Hooke's Law Calculator
Calculate spring force, spring constant, or displacement using Hooke's law (F = k·x). Enter the two known quantities, pick your units, and get the third value instantly.
Find Spring Force, Spring Constant, or Displacement
Hooke's Law Formula
Where:
- \(F\) — spring force exerted by (or applied to) the spring, in newtons (N)
- \(k\) — spring constant (stiffness), in newtons per meter (N/m)
- \(x\) — displacement from the natural length, in meters (m); positive for extension, negative for compression
The formula states that the force a spring exerts is directly proportional to how far it is stretched or compressed from its natural length. Rearranged forms let you solve for any of the three quantities: \( k = F/x \) and \( x = F/k \).
Hooke's Law – Calculation Example
Example: a spring with a stiffness of 250 N/m is stretched 0.2 m beyond its natural length. What force does it exert?
\[ F = k \cdot x = 250 \, \text{N/m} \times 0.2 \, \text{m} \]
\[ F = 50 \, \text{N} \]
Result: the spring exerts a restoring force of 50 N (about 11.24 lbf), pulling back toward its natural length.
How to Use the Hooke's Law Calculator
- In the Solve for dropdown, choose the quantity you want to calculate: Spring force (F), Spring constant (k), or Extension / compression (x). The calculator hides the field you are solving for.
- Enter the two known values in the remaining fields: Spring force (F), Spring constant (k), and Displacement (x).
- Pick the unit for each input from the dropdown next to it — force in N, lbf, or kgf; stiffness in N/m, N/cm, lbf/in, or lbf/ft; displacement in m, cm, mm, ft, or in.
- Read the result: the solved quantity is shown in your chosen unit with common alternatives, and a table lists all three quantities (force, stiffness, displacement) with their directions.
What Is Hooke's Law?
Hooke's law, formulated by Robert Hooke in 1676, describes the behavior of an ideal elastic spring: the force needed to stretch or compress a spring is directly proportional to the distance you deform it. Double the stretch, and you need double the force. This linear relationship holds for most springs, elastic bands, and many elastic materials — as long as you stay within their elastic limit.
The proportionality constant \(k\), called the spring constant or stiffness, tells you how hard a spring resists deformation. A stiff car suspension spring might have \(k\) in the hundreds of thousands of N/m, while a soft toy spring can be just a few N/m. Importantly, \(k\) depends on the material, the coil diameter, the wire thickness, and the number of coils — it is a property of the specific spring, not of the material alone.
Note that the force in Hooke's law is a restoring force: it always points opposite to the displacement. In this calculator, a positive displacement means the spring is stretched, and the resulting positive force is the magnitude of the restoring pull. If you enter a negative displacement (compression), the force is negative, indicating the spring pushes back outward. If you want to explore the energy stored in a deformed spring, see the elastic potential energy calculator; for force in general, try the force calculator.
Because the formula is a simple proportion, you can use it in any consistent unit system. The calculator converts your inputs to SI units internally, so mixing, say, lbf with inches works correctly — no manual conversions needed.
Solving for Each Variable
- Find the force: \( F = k \cdot x \). Multiply the stiffness by the displacement. This is what happens when you know the spring and how far it is deformed.
- Find the spring constant: \( k = F / x \). Divide the applied force by the resulting deformation. This is a common way to measure an unknown spring's stiffness experimentally: hang a known weight and measure how far it stretches.
- Find the displacement: \( x = F / k \). Divide the force by the stiffness. This tells you how far a spring will deform under a given load.
Assumptions, Limitations, and Practical Tips
- Elastic limit: Hooke's law only works while the deformation is reversible. Stretch a spring too far and it deforms permanently; the relationship stops being linear well before that point.
- Massless, ideal spring: the model ignores the spring's own weight and internal friction (hysteresis). Real springs dissipate a little energy on each cycle.
- Sign conventions: enter positive values for stretching and negative values for compressing. When solving for \(k\), the force and displacement must have the same sign — a stretching force with a negative displacement would imply a negative, physically impossible spring constant, and the calculator will warn you.
- Zero inputs: a force or displacement of zero cannot determine \(k\) (any stiffness produces zero deformation under zero load), and a spring constant must be greater than zero.
- Measuring k in practice: when testing a real spring, take several force–displacement pairs and check that the ratio \(F/x\) is constant — if it drifts, the spring is leaving its linear range.
Typical Spring Constants
| Spring / system | Approximate k |
|---|---|
| Ballpoint pen spring | ~50–200 N/m |
| Car suspension spring | ~20,000–100,000 N/m |
| Rubber band (small) | ~10–100 N/m |
| Mattress coil | ~2,000–10,000 N/m |
Hooke's law also underlies oscillation problems: a mass on a spring exchanges kinetic energy and elastic potential energy, a case you can analyze with the conservation of energy calculator.
Frequently Asked Questions
How do I calculate Hooke's law by hand?
Use \( F = k \cdot x \): multiply the spring constant by the displacement. To find stiffness, divide force by displacement: \( k = F/x \). To find displacement, divide force by stiffness: \( x = F/k \). Always measure displacement from the spring's natural (unstretched) length, not its total length, and keep the units consistent.
What is a good or normal spring constant value?
It depends entirely on the application. Small household springs (pens, toys) run about 10–200 N/m, mattress coils around 2,000–10,000 N/m, and car suspension springs from 20,000 up to 100,000 N/m. There is no universally "good" value — a stiffer spring (higher \( k \)) resists deformation more, so the right value depends on the load and how much travel you need.
How do I convert spring constant units, like N/m to lbf/in?
Multiply N/m by 0.0057102 to get lbf/in; conversely, 1 lbf/in ≈ 175.13 N/m. For example, a 250 N/m spring equals about 1.43 lbf/in. Also handy: 1 N/cm = 100 N/m, and 1 lbf/ft ≈ 14.59 N/m. The calculator handles these conversions automatically, so you can enter values in mixed units.
What does a negative force or displacement mean?
The sign shows direction. A positive displacement means the spring is stretched, so the restoring force is negative (it pulls back inward). A negative displacement means compression, so the spring pushes outward. In this calculator, entering a negative displacement returns a negative force, indicating a push rather than a pull. When solving for \( k \), force and displacement must have the same sign, since a spring constant cannot be negative.
How can I measure the spring constant of a real spring at home?
Hang a known mass from the spring and measure the extra stretch. The force equals the weight: \( F = m \cdot g \) (with \( g \approx 9.81 \, \text{m/s}^2 \)), so \( k = mg/x \). For instance, a 1 kg weight stretching a spring by 0.15 m gives \( k \approx 65.4 \, \text{N/m} \). Repeat with two or three masses — if the ratio \( F/x \) stays constant, the spring is behaving linearly.
What are the most common mistakes when using Hooke's law?
Three stand out. First, using the spring's total length instead of the displacement from its natural length. Second, confusing mass with force — a mass in kilograms must be multiplied by \( g \) to get newtons. Third, forgetting the elastic limit: the formula only works while the spring returns to its original shape, and the relationship goes non-linear well before permanent deformation begins.
How is spring potential energy related to Hooke's law?
The energy stored in a deformed spring is \( E = \tfrac{1}{2} k x^2 \), where \( x \) is the displacement. Note that energy grows with the square of displacement — doubling the stretch stores four times the energy. This is why a heavily compressed spring can be dangerous. You can compute this value with an elastic potential energy calculator once you know \( k \) and \( x \).