Hooke's law describes one of the simplest and most useful relationships in physics: for an ideal spring, the force needed to stretch or compress it is directly proportional to how far it moves. That single proportionality — F = kx — underlies everything from car suspensions to mechanical watches to the springs in a ballpoint pen.
The Linear Relationship Behind F = kx
Robert Hooke discovered in the 1660s that a spring's restoring force grows in direct proportion to its displacement — stretch a spring twice as far, and it pulls back with twice the force. This linear behavior holds for most springs and elastic materials as long as the deformation stays small relative to the material's overall size. The constant of proportionality, k, is a property of the specific spring: its material, wire thickness, coil diameter, and number of coils all determine how stiff or soft it is. A car's suspension spring might have k around 20,000–50,000 N/m, while a soft pen-clip spring might be closer to 100 N/m — three orders of magnitude apart, yet both obey the same F = kx formula.
Why Stored Energy Grows With the Square of Displacement
Because the force needed to stretch a spring increases as you stretch it further, the work done — and therefore the energy stored — is not simply force times distance at a single value. Instead, it's the area under the force-versus-displacement line, which for a linear spring is a triangle: PE = ½kx². This quadratic relationship has a real consequence: doubling how far you stretch a spring quadruples the stored energy, not doubles it. A slingshot pulled back twice as far launches its projectile roughly four times faster in kinetic energy terms — one reason archery and slingshot power is so sensitive to draw length.
When Hooke's Law Stops Working
Every real spring has an elastic limit — a maximum displacement beyond which the material stops behaving linearly and starts to deform permanently. Push a spring past this point and it won't spring back to its original shape; the metal has yielded. Engineers design springs with a working range well inside the elastic limit specifically so the F = kx formula stays accurate throughout normal use, and they specify a maximum safe displacement for exactly this reason. Beyond the elastic limit, predicting force and energy requires more complex models of plastic deformation that this calculator does not cover.