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.
What a learner can do afterwards
- 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
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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.
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.
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.
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.
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.
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Input work and stored energy
A spring's required stretching force usually changes as it extends. Plot applied force vertically and extension horizontally. During slow loading, the area below the curve measures work done on the spring. Additional area under the force-extension curve means additional work.
For an ideal Hooke's law spring, F = kx. Its straight graph begins at the origin. The area is a triangle whose base is extension x and whose height is final force F. Its area is half F x. Replacing F with kx gives half k x².
The ideal model stores all input work as recoverable strain energy. A real sample may dissipate some energy during loading. Its loading area still gives input work, but it does not automatically give the amount available for return. Keep these quantities separate when interpreting experimental curves.
Two checks after unloading
After releasing a stretched sample, check its shape and its energy return separately. A sample can regain its original shape while returning less mechanical work than was put in. Rubber can show this behaviour.
For a repeatable closed cycle, the loading area gives input work and the unloading area gives returned work. Subtracting them gives the enclosed loop area. This represents energy dissipated during the cycle, usually as thermal energy in the sample and surroundings.
Permanent deformation is another observation: the original shape is not fully restored. A curved loading graph does not alone prove permanent deformation. It can mark a loss of proportionality while recovery remains elastic. This is why the proportionality limit and elastic limit must be distinguished.
Find the energy that reaches the object
Before calculating launch speed, identify how much energy reaches the launched object. Friction, sound or motion remaining in the spring can reduce the transfer. You must also account for any change in height.
For an object initially at rest with no height change, suppose all spring energy E becomes its kinetic energy. Then E = half m v². Multiply by two and divide by mass to obtain v² = 2E/m. The positive square root gives speed. Use joules and kilograms to obtain metres per second.
An ideal spring force is conservative: its stored energy depends on extension. This does not mean any rapid or uneven pulling process puts all external work into stored energy. Motion and dissipation can also occur. State the model and transfer conditions before applying the energy equation.
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