Result
Enter any two values
Hooke's Law Calculator: Spring Constant and Elastic Energy
Solve Hooke's law F = -kx for force, spring constant, or displacement, plus elastic energy U = (1/2)kx^2. SI and imperial units, signed restoring force, and the sign convention named.
Model
Hooke's law F = -kx
| Quantity | Value | Unit | Status |
|---|
|F| = k|x|; F = -kx; U = (1/2) k x^2
How?
How this is calculated
The solver treats force, spring constant, displacement, and elastic energy as one coupled system. Supply any two and it solves the rest from Hooke's law and the elastic energy relation U = (1/2)kx^2. When you supply more than two, the extra values are checked against the model and a contradiction is reported rather than silently overwritten.
Internal math runs in SI units: newtons, newtons per metre, metres, and joules. Each field carries its own unit toggle (lbf, lbf/in, N/mm, cm, mm, in, mJ, kJ) so imperial and metric-prefix inputs convert before the solve. Spring constant must be greater than zero: a value of zero is not a spring and a negative value is unphysical. Displacement is signed and has no lower bound so a compression can be entered directly.
Assumptions: an ideal Hookean spring, displacement inside the elastic (proportional) regime, constant k, a massless spring, and one-dimensional displacement measured from equilibrium. Symbols follow x for displacement and U for elastic energy; the standard equivalents are the change in length and PE_el.
Formula: |F| = k|x|; F = -kx; U = (1/2) k x^2
How to calculate the spring constant (worked example)
A person of mass 80.0 kg gets into a car, adding a load of |F| = 784 N, and the car settles by 1.20 cm. The car moves down, so the displacement is a compression: x = -0.0120 m.
Spring constant: k = |F| / |x| = 784 N / 0.0120 m = 6.533e4 N/m.
Elastic energy stored: U = (1/2) |F| |x| = (1/2)(784 N)(0.0120 m) = 4.704 J.
Restoring force: F = -kx = +784 N. The load presses down while the spring pushes up, so F and x carry opposite signs. That positive sign is the piece a physics problem grades, and the piece most spring calculators discard by writing only F = kx.
A spring always pushes back toward rest. Stretch it and it pulls inward; compress it and it shoves out. Hooke's law puts a number on that push: F = -kx, where the minus sign records the direction. Enter any two of force, spring constant, displacement, or stored energy and this calculator returns the rest, including the signed restoring force and the elastic energy the spring holds. That sign is the piece most spring calculators quietly discard, and it is the piece a physics problem grades: a restoring force of +784 N tells you the spring pushes up while the load settles down.
Force is entered as a magnitude and displacement is signed, so the direction lives in x. When x is one of your inputs the tool reports both the signed restoring force and its magnitude. When x is solved from a magnitude-only pair such as force and spring constant, only the size of x is fixed, so the tool reports the magnitude and states that the direction depends on whether the spring is stretched or compressed.
Sources
- OpenStax College Physics 2e, Hooke's Law - Stress and Strain Revisited. OpenStax. Retrieved .