Library / Foundations: Physics / FPH-10
Foundations: Physics
Spring force and stored energy
Energy stored in a stretched or compressed spring.
- Ref
- FPH-10
- Rev
- v1
- Field
- Foundations: Physics
- Result
- E [J]
x is extension from the natural length in m. Use in the proportional region, with spring constant k in N/m.
The force of a spring grows in proportion to how far it is stretched, and the stored energy grows with the square of the stretch.
Calculator
J
N/m
m
- Force kx60 N
- Work done = energy4.5 J
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
Elastic energy (J) current value 4.5 J
xExtensionKey driver
0.02200
kSpring constant
0.112113
- The energy is the area under the force–extension graph, a triangle of base x and height kx.
- Beyond the elastic limit, the force no longer follows Hooke’s law.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (J) | Local step | Change in result |
|---|---|---|---|---|---|---|
| x Extension | 0.01 to 1 m | 0.02 | 200 | 200 | +10% | +21 % |
| k Spring constant | 10 to 10,000 N/m | 0.1125 | 112.5 | 112.4 | +10% | +10 % |
Sweep of Extension
| Extension (m) | Elastic energy (J) |
|---|---|
| 0.01 | 0.02 |
| 0.01778 | 0.06325 |
| 0.03162 | 0.2 |
| 0.05623 | 0.6325 |
| 0.1 | 2 |
| 0.1778 | 6.325 |
| 0.3162 | 20 |
| 0.5623 | 63.25 |
| 1 | 200 |
Sweep one variable
Elastic energy (J)Hover, touch, or use the arrow keys on the chart
Variables and how each one contributes
E
Elastic energy
What the equation gives you. Everything else feeds it.
k
Spring constant
Stiffness of the spring. Directly proportional.
x
Extension
Stretch from the natural length. Squared: doubling the stretch quadruples the energy.
Worked example with the values above
- Force: F = kx = 400 × 0.15 = 60 N.
- Energy: ½kx² = 0.5 × 400 × 0.15².
- E = 4.5 J.
Assumptions and limits
- Within the proportional (elastic) region.
- Extension measured from the natural length.
Common mistakes
- Using total length instead of extension.
- Forgetting to square x.
Statistics and uncertainty
Extension appears squared, so a 2% error in x gives about a 4% error in energy.
Where it is used
- Springs in recoil and suspension systems.
- Energy storage in latches and launchers.
Maths you need
- Hooke’s law: F = kx
- Area under a graph: work done