What is Hooke's Law?
Hooke's Law, named after the English scientist Robert Hooke who published it in 1678, describes the behaviour of elastic materials under deformation. It states that the force a spring exerts is directly proportional to the distance it is displaced from its natural (rest) position. The relationship is written as:
F = k × x
Where:
- F = restoring force in newtons (N)
- k = spring constant (also called stiffness coefficient) in newtons per metre (N/m)
- x = displacement from natural length in metres (m)
The force F is the restoring force that acts in the opposite direction to the displacement. When you stretch a spring, it pulls back; when you compress it, it pushes back. This is why Hooke's Law is sometimes written as F = -kx, with the negative sign indicating the opposing direction. For magnitude calculations (how strong is the spring force?), you use F = kx.
The Spring Constant (k)
The spring constant k describes how stiff a spring is. A high k value means the spring is stiff and requires a large force to stretch it a small amount. A low k value means the spring is soft and stretches easily with a small force.
| Spring Type | Typical k Value | Notes |
|---|
| Slinky toy | ~1 to 10 N/m | Very soft, long extension |
| Pen click spring | ~150 N/m | Small, stiff for its size |
| Typical physics lab spring | 100 to 400 N/m | General demonstration use |
| Trampoline springs | ~2,500 N/m | Stiff, high load capacity |
| Automotive coil spring | 15,000 to 30,000 N/m | Must support vehicle weight |
Elastic Potential Energy
When a spring is stretched or compressed, it stores energy. This stored energy is called elastic potential energy (also called spring potential energy) and is given by:
E = 0.5 × k × x²
The energy is in joules (J). Notice that energy depends on the square of displacement, so doubling the stretch quadruples the stored energy. This is why compressed springs can release energy quickly and powerfully, as in spring-loaded mechanisms, toys, and shock absorbers.
The Elastic Limit
Hooke's Law holds only within the elastic limit of a spring. If you stretch a spring beyond this point, the spring deforms permanently and no longer returns to its original length. The relationship between force and displacement becomes non-linear. In practical applications, springs are always designed to operate well within their elastic limit to ensure reliable, repeatable behaviour.
Rearranging Hooke's Law
You can rearrange the formula to solve for any of the three variables:
- To find force: F = k × x
- To find spring constant: k = F / x
- To find displacement: x = F / k
This calculator handles all three arrangements. Select what you want to solve for, enter the other two values, and the result updates instantly.
Related Calculators
Sources and method: Hooke, R. (1678). De Potentia Restitutiva. The formula F = kx and elastic potential energy E = 0.5kx² are standard results from classical mechanics. Spring constant examples are illustrative only; actual values depend on spring geometry and material.
This calculator provides results based on ideal Hooke's Law behaviour within the elastic limit. Real springs may deviate from this model when subjected to high loads, temperature extremes, or fatigue. For engineering applications, consult manufacturer specifications and appropriate standards.
