Hooke's Law and Spring Constant
Hooke's law deals with springs and their main property — elasticity. Each spring can be deformed (stretched or compressed) to some extent. When the force that causes the deformation disappears, the spring comes back to its initial shape, provided the elastic limit was not exceeded.
Hooke's law states that for an elastic spring, the force and displacement are proportional to each other. It means that as the spring force increases, the displacement increases too. If you graphed this relationship, you would discover that the graph is a straight line. Its inclination depends on the constant of proportionality, called the spring constant. It always has a positive value.
Spring Force Equation
Knowing Hooke's law, we can write it in the form of a formula:
F = −k × Δx
where:
- F — the spring force (in N for metric, lbf for imperial)
- k — the spring constant (in N/m for metric, lbf/in for imperial)
- Δx — the displacement, positive for elongation and negative for compression (in m for metric, in for imperial)
The minus sign reflects that the spring force always opposes the displacement. If you pull a spring to the right, the restoring force acts to the left — this is what brings the spring back to its natural length.
How to Use the Hooke's Law Calculator
- Select your unit system — Metric (SI) uses Newtons, N/m, and meters; Imperial (US) uses pound-force, lbf/in, and inches.
- Choose what to solve for — spring force (F), spring constant (k), or displacement (Δx).
- Enter the known values — the calculator will fill in the two inputs you provide and compute the unknown.
- Click Calculate — results appear instantly with full unit conversions and the applied formula.
Unit Conversions
The calculator automatically converts results between metric and imperial units:
| Quantity | Metric | Imperial |
|---|---|---|
| Force | Newton (N), kilonewton (kN), dyne (dyn) | Pound-force (lbf) |
| Spring constant | N/m | lbf/in |
| Displacement | meters (m), centimeters (cm), millimeters (mm) | inches (in), feet (ft) |
Spring Constant Reference Values
| Spring type | Typical k (N/m) | Typical k (lbf/in) |
|---|---|---|
| Soft rubber band | ~100–500 | ~0.57–2.85 |
| Pen spring | ~200–300 | ~1.14–1.71 |
| Car suspension spring | ~15,000–30,000 | ~85–171 |
| Stiff industrial spring | ~100,000+ | ~570+ |
FAQs
What is the spring constant?
The spring constant (k) is a measure of a spring's stiffness. A higher spring constant means the spring is stiffer and requires more force to stretch or compress by the same amount. It is measured in N/m (newtons per meter) in the metric system, or lbf/in (pound-force per inch) in the imperial system.
Does Hooke's law apply to compression too?
Yes. Hooke's law applies to both extension and compression, as long as the spring is not deformed beyond its elastic limit. For compression, the displacement Δx is negative, but the restoring force magnitude is still F = k × |Δx|.
What is the elastic limit?
The elastic limit (or yield point) is the maximum deformation a spring can undergo and still return to its original shape. Beyond this point, the spring deforms permanently and Hooke's law no longer applies.
What is elastic potential energy?
The energy stored in a stretched or compressed spring is the elastic potential energy: E = ½ × k × Δx². This energy is fully released when the spring returns to its natural length.
What is rotational stiffness?
The rotational analog of the spring constant is rotational stiffness (also called torsional stiffness), measured in N·m/rad. It describes how much torque is needed per unit of angular displacement in a torsional spring.