Mechanics
How To Calculate A Spring Force
Stretch a spring and it pushes back in proportion to how far you stretched it. That single sentence is Hooke's law, and it is the reason springs appear in suspension, in scales, in mattresses and in the shock absorbers that keep a door from slamming. The force is easy to compute; the subtlety is knowing how far the linear rule actually holds.
Quick Answer
F = -k x
- k
- Spring constant in newtons per metre
- x
- Displacement from the natural length in metres
- m
- Mass hung on the spring in kilograms
Spring force is the spring constant times the displacement from the natural length. A 250 newton per metre spring stretched 0.12 metres pulls back with 30 newtons, holds 1.8 joules, and with 1.5 kilograms attached oscillates every 0.487 seconds.
What Is A Spring Force?
Hooke's law says the restoring force of a spring is the spring constant multiplied by the displacement from its relaxed length, with a minus sign to show the force points back toward equilibrium. In practice almost everyone computes the magnitude first, because the direction is obvious: the spring pulls back when stretched and pushes out when compressed. The minus sign only matters when you are writing equations of motion, not when you want to know how hard the spring is pulling.
The spring constant, written k, is the property of the individual spring. A stiff spring has a large k, a soft spring a small one, and the unit is newtons per metre because the law multiplies a force per unit length by a length to get a force. Manufacturers quote it for shocks and coilovers; for an unknown spring you measure it by hanging a known mass and seeing how far the spring settles.
Displacement is measured from the spring's natural length, not from where it happens to sit in the assembly. A spring that is already compressed by 40 millimetres inside a mechanism and then compressed a further 10 millimetres has a displacement of 50 millimetres, and a force proportional to that 50. This is the single most common error in spring work, and it comes from measuring the position of the end rather than the change in length.
The same stretch that produces a force also stores energy. The energy in a spring is one half times k times the displacement squared, which means doubling the stretch quadruples the stored energy. That quadratic is why over-stretching a spring is so much more dangerous than it looks: a little more travel suddenly holds a lot more energy.
Attach a mass to a spring and let go, and the force becomes a restoring force that produces oscillation. The period of that oscillation depends only on the mass and the spring constant, not on how far you pulled it back. That amplitude independence is the defining property of simple harmonic motion, and it is what makes spring clocks and tuning forks keep steady time.
Formula
F = -k x
The restoring force is the spring constant times the displacement from the natural length, directed back toward equilibrium.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| F | Restoring force | — | The force the spring exerts, in newtons. |
| k | Spring constant | — | Stiffness of the spring, in newtons per metre. |
| x | Displacement | — | Distance from the natural length, in metres. |
U = 0.5 k x squared
The work stored in the spring by stretching or compressing it.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| U | Stored energy | — | Energy held in the spring, in joules. |
| k | Spring constant | — | Stiffness of the spring, in newtons per metre. |
| x | Displacement | — | Distance from the natural length, in metres. |
T = 2 pi x sqrt(m / k)
The time for one full oscillation, independent of how far the mass was pulled back.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| T | Period | — | Time for one complete cycle, in seconds. |
| m | Attached mass | — | Mass hung on the spring, in kilograms. |
| k | Spring constant | — | Stiffness of the spring, in newtons per metre. |
How To Calculate A Spring Force
- 1
Measure the displacement from the natural length
Record how far the spring is stretched or compressed away from the length it settles at with nothing attached. Take this in the same unit you want the answer in; metres is the convenient choice because the spring constant is quoted per metre.
- 2
Read or measure the spring constant
Take k from the manufacturer's datasheet, or find it by hanging a known mass and dividing the weight by the extension. Newtons per metre is the standard unit; a car coilover is in the tens of thousands, a pen spring in the single digits.
- 3
Multiply to get the force
Restoring force equals k times displacement. A 250 newton per metre spring pulled 0.12 metres gives 30 newtons. If the spring is compressed rather than stretched, the magnitude is identical and only the direction flips.
- 4
Compute the stored energy
Energy equals one half times k times the displacement squared. For the numbers above that is 0.5 x 250 x 0.12 x 0.12, or 1.8 joules, which is the work you did stretching it.
- 5
Find the oscillation period if a mass is attached
Period equals two pi times the square root of mass over k. With 1.5 kilograms on the 250 newton per metre spring the period is about 0.487 seconds, so the mass completes roughly two cycles per second.
Examples
Example 1: A door spring holding a door shut
- Spring constant
- 250 N/m
- Displacement
- 0.12 m
| Step | Calculation | Result |
|---|---|---|
| Substitute | k x x | 250 x 0.12 |
| Spring force | F | 30 |
Result: The spring pulls the door closed with 30 newtons, roughly the weight of a 3 kilogram bag of flour hanging from the handle.
Example 2: Stored energy in a stretched spring
- Spring constant
- 250 N/m
- Displacement
- 0.12 m
| Step | Calculation | Result |
|---|---|---|
| Displacement squared | x squared | 0.12 x 0.12 |
| Substitute | 0.5 x k x x squared | 0.5 x 250 x 0.0144 |
| Stored energy | U | 1.8 |
Result: The spring stores 1.8 joules, enough to lift a 1.8 newton object one metre, and the same energy would be released if it snapped back.
Example 3: Oscillation with a mass attached
- Spring constant
- 250 N/m
- Attached mass
- 1.5 kg
| Step | Calculation | Result |
|---|---|---|
| Mass over k | m / k | 1.5 / 250 |
| Square root | sqrt(m / k) | 0.0774596669 |
| Oscillation period | T | 0.4866934411 |
Result: The mass completes one cycle in 0.4866934411 seconds, a frequency of about 2 hertz, and that period does not change if the mass is pulled back further.
Calculator
Spring force
30
- Stored energy
- 1.8
- Angular frequency
- 12.9099
- Oscillation period
- 0.4867
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the A Spring Force calculator page.
Common Mistakes
Measuring from the wrong reference
Displacement is the change from the spring's natural length, not the absolute position of its end. A spring already preloaded 40 millimetres and then compressed another 10 has a displacement of 50 millimetres and three times the force of the additional travel alone.
Forgetting the spring constant is per metre
A constant of 250 newtons per metre multiplied by a displacement in centimetres gives a force 100 times too large. Convert the displacement to metres before multiplying, which is the same discipline as converting to joules before reporting energy.
Assuming Hooke's law holds forever
The linear rule fails once the spring is compressed enough that its coils touch, or stretched past its elastic limit so it takes a permanent set. Reporting a force computed from a displacement beyond the datasheet range is reporting a number the spring will never produce.
Using the wrong spring constant for a pair of springs
Two springs in parallel add their constants, while two in series combine as the reciprocal of the sum of reciprocals. Treating a pair of springs as if they were one springs with a single datasheet value is a common source of large errors in suspension calculations.
Confusing period with frequency
The period is the seconds per cycle and the frequency is the cycles per second; they are reciprocals. A period of 0.487 seconds is a frequency of about 2 hertz, and quoting one when the other was asked for is out by the full value of the other.
FAQ
What is Hooke's law in one sentence?
The force a spring exerts is its spring constant times how far it is displaced from its natural length, and it always points back toward that natural length. The relation holds as long as the spring stays within its elastic range.
How do I find the spring constant of an unknown spring?
Hang a known mass from it and measure how far it stretches. Divide the weight of the mass, which is mass times 9.81, by the extension in metres. The result is the spring constant in newtons per metre.
Why does the period not depend on how far I pull the mass back?
Because the restoring force is proportional to the displacement, the acceleration is proportional to the displacement too, and the two proportionalities cancel in the period. This independence is what makes a spring clock keep the same rate regardless of how tightly it is wound.
What happens if I stretch a spring too far?
Past the elastic limit the coils separate or the material deforms permanently, so the spring no longer returns to its original length and Hooke's law no longer describes it. Past the solid height, where the coils touch, the force rises far more steeply than the linear law predicts.
Is the energy in a spring the same as the work I put in?
Yes, in an ideal spring with no losses the stored energy equals the work you did stretching it, which is why the energy formula is the area under the force against displacement graph. Real springs warm up and lose a little of that energy to internal friction.
References
- [1]Encyclopaedia Britannica, Hooke's law — https://www.britannica.com/science/Hookes-law
- [2]OpenStax University Physics, Hooke's Law and Simple Harmonic Motion — https://openstax.org/books/university-physics-volume-1/pages/15-1-simple-harmonic-motion
- [3]Wikipedia, Elastic potential energy — https://en.wikipedia.org/wiki/Elastic_potential_energy