What this calculator does
Find how much force a spring takes, how stiff it is or how far it stretches, and how much energy it stores. Hooke's law describes how springs and other stretchy objects respond to a load. Within their elastic range, the force to stretch or squeeze them grows in step with the distance moved. Pull twice as far, and it takes twice the force. As a formula, that's F = k × x. F is the force and x is the displacement, the distance from the spring's natural length. k is the spring constant, a measure of stiffness in newtons per meter.
The spring pushes back with a restoring force of the same size, in the opposite direction. That's why the law is often written F = −k × x. This calculator solves for the force, the spring constant or the displacement. It also shows the elastic potential energy stored in the spring, ½ × k × x². That's the energy the spring can give back when you let it go.
Where it's used
Why this math matters outside the classroom:
- Spring scales. Spring scales work because stretch is proportional to force, so a spring with k = 200 N/m stretches 5 cm under 10 N.
- Car suspension. Suspension engineers pick spring rates so a car's springs compress a set amount under its weight.
- Stored energy. A spring with k = 500 N/m compressed 0.1 m stores ½ × 500 × 0.1² = 2.5 J of energy.
- Lab graphs. Students hang weights on a spring, plot force against stretch and read k from the slope of the line.
Grade level and standards: Grades 9–10; Common Core State Standards for Mathematics HSA.CED.A.4. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- Choose whether to solve for the force, the spring constant or the displacement.
- Enter the values you know, with their units. Use negative values for compression (squeezing).
- Choose the output unit and read the result. Then check the stored energy and the spring's restoring force.
What the inputs mean
- Applied force (F)
- The force you apply to hold the spring at that displacement. The spring pushes back on you with an equal and opposite force.
- Spring constant (k)
- Spring catalogs often list the rate in N/mm or lbf/in (pounds of force per inch). 1 N/mm = 1,000 N/m, and 1 lbf/in ≈ 175.127 N/m.
- Displacement from rest length (x)
- Measure from the spring's relaxed (unloaded) length, not from its total length. A spring that is 10 cm long at rest and 14 cm when stretched has x = 4 cm.
Formula and method
F = k × x k = F ÷ x x = F ÷ k Restoring force = −k × x Elastic potential energy = ½ × k × x²
F is the force in newtons, k is the spring constant in newtons per meter, and x is the displacement in meters. The calculator changes your inputs to these units first, so the energy comes out in joules.
As you stretch the spring, the force grows steadily from zero up to k × x. The work you do is the area under that rising force line, which is a triangle. A triangle's area is half its base times its height, and that's where the ½ comes from.
The energy depends on x², x times itself, so it's the same for stretching and squeezing by the same amount. It's never negative. Doubling the displacement makes the stored energy four times as big.
Worked example
A spring with a constant of 200 N/m is stretched 15 cm beyond its natural length.
- Convert the displacement: 15 cm × 0.01 = 0.15 m.
- F = 200 × 0.15 = 30 N.
- Stored energy = ½ × 200 × 0.15² = 2.25 J.
Result: Holding the spring there takes 30 N, and it stores 2.25 J of elastic potential energy.
Understanding the result
A large k means a stiff spring that needs a lot of force for a small stretch, like a car suspension spring. A small k means a soft spring, like the one in a retractable pen.
You might work out the spring constant from several loads and get different answers. If so, the spring isn't behaving in a straight-line way over that range. Or it's been stretched past its elastic limit, the point where it no longer springs fully back.
Assumptions and limitations
- The spring is ideal and linear: the force is exactly in step with the displacement.
- The spring stays within its elastic limit and returns to its original length when unloaded.
- The spring's own mass is small enough to ignore.
- Real springs stop following Hooke's law near their elastic limit. Beyond it, they're permanently bent out of shape.
- Rubber bands and many materials don't follow a straight line, even at small stretches.
- Motion effects, such as bouncing (oscillation) and slowing down (damping), aren't included.
Frequently asked questions
Why is there a minus sign in some versions of Hooke's law?
F = −kx describes the spring's restoring force, which always points back toward the rest position. F = kx describes the force you apply to hold the spring stretched or squeezed. The two are equal in size and opposite in direction.
How do I find a spring constant experimentally?
Hang known weights from the spring and measure the stretch each time. Plot force against stretch. The slope (steepness) of the straight line is k. Solving here with one weight gives a single estimate.
What happens to the spring constant when springs are combined?
Two identical springs side by side (in parallel) are twice as stiff. End to end (in series), they're half as stiff. In general, the constants add in parallel. In series, the reciprocals (1 divided by each constant) add.
Where does the stored energy go when the spring is released?
It turns into kinetic energy, the energy of motion, of whatever the spring pushes. Some becomes heat and sound. An ideal spring launching a mass on a frictionless track gives the mass kinetic energy equal to ½kx².
How this calculator is checked
Method: Hooke's law for a linear elastic spring (F = kx) and the elastic potential energy ½kx² from integrating the force over the displacement. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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