Hooke’s Law Calculator (F = k × x)
Solve the spring equation for force, spring constant, or displacement.
Elastic potential energy: ½kx² = 1 J.
Hooke’s law is F = k × x: the force a spring exerts equals its spring constant times how far it is stretched or compressed. A spring with k = 200 N/m pulled 0.1 m needs F = 200 × 0.1 = 20 N. Rearrange to k = F ÷ x or x = F ÷ k for the other two.
What Hooke’s law says
Within its elastic range, a spring resists being deformed by an amount proportional to the deformation: F = k × x. Here x is the displacement from the spring’s natural (unstretched) length and k is the spring constant — a measure of stiffness. Double the stretch and you double the force; the relationship is perfectly linear until the material starts to give. Because it is one equation with three variables, knowing any two yields the third.
F = spring force (N), k = spring constant (N/m), x = displacement (m)
Worked example
A spring has a stiffness of k = 200 N/m and is stretched x = 0.1 m. How much force does it take?
- 1 Write Hooke’s law. F = k × x, with k in N/m and x (the displacement from rest length) in metres.
- 2 Substitute the known values. F = 200 N/m × 0.1 m.
- 3 Compute the force. F = 20 N — the force needed to hold the spring at that stretch, equal in size to the spring’s restoring force.
Spring quantities at a glance
A stiffer spring has a larger k; the energy stored while deforming it is the elastic potential energy ½kx².
| Quantity | Formula | SI unit |
|---|---|---|
| Spring force | F = k × x | newton (N) |
| Spring constant | k = F ÷ x | newton per metre (N/m) |
| Displacement | x = F ÷ k | metre (m) |
| Elastic potential energy | E = ½kx² | joule (J) |
The restoring force and where it breaks down
The spring pushes back. A stretched or compressed spring exerts a force that tries to return it to its rest length — opposite to the displacement. That is why Hooke’s law is often written F = −kx: the minus sign marks the restoring force as pointing against x. The plain F = k × x gives its magnitude.
Linearity has limits. Hooke’s law holds only up to the elastic limit. Stretch a spring too far and it deforms permanently or the force stops rising in step with x; beyond that point the simple formula no longer applies and the spring may not return to its original shape.