--- title: "Elastic Strain Energy and Force-Extension Graphs" description: "Area under an applied force-extension loading graph gives work done on a sample. An ideal elastic spring stores this work as strain energy. A closed loading-unloading loop measures energy dissipated p" canonical: https://lightmysky.com/learn/science/elastic-strain-energy-and-force-extension-graphs-mt_4WnFUmR13O source: https://lightmysky.com/learn/science/elastic-strain-energy-and-force-extension-graphs-mt_4WnFUmR13O.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Elastic Strain Energy and Force-Extension Graphs Area under an applied force-extension loading graph gives work done on a sample. An ideal elastic spring stores this work as strain energy. A closed loading-unloading loop measures energy dissipated per cycle. Energy transferred from a spring can determine an object's launch speed under stated conditions. Subject: Science · Area: Matter & Materials · Ages 17 to 18 Page: https://lightmysky.com/learn/science/elastic-strain-energy-and-force-extension-graphs-mt_4WnFUmR13O ## Ready when they can - Finds work from force-extension graph area and stored energy as half F x or half k x² for an ideal Hooke's law spring - Uses loading and unloading areas for a closed cycle to calculate energy not returned - Predicts an object's launch speed when the stated spring energy becomes its kinetic energy ## Lesson: Banking energy in a stretch Here graph area differs from a particle's exposed surface area. Plot applied stretching force F vertically in newtons and extension x horizontally in metres. During slow loading, area under the curve gives work done on the sample in joules: N × m = J. An ideal elastic spring stores all this work as recoverable strain energy. If the sample dissipates energy, not all loading work is recoverable. For an ideal Hooke's law spring, F = kx. Its graph is a straight line from the origin, so stored energy E = half F x = half k x². Here F is final force and k is stiffness in N/m. F x alone gives the surrounding rectangle's area, twice the triangle's area. Beyond the proportionality limit, a sample may still recover its shape even though its graph curves. **Example.** An ideal spring's straight graph runs from the origin to 12 N at an extension of 0.5 m. Its triangle area is half × 12 × 0.5 = 3 J. Using 12 × 0.5 gives 6 J, the surrounding rectangle's area. *(drawing: These bars compare energies. They are not a force-extension graph.)* In a repeatable closed loading-unloading cycle, the loading area measures input work and the unloading area measures returned work. Their difference is the enclosed loop area: energy dissipated per cycle, usually as thermal energy in the material and surroundings. When unloading lies below loading, less work comes back. Subtract returned work from input work to find the loss. Elastic recovery means regaining the original shape after unloading. It does not guarantee full energy return: rubber can recover its shape while showing hysteresis. An ideal elastic metal wire with negligible dissipation retraces its loading graph and returns the input work. Beyond the elastic limit, permanent deformation remains. The proportionality limit and elastic limit need not coincide. Kinetic energy K = half m v². If an object starts at rest and receives all spring energy E, with no height change, losses or energy left in spring motion, E = half m v². An ideal Hooke's law spring has E = half F x, so half F x = half m v². Rearrange to v² = 2E/m and take the positive square root. Use joules, kilograms and m/s. **Example.** A spring transfers all 4 J of its stored energy to a 0.5 kg object initially at rest. There is no height change or energy loss. Then v² = 2 × 4 / 0.5 = 16, giving speed 4 m/s. Less transferred energy would give a lower speed. **Recap.** Loading area gives input work. An ideal Hooke's law spring stores half F x. A closed loop measures dissipated energy. Launch speed depends on energy reaching the object. *Lesson, questions, worked solutions and flashcards written for this dataset. Physics references informed the facts and calculation design.* ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Moments, Pressure & Hooke's Law](https://lightmysky.com/learn/science/moments-pressure-and-hookes-law-mt_vuNjYx3qOy) ## Opens up - [Simple Harmonic Motion: the Defining Relation and Its Graphs](https://lightmysky.com/learn/science/simple-harmonic-motion-the-defining-relation-and-its-graphs-mt_NTDayHkdfh) - [Stress, Strain and the Young Modulus](https://lightmysky.com/learn/science/stress-strain-and-the-young-modulus-mt_uWdqhWGT7Y)