Elastic Potential Energy Calculator finds spring energy from k and stretch or force, solves for k or stretch, and works out preloaded springs and the energy density of solid parts.
Enter the spring constant and how far the spring is stretched or compressed.
Mass and Units
Force–Stretch Graph
The shaded triangle is the stored energy: half of the base (Δx) times the height (F). Both axes rescale to fit your values.
Step-by-Step Working
- Formula: U = ½ × k × Δx²
- Square the stretch: (0.15 m)² = 0.0225 m²
- Multiply: U = ½ × 80 N/m × 0.0225 m² = 0.9 J
Energy by Stretch
| Stretch (m) | Force (N) | Energy (J) |
|---|---|---|
| 0.0375 | 3 | 0.05625 |
| 0.075 | 6 | 0.225 |
| 0.1125 | 9 | 0.5063 |
| 0.15 | 12 | 0.9 |
| 0.225 | 18 | 2.025 |
| 0.3 | 24 | 3.6 |
| 0.45 | 36 | 8.1 |
What the Elastic Potential Energy Calculator Solves
The Elastic Potential Energy Calculator works out the energy a spring stores when you stretch it or squeeze it. It can also run the formula backward, to find the spring constant or the stretch when you already know the energy.
A menu at the top picks one of six jobs. The first finds energy from the spring constant and the stretch. The second finds energy from the force and the stretch, for when you have no spring constant. The next two solve for the spring constant or for the stretch.
The last two go beyond a single spring at rest length. One finds the energy added or released when a spring that is already stretched moves from one length to another. The other finds the energy stored per unit volume in a solid part from its stress and Young’s modulus.
Spring constants can be typed in N/m, N/cm, N/mm, kN/m, lbf/in or lbf/ft. Stretch can be in meters, centimeters, millimeters, inches or feet. Energy comes out in joules by default, and the options panel switches it to millijoules, kilojoules, foot-pounds, inch-pounds or calories.
The default spring has k = 80 N/m and is stretched 0.15 m. The Elastic Potential Energy Calculator shows 0.9 J stored, along with the peak spring force, the launch speed it could give a mass, and what the energy would be at double or half the stretch. An optional elastic limit field in the options panel warns you when the stretch goes past the point where the spring stops behaving linearly.
Elastic Potential Energy Formula
A spring that obeys Hooke’s law stores energy in proportion to its stiffness and to the square of how far it has moved from its rest length. The starting length of the spring does not enter the formula.
$$U = \frac{1}{2} k \Delta x^2$$
U is the elastic potential energy in joules, k is the spring constant in N/m and Δx is the stretch or compression in meters. The symbol U is the one physics textbooks use, but you will also see PEel, Us or Ee for the same quantity.
The formula comes straight from the work needed to deform the spring. OpenStax University Physics writes the potential energy of a perfectly elastic spring as one half k x squared, measured from the rest length.
The one half has a simple reason. The spring force starts at zero and climbs in a straight line to F = kΔx at full stretch. The average force over the move is therefore half the peak, and average force times distance gives ½kΔx². The force–stretch graph in the results shows this as the area of a triangle.
How to Calculate Elastic Potential Energy of a Spring
Take a spring with k = 200 N/m pulled 0.10 m past its rest length. First square the stretch, which gives 0.01 m². Then multiply by the spring constant and by one half.
$$U = \frac{1}{2} \times 200 \times 0.10^2 = 1.0$$
The spring stores 1.0 J, and it pulls back with 200 × 0.10 = 20 N at that point. Both figures match the spring example in the OpenStax chapter on work. The step list under the results prints the same lines with your numbers, including any unit conversions.
Compression works exactly the same way. Because Δx is squared, pushing a spring in by 0.15 m stores the same energy as pulling it out by 0.15 m. The Elastic Potential Energy Calculator accepts a negative stretch and uses its size, so you can type compression as a negative number if you prefer.
Solving for Spring Constant or Stretch
Rearranging the formula gives the other two variables. The spring constant mode uses the first form, and the stretch mode uses the second. Both need the energy in joules and the other value in SI units before dividing.
$$k = \frac{2U}{\Delta x^2}$$
$$\Delta x = \sqrt{\frac{2U}{k}}$$
A spring that stores 2 J when compressed 0.05 m has k = 2 × 2 ÷ 0.0025 = 1,600 N/m. A 500 N/m spring has to move √(2 × 10 ÷ 500) = 0.2 m to store 10 J. In both modes the result cards still list the force, the launch speed and the energy at other stretches.
Formula Without the Spring Constant
Often you can measure the force on a spring but do not know its constant. A luggage scale, a force gauge or a known weight hanging from the spring gives the force, and a ruler gives the stretch.
$$U = \frac{1}{2} F \Delta x$$
F is the force at full stretch. This is the elastic potential energy formula without the spring constant, and it is the second mode in the Elastic Potential Energy Calculator menu. A spring that reads 20 N when pulled 0.10 m stores ½ × 20 × 0.10 = 1.0 J. The first result card then reports its constant, k = F ÷ Δx = 200 N/m.
The same mode gives a quick estimate for a bow, as long as you treat the draw as linear. A bow that reaches 50 lbf over a 21 inch power stroke stores about ½ × 50 × 21 = 525 in·lbf, which is 43.75 ft·lbf or 59 J. Real draw curves are not straight lines, and the Arrow Speed Calculator handles bows from their IBO rating instead.
The link between elastic potential energy and force is Hooke’s law, F = kΔx. If the force is what you need, the Hooke’s Law Calculator solves for force, spring constant or deflection directly.
Energy Between Two Stretches
Many springs are never at their rest length in use. Valve springs, door closers and the springs in a click pen sit partly compressed before they do any work. The energy they give back is the difference between two positions, not ½kΔx² from zero.
$$\Delta U = \frac{1}{2} k (x_2^2 – x_1^2)$$
OpenStax works an example with a 4 N/cm spring. At 3 cm of stretch it holds 0.18 J, and at 6 cm it holds 0.72 J. Pulling it from 3 cm to 6 cm stores 0.54 J more, three times what the first 3 cm stored, even though the extra distance is the same.
The Energy Change mode of the Elastic Potential Energy Calculator does this in one step. Enter the spring constant, the start stretch x₁ and the end stretch x₂, both measured from the rest length. Type in 4 N/cm, 3 cm and 6 cm and the result reads 0.54 J Added, with 0.18 J stored at the start and 0.72 J at the end.
Swap the two stretches and the same 0.54 J shows as Released, which is the energy a pre-loaded spring hands back as it relaxes from 6 cm to 3 cm. The force card gives 12 N at the start, 24 N at the end and 18 N on average, and the graph shades the band between the two stretches.
Two entries are refused in this mode. A start stretch of zero is just the plain energy case, so the tool points you to the first mode. A start and end on opposite sides of the rest length are also rejected, because the force changes direction there. In that case work out each side on its own and add the two energies.
Elastic Energy Into Kinetic Energy
When a spring lets go, its stored energy turns into motion. If all of it goes into one mass, the elastic and kinetic energy formulas can be set equal.
$$\frac{1}{2} k \Delta x^2 = \frac{1}{2} m v^2$$
Solving for speed gives v = Δx √(k ÷ m). Set a launched mass in the options panel of the Elastic Potential Energy Calculator and the second result card does this. The default spring with a 0.1 kg mass gives 4.243 m/s, which is 13.92 ft/s.
The same card gives the height the mass would reach if fired straight up. Gravitational potential energy is mgh, so h = U ÷ (mg). The default 0.9 J lifts 0.1 kg by 0.918 m. The Maximum Height Calculator takes that launch speed further if you fire at an angle.
The stored energy itself does not depend on the mass at all, only on k and Δx. Mass changes how that energy is shared out as speed. Double the stretch of the default spring to 0.3 m and the energy rises to 3.6 J, so the same 0.1 kg mass leaves at 8.485 m/s, exactly twice as fast.
Real launches fall short of these figures. Friction, air drag and the moving mass of the spring itself all take a share, so treat the launch card as the upper limit for a given spring.
What is the formula for PE and KE?
Kinetic energy is KE = ½mv². Gravitational potential energy near the ground is PE = mgh, and elastic potential energy is U = ½kΔx². All three are in joules, which is why energy can move from one form to another and be tracked with a single sum.
For a mass bouncing on a spring with no friction, the total ½mv² + ½kΔx² stays fixed. At the ends of the bounce all the energy is elastic, and as the mass passes the rest point all of it is kinetic.
Elastic Potential Energy Density
Engineers working with solid parts rather than coil springs often need the energy stored per unit volume. For a material loaded within its linear range, this energy density depends on the stress and the strain.
$$u = \frac{1}{2} \sigma \epsilon = \frac{\sigma^2}{2Y}$$
σ is the stress in pascals, ε is the strain and Y is Young’s modulus. The result is in joules per cubic meter. Both forms appear in the strain energy notes on the University of Nebraska–Lincoln engineering mechanics pages.
Steel has a Young’s modulus of 2.0 × 10¹¹ Pa in the OpenStax table of elastic moduli. A steel rod loaded to 200 MPa has a strain of 0.001 and stores 100,000 J/m³, which is 0.1 J in every cubic centimeter.
When the stress reaches the proportional limit, this value is called the modulus of resilience. It is the most energy a material can absorb per unit volume and still spring back fully.
The Energy Density mode of the Elastic Potential Energy Calculator takes the stress, the Young’s modulus and the volume under stress. Stress and modulus can be typed in Pa, kPa, MPa, GPa, psi or ksi. With the steel figures above and a 1 cm³ volume, it returns 100 kJ/m³, a strain of 0.1 percent, a stretch of 1 mm per meter and 0.1 J of total energy.
Total energy is the density times the stressed volume. For a wire or rod, that volume is the cross-section area times the length. The density figure holds only while stress stays proportional to strain, so keep the stress below the material’s proportional limit.
Examples of Elastic Potential Energy
Any object that springs back after being bent, stretched, squeezed or twisted stores elastic potential energy. Coil springs are the simplest case, but the same physics runs through archery bows, trampolines, rubber bands, pole vault poles, diving boards, car suspension and the torsion springs on a garage door.
A car coil spring makes a good worked example in US units. A spring rated at 300 lbf/in and compressed 2 in pushes back with 600 lbf and stores 600 in·lbf, which is 50 ft·lbf or 67.8 J. Enter the rate in lbf/in in the Elastic Potential Energy Calculator and the first result card switches the force row to pounds-force.
We use this stored energy in two opposite ways. Clocks, toys, pens and bows store energy to release it on demand. Suspension springs, bumpers and climbing ropes store energy so that a sudden load is spread out gently, and the energy goes back out slowly or turns into heat.
Rubber bands and bungee cords are elastic but not linear. Their force does not rise in a straight line with stretch, so U = ½kΔx² gives only a rough figure for them. For those, the force mode with an averaged force is the better estimate.
Elastic Potential Energy Basics
What is elastic potential energy?
Elastic potential energy is energy stored because an elastic object has been deformed. HyperPhysics at Georgia State University describes it as energy stored as a result of deformation of an elastic object, such as the stretching of a spring.
It is the work you did to stretch or compress the object, held in its shape until it is released. As long as the object stays below its elastic limit, it gives that energy back when it returns to its rest length.
What are the units for elastic potential energy?
The SI unit is the joule, the same unit used for work and every other form of energy. One joule is one newton acting over one meter, or 1 kg·m²/s². A spring with k in N/m and Δx in meters gives U in joules directly.
US spring data often uses lbf/in and inches, which gives inch-pounds. One joule equals 0.7376 ft·lbf, 8.851 in·lbf or 0.239 calories, using the factors in NIST Special Publication 811. The fourth result card in the Elastic Potential Energy Calculator lists the energy in three of these units at once.
How is elastic energy measured?
Energy itself is not read off a meter. You measure the force and the stretch, then calculate. Hang known weights on a spring, record the stretch for each, and the slope of force against stretch is k.
The area under that force–stretch line is the stored energy. For a linear spring it is a triangle, so ½kΔx² and ½FΔx give the same answer. For a non-linear part such as a bow or a rubber band, the area has to be measured from the real curve.
Can elastic potential energy be negative?
No, not when measured from the rest length. The stretch is squared and k is positive, so U is zero at the rest length and positive everywhere else. The Elastic Potential Energy Calculator rejects a negative spring constant for that reason.
A negative sign does appear in Hooke’s law, F = −kΔx, but it only shows that the force points back toward the rest position. It says nothing about the energy, which is the same for a pull or a push of equal size.
What happens past the elastic limit?
Beyond the elastic limit a spring or a part no longer returns to its original length. Some of the work you put in goes into permanent bending or stretching, so less energy comes back than ½kΔx² predicts, and the spring constant you measured stops applying.
If the maker lists a maximum deflection or solid height, enter it in the Elastic Limit Stretch field. Leave it empty and nothing changes. Fill it in and the tool shows an amber warning whenever your stretch goes past it, so you know the energy and force figures are no longer reliable.
What are the main types of potential energy?
The forms most courses list are gravitational, elastic, electric, chemical and nuclear. Each is energy stored by position or arrangement, released when the object or the particles move to a lower-energy state.
Elastic potential energy is the one stored by changing an object’s shape. The energy, work and power calculators hub sets the spring formula beside the gravitational and kinetic ones, with the unit factors that tie them together.