What Is Elastic Potential Energy?
Elastic potential energy is the energy stored when an elastic object such as a spring, rubber band, bow, or trampoline is deformed. In physics problems the ideal spring model is most common: a spring that obeys Hooke's law stores energy proportional to the square of its displacement from equilibrium.
Elastic Potential Energy Formula
Where U is elastic potential energy in joules, k is the spring constant in N/m, and x is displacement from the equilibrium position in metres. Because x is squared, stretching or compressing by the same distance stores the same energy, and larger displacements increase energy rapidly.
Hooke’s Law and Spring Force
Hooke's law states that the restoring force of an ideal spring is directly proportional to displacement. The spring constant k tells you how stiff the spring is. A higher k means more force and more stored energy for the same displacement.
Why There Is a ½ in the Formula
When you stretch a spring, the force starts at 0 (when x = 0) and increases linearly to F = kx. The work stored is the area under the force-displacement graph, which is a triangle: ½ × base × height = ½ × x × kx = ½kx². This section is key for understanding where the formula comes from.
Spring Constant Units
The SI unit is N/m. Other common units must be converted before calculation:
| Unit | Conversion |
|---|---|
| 1 N/cm | 100 N/m |
| 1 N/mm | 1 000 N/m |
| 1 kN/m | 1 000 N/m |
| 1 lbf/in | ≈ 175.13 N/m |
How to Use the Calculator
- Choose what you want to calculate: energy, spring constant, displacement, or force.
- Enter the known values with appropriate units.
- Click Calculate.
- Review the result, formula substitution, and unit conversions.
- Copy the answer if needed.
Example Calculations
k = 200 N/m, x = 0.10 m
U = ½ × 200 × 0.10² = 1 J
k = 500 N/m, x = 5 cm
x = 0.05 m → U = ½ × 500 × 0.05² = 0.625 J
U = 2 J, x = 0.20 m → k
k = 2 × 2 / 0.20² = 100 N/m
U = 4 J, k = 800 N/m → x
x = √(2 × 4 / 800) = 0.10 m
Common Mistakes
- Using centimetres without converting to metres.
- Using total spring length instead of displacement from equilibrium.
- Forgetting to square the displacement.
- Forgetting the ½ factor.
- Using mass instead of spring constant.
- Applying the formula beyond the elastic limit.
- Assuming rubber bands always behave like ideal springs.
- Confusing elastic PE with gravitational PE.
Accuracy and Limitations
This calculator assumes an ideal linear spring obeying Hooke's law. Real springs become non-linear at large extensions and can permanently deform beyond the elastic limit. Rubber bands and biological tissues may not follow simple Hooke's law. Energy losses from friction, heat, and damping are not included. This tool is educational and should not replace engineering safety calculations for real springs or stored-energy devices.
FAQ
What does an Elastic Potential Energy Calculator do?›
It calculates the energy stored in a stretched or compressed spring using U = ½kx², and can also solve for the spring constant, displacement, or restoring force.
What is the formula for elastic potential energy?›
U = ½kx², where k is the spring constant in N/m and x is the displacement from equilibrium in metres.
What does k mean in U = ½kx²?›
k is the spring constant, measured in N/m. A larger k means a stiffer spring that requires more force per unit extension.
Why is displacement squared?›
Because the restoring force increases linearly with displacement. The work done (area under the F-vs-x graph) grows as x².
Why is there a ½ in the formula?›
The force starts at 0 and increases linearly to F = kx. The work stored equals the triangular area ½ × x × kx = ½kx².
Can displacement be negative?›
Yes. Negative displacement indicates direction (compression vs extension), but energy depends on x², so it is always non-negative.
Is elastic potential energy always positive?›
Yes, for an ideal spring. Energy depends on x², which is always ≥ 0.
What unit should spring constant use?›
The SI unit is N/m. You can enter N/cm, N/mm, kN/m, or lbf/in and the calculator converts automatically.
Can this formula be used for rubber bands?›
Only approximately for small deformations. Rubber bands and biological tissues often have non-linear force-extension behaviour.
What happens if the spring is stretched beyond its elastic limit?›
The spring deforms permanently and Hooke’s law (and this formula) no longer applies.
Sources
Author & technical reviewer
Manish Kumar
PhysicsCalcs tools are reviewed with an educational focus: clear formulas, transparent assumptions, and practical context for students and science learners.
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