Elastic Potential Energ Calculator
Find elastic potential energy stored in a spring from spring constant and stretch distance. See how energy rises with extension.
Elastic Potential Energy Calculator
U = ½ kΔx²
Result will appear here...
What the elastic potential energy calculator does
Stretch or squeeze a spring and it stores energy, ready to spring back. This calculator works out that stored elastic potential energy from the spring's stiffness and how far it is stretched, and it can also solve for either of those from the energy.
Below is how a spring stores energy, the equation behind it, why the stored energy climbs so steeply with stretch, and a worked example.
How to use it
- Choose what to find: the stored energy, the spring constant, or the stretch length.
- Enter the other two. The spring constant is in newtons per metre, and the stretch can be entered in millimetres, metres, inches, and more.
- Press Calculate for the answer across the energy units, or Reset to clear it.
Springs, Hooke's law, and stored energy
A spring resists being stretched or compressed, and for a wide range it does so in a simple way: the force it pushes back with is proportional to how far you have moved it from its natural length. Pull it twice as far and it pulls back twice as hard. That neat proportionality is Hooke's law, and the constant of proportionality, the spring constant, measures the spring's stiffness. A stiff spring has a large constant, a floppy one a small constant.
Because you are pushing against a force that grows as you go, the work you do builds up, and that work is stored as elastic potential energy. Release the spring and it gives every bit back, which is what makes springs so useful: in a clock, a trampoline, a car's suspension, or a drawn bow, the energy goes in on the stretch and comes out on the return.
The equation it solves
The elastic potential energy U stored in a spring depends on the spring constant k and the stretch distance Δx from the natural length:
U = ½ k Δx²
Rearranged, the same relationship gives the other two quantities the calculator can find: the stretch for a given energy, Δx = √(2U ÷ k), and the spring constant implied by an energy and a stretch, k = 2U ÷ Δx².
Why stretching twice as far stores four times the energy
The stretch appears squared in the formula, and that has a real consequence. The force needed grows steadily as you pull, so the energy you bank is the average force times the distance, and both of those rise together as you stretch further. The upshot is that doubling the stretch does not double the stored energy, it quadruples it.
So the last centimetre of a long pull stores far more than the first. This is why a bow drawn to full draw holds so much more energy than one drawn halfway, and why a spring compressed hard stores energy out of proportion to how far it has moved. The same squaring sets a limit too: stretch a real spring too far and it stops obeying Hooke's law, bending out of shape, so the formula holds while the spring springs back cleanly.
Units and precision
The spring constant is entered in newtons per metre, the SI unit of stiffness, and the stretch is converted to metres internally, so the stored energy comes out in joules and a range of other energy units. The stretch distance is measured from the spring's relaxed, natural length, whether you are stretching or compressing. Some results are shown with many digits, so the precision is in the arithmetic, and you can read the unit and rounding that suit you.
A worked example
Take a spring with a constant of 200 N/m, stretched 0.1 metres, that is 10 centimetres, from its natural length.
The stored energy is U = ½ × 200 × 0.1² = ½ × 200 × 0.01 = 1 J. Now stretch it to 0.2 metres instead, and the energy becomes ½ × 200 × 0.04 = 4 J, four times as much for twice the stretch, the squaring in action.
Questions people ask
What is the formula for elastic potential energy?
It is one-half the spring constant times the square of the stretch, U = ½kΔx². Knowing any two of energy, spring constant, and stretch fixes the third.
What is the spring constant?
It is a measure of a spring's stiffness, in newtons per metre, the force the spring pushes back with for each metre of stretch. A larger value means a stiffer spring.
What happens if I stretch the spring twice as far?
The stored energy quadruples. Because the stretch is squared, doubling it multiplies the energy by four, which is why the last part of a long pull stores so much.
Does it work for compression too?
Yes. The stretch distance is measured from the spring's natural length, so compressing it stores energy the same way stretching does, and the formula is unchanged.
References
A quick note on where the physics comes from. Hooke's law and the elastic potential energy ½kΔx² that follows from it are standard mechanics, set out in OpenStax's University Physics and in Georgia State University's HyperPhysics. The joule, the newton, and the other SI units follow the US National Institute of Standards and Technology.
- OpenStax, University Physics Volume 1, Section 8.1, Potential Energy of a System (elastic potential energy). https://openstax.org/books/university-physics-volume-1/pages/8-1-potential-energy-of-a-system
- HyperPhysics, Georgia State University, Elastic Potential Energy. http://hyperphysics.phy-astr.gsu.edu/hbase/pespr.html
- National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI). https://www.nist.gov/pml/special-publication-811
Bibek Lal Karna is a PhD student and graduate teaching assistant at the University of Mississippi, with deep interests in theoretical and gravitational physics. He is also the founder of NRCC and is strongly engaged in scientific teaching and communication. At Eon Tools, he reviews physics tools.