Elastic Potential Energy Calculator

Using the elastic potential energy calculator

The Calculate buttons let you choose the unknown. Each mode asks for the other two values:

• Potential energy: enter the spring constant (in N/m) and the change in length. The energy is listed in joules, watt-hours, calories and foot-pounds-force; kilojoules or megajoules take over for large amounts.
• Length change: enter the spring constant and the stored energy. The distance is shown in several length units, and the set changes with the size of the answer (millimetres and centimetres for small stretches, metres and feet for large ones, kilometres and miles at the extreme).
• Spring constant: enter the change in length and the energy; the answer is in N/m.

The length can be entered in mm, cm, dm, m, in, ft or yd, and the energy in J, kJ, MJ, Wh, kWh, kCal or ft-lbf. Every value has to be greater than zero: a spring that has not moved stores no energy, and the calculator needs a real displacement to work with.

Hooke's law and the elastic energy formula

Within its elastic range a spring pulls or pushes back with a force proportional to how far it has been moved from its resting length. This is Hooke's law, with the spring constant k as the stiffness: a spring with k = 200 N/m needs 200 newtons to move it by one metre, or 2 newtons to move it by one centimetre.

F = k × xF in newtons, k in N/m, x in metres

U = ½ × k × x²elastic potential energy in joules

The energy is the work done in stretching the spring. The force grows steadily from 0 at the start to k × x at the end, so the average force is half of the final one, and work is average force times distance: ½ × k × x × x. On a force–extension graph this is the area of a triangle.

Rearranging the same formula gives the other two modes:

x = √(2 × U ÷ k)

k = 2 × U ÷ x²

Compressing a spring by a distance x stores exactly the same energy as stretching it by x, because only x² appears in the formula.

Worked examples

Energy. A spring with k = 200 N/m is stretched by 10 cm. In metres, x = 0.10, so U = ½ × 200 × 0.10² = 1 J. The same energy is about 0.000278 Wh or 0.7376 ft-lbf.

Length change. A 500 N/m spring stores 4 J. Then x = √(2 × 4 ÷ 500) = √0.016 = 0.1265 m, so the spring is moved by about 12.65 cm.

Spring constant. Pushing a plunger in by 8 cm stores 2.4 J. Convert 8 cm to 0.08 m, then k = 2 × 2.4 ÷ 0.08² = 4.8 ÷ 0.0064 = 750 N/m.

Launching a block. All 1 J from the first example is handed to a 0.5 kg block on a smooth surface. Setting ½mv² equal to ½kx² gives v = x × √(k ÷ m) = 0.10 × √(200 ÷ 0.5) = 2 m/s. The kinetic energy calculator does the same sum in the other direction.

Combining springs

To use the calculator on a system of several springs, replace them with one equivalent spring constant. Springs side by side (in parallel) share the load, so their constants add: 200 + 300 = 500 N/m. Springs end to end (in series) behave more softly: 1 ÷ k = 1 ÷ k₁ + 1 ÷ k₂, so 200 N/m and 300 N/m in series give k = 120 N/m.

Assumptions and limits

• Hooke's law holds only up to the elastic limit. A spring stretched too far deforms permanently, and its force no longer follows k × x.
• The spring is treated as ideal and massless; the energy that goes into moving the coils themselves is ignored.
• Friction, air resistance and heating are not included, so the energy released from a real spring is a little lower than the stored energy.
• The spring constant must refer to the axis along which you stretch or compress. Rubber bands, bungee cords and some materials are not linear at all, and the formula then gives only a rough guide.

Gravitational energy is a different kind of stored energy; for lifted objects use the potential energy calculator. If a spring-loaded object strikes an obstacle, the impact force calculator estimates the force involved.