Elastic Potential Energy Calculator

Calculate elastic potential energy stored in a spring using E = ½kx².

Soft spring: 10-100, Car spring: 10,000-50,000
For natural frequency calculation
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How Springs Store Elastic Potential Energy

An ideal spring stores energy because work must be done against a restoring force that grows with displacement. Hooke's law gives F=kx, so the force is zero at the equilibrium length and rises linearly as the spring is stretched or compressed. The work required is the area under the force-versus-displacement graph, a triangle with area one-half times base times height. That is why the energy formula contains the factor 1/2.

Elastic potential energy depends on x2, so stretching or compressing by the same distance stores the same ideal energy. Doubling the displacement stores four times as much energy. The spring constant k measures stiffness: a larger k means more force and more stored energy for the same displacement.

Us = ½kx2    with    F=−kx
SymbolMeaningWhy it appears / units
UsElastic potential energyJoules (J).
kSpring constantN/m; larger k means a stiffer spring.
xDisplacement from equilibriumMeters; its square makes compression and extension store positive energy.
FRestoring forceThe minus sign in F=−kx indicates force points toward equilibrium.

The formula is valid only while the object behaves elastically and approximately follows Hooke's law. Real springs, rubber bands, bows, and biological tissues can become nonlinear, dissipate energy, or permanently deform. When a problem states an ideal spring, however, the stored energy can be converted into kinetic, gravitational, or other forms while total energy is conserved.

Worked Examples

Example 1: Car suspension: k=20000N/m, x=0.05m
E=0.5×20000×0.0025
Result: 25 J stored
Energy absorbed hitting a pothole
Example 2: Archery bow: k=500N/m, x=0.6m
E=0.5×500×0.36
Result: 90 J
Converted to arrow kinetic energy
Example 3: Compression stores positive energy too
k=800N/m, x=−0.030m → U=½(800)(0.030)2
Result: U=0.36 J
The displacement sign disappears when x is squared, so equal compression and extension store the same ideal energy.
Example 4: Spring energy converted to speed
k=300N/m, x=0.12m, m=0.50kg → U=2.16J and ½mv2=U
Result: v≈2.94 m/s
If all stored elastic energy becomes kinetic energy and losses are neglected, energy conservation gives the maximum speed.

Common Mistakes

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Using U=kx instead of U=½kx2

kx is the magnitude of spring force. Energy is the work accumulated while the force increases from zero to kx.

⚠️
Leaving displacement in centimeters

With k in N/m, x must be in meters. A centimeter-to-meter mistake is squared in the energy calculation, making the error especially large.

⚠️
Assuming Hooke's law remains valid for any stretch

Real elastic objects have a limited linear range. Beyond it, k may not stay constant and permanent deformation or damage can occur.

Frequently Asked Questions

Hooke's law validity?
F=kx is valid only in the elastic region (below elastic limit). Beyond this, permanent deformation occurs. The spring constant k is the slope of the linear F-x graph.
Spring in series vs parallel?
Series: 1/k_total = 1/k₁+1/k₂ (softer). Parallel: k_total = k₁+k₂ (stiffer). Weight on two parallel springs shares load; series springs each extend fully.
Why is elastic potential energy always positive in the ideal formula?
The reference energy is usually chosen as zero at equilibrium, x=0. Because U=½kx2 and k is positive, either positive extension or negative compression gives positive stored energy. The sign of x instead matters in the restoring force F=−kx, which points toward equilibrium.
Why is there a one-half in the spring energy formula?
The spring force is not constant while the spring is stretched. It rises linearly from zero to kx. On a force-versus-displacement graph, the work done is the triangular area under F=kx: one-half times the displacement x times the final force kx, giving ½kx2.
What happens to stored energy if displacement doubles?
Because U is proportional to x2, doubling displacement multiplies the energy by four. Tripling displacement multiplies it by nine. This quadratic growth is why modest increases in spring compression or stretch can produce much larger changes in stored energy and launch speed.
Can all spring energy become kinetic energy?
In an ideal frictionless system, yes: conservation of mechanical energy can convert elastic potential energy completely into kinetic energy at equilibrium. Real systems lose some energy to heat, sound, internal damping, air resistance, or deformation, so measured kinetic energy is often lower than the ideal prediction.

Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

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