Elastic Potential Energy Calculator - Solve U = ½ k x² for Spring Energy

The elastic potential energy calculator solves U = ½ k x² for the energy stored in a stretched or compressed spring. Enter the spring constant and displacement; the form returns U in joules, kilojoules, millijoules, foot-pounds, calories, or BTU, alongside the restoring force F = kx. The same formula rearranged to k = 2U / x² or x = √(2U / k) handles the inverse cases on paper.

• • Mechanics homework and exam checks: Confirm U = ½ k x² worked problems for a Hooke's-law question on a quiz or lab.
• • Spring selection for mechanical design: Estimate the energy a candidate spring stores at its working deflection when comparing options for a latch or counterbalance.
• • Physics demonstrations and lab write-ups: Translate a measured force and deflection into joules so a lab group can plot stored energy versus x².
• • Toy, slingshot, or trigger prototypes: Estimate the energy released by a small spring-driven mechanism before scaling it up.

The model assumes an ideal linear spring, so the same form works for compression and extension as long as the spring stays inside its elastic range.

For a deeper look at the same Hooke's-law relationship between force and deflection, the Spring Constant & Deflection Calculator solves for k, force, or displacement from any two of those quantities.

The page is built around the Hooke's-law definition U = ½ k x². It converts k to newtons per meter and x to meters, multiplies k by x² and halves the result, and reports that energy in the unit you select. The same formula rearranged to k = 2U / x² or x = √(2U / k) covers the inverse problems when you have a measured energy and want the missing stiffness or deflection by hand.

• U: Elastic potential energy stored in the spring (joules in SI, or whatever energy unit is selected).
• k: Spring constant, the slope of force versus deflection in the linear range (newtons per meter after unit conversion).
• x: Magnitude of the displacement from the spring's relaxed length (meters after unit conversion). The sign of x is ignored because U depends on x².
• F: Restoring force F = kx that the spring exerts to return to its relaxed length (newtons).

The energy is stored as elastic potential energy while the spring is held displaced; once released, it converts into kinetic energy, work on the attached object, or heat.

According to HyperPhysics, elastic potential energy stored in a spring that obeys Hooke's law is U = ½ k x², where k is the spring constant and x is the displacement from the relaxed length.

According to Britannica, elastic energy is the potential energy stored in a configuration of a system that can be released to do work, and for a linear spring this energy equals ½ k x².

To see how the stored elastic potential energy is converted into motion once the spring is released, the Work, Energy & Power Calculator tracks work, kinetic energy, and mechanical power over the same event.

Four ideas make the formula behave the way it does, and they explain why the same equation describes everything from a pogo stick to a watch mainspring.

These four ideas carry over to gravitational and electrostatic potential energy, which are also defined relative to a chosen reference point.

When a released spring trades its elastic potential energy for motion, the Kinetic Energy Calculator computes the kinetic energy of the moving mass from the same velocity input.

Enter spring constant and displacement; the form returns U = ½ k x² in the unit you pick and shows the restoring force F = kx in newtons. For a target U, solve for the missing k or x on paper using the rearranged forms.

1. 1 Enter the spring constant: Type k and choose its unit (N/m, kN/m, N/cm, lbf/in, or lbf/ft). Converted to newtons per meter.
2. 2 Enter the displacement: Type the deflection from the relaxed length and choose its unit (m, cm, mm, in, or ft). Negative values are accepted and treated as the magnitude, since U depends on x².
3. 3 Pick the energy output unit: Choose J, kJ, mJ, ft·lb, cal, or BTU. The energy result is converted from joules into this unit.
4. 4 Read the stored energy: The primary output shows U = ½ k x² in the unit you selected. The energy-per-centimeter row lets you compare springs at the same deflection.
5. 5 Work the inverse problems on paper: To find k that stores a target U at a given x, use k = 2U / x². To find x that produces a target U at a given k, use x = √(2U / k).

An engineer with a spring rated 4 N/mm (4000 N/m) needs the energy at 12 mm compression: k = 4000 N/m, x = 0.012 m, so U = ½ × 4000 × (0.012)² = 0.288 J. That is the work the spring does on release.

For the height-based counterpart to spring energy in a mechanics problem, the Gravitational Potential Energy Calculator returns U = mgh in joules, kilocalories, and similar energy units.

The page condenses the textbook derivation and the unit conversions into one form so the focus stays on the physics rather than the arithmetic.

• • U and force on one screen: The form shows U = ½ k x² next to F = kx and an energy-per-centimeter row, so the energy and the force at the same deflection sit together.
• • Energy unit flexibility: Switch between joules, kilojoules, millijoules, foot-pounds, calories, and BTU to match the units in your textbook or data sheet.
• • Mixed unit inputs: Take k in lbf/in and x in cm and still see U in joules; the page handles the conversions so the focus stays on the physics.
• • Direction-independent energy: The displacement is squared, so compressing or stretching the spring by the same magnitude yields the same stored energy, matching the symmetry of Hooke's law.
• • Stiffness comparison: The energy-per-centimeter output ranks springs by how much energy they store at the same small deflection, the relevant number for selection.
• • Edge-case safe: Zero spring constant, zero displacement, and unit-mix slips are caught with a clear validation message instead of returning NaN or infinity.

The same energy budget that powers a pogo stick also sets the recoil of a mechanical latch, so the page is as useful for a hobby project as for a homework set.

To see how the restoring force F = kx feeds back into Newton's second law, the Forces & Newton's Laws Calculator solves F = ma problems with the same force variable.

Three inputs and three conversions feed the result. Each is small on its own, but together they set the magnitude of U and the units you see.

• • The model assumes an ideal linear spring. Real springs show softening or hardening as they approach the elastic limit, and the page does not know the spring's material or geometry.
• • Internal damping, end-coil friction, and any preload or initial tension are ignored. The page returns the ideal energy, not the energy that dissipates inside a real spring.
• • The unit conversions use the conventional SI values listed in NIST SP 811. A national metrology institute might publish a slightly different conversion (rare for these units); the page still uses the conventional value.

For most classroom and lab uses the ideal linear model is what you want; the caveats start to matter only when a spring is pushed near its yield point.

According to OpenStax University Physics, the elastic potential energy of a spring stretched or compressed by x from equilibrium is U = ½ k x², and the same derivation only holds while Hooke's law is a good model for the spring.

For the kinetic energy a spring-driven projectile carries the instant it leaves the spring, the Bullet Energy Calculator takes the mass and muzzle velocity and returns foot-pounds and joules.