts with formulas, diagrams, and worked examples. Practice mode generates random problems, and Quiz mode tests your understanding with 5 questions per session.

Hooke’s Law & Spring Formulas

Property	Formula	Unit
Hooke’s Law	F = k × x	N
Spring Constant	k = F / x	N/m
Elastic Potential Energy	PE = ½ k x²	J
Springs in Series	1/k_eff = 1/k₁ + 1/k₂	N/m
Springs in Parallel	k_eff = k₁ + k₂	N/m
Natural Frequency (mass-spring)	f = (1/2π) √(k/m)	Hz

Typical Spring Constants by Application

Application	Approx. k (N/m)	Material
Ballpoint pen spring	100 – 300	Music wire
Screen door spring	500 – 1 500	Galvanised steel
Automotive valve spring	15 000 – 50 000	Chrome vanadium
Car suspension spring	20 000 – 60 000	Chrome silicon
Industrial die spring	50 000 – 200 000	Alloy tool steel
Railroad buffer spring	500 000+	High-carbon steel

Explore Related Simulators

If you found this Hooke’s Law simulator helpful, explore our Spring Design simulator, Stress–Strain Curve simulator, Simple Harmonic Motion simulator, and Universal Testing Machine simulator for more hands-on practice.