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Spring constant calculator

A 10 N pull stretches a spring 4 cm, and the spring constant calculator turns that into k. Change the force and the stretch, or switch to solve for the force or the extension instead.

Spring mode, and Hooke's law when not solving for k.

Spring and time modes; also used for energy.

Spring and time modes: the largest displacement from equilibrium.

For the amplitude and for the extension x.

Pendulum mode. From the pivot to the center of the bob.

Pendulum mode, and the hanging stretch in spring mode.

m/s²Used only when gravity is set to custom.

degreesPendulum mode. The angle from vertical it swings out to; sets the large-amplitude correction.

Hooke's law mode only. The box for the quantity you pick is ignored.

Hooke's law mode. The force stretching or compressing the spring.

Hooke's law mode. Stretch from the spring's natural length, in the displacement unit; negative for compression.

HzTime mode.

degreesTime mode. 0 means released from x = +A at t = 0.

sTime mode.

Spring constant k250 N/m

Spring constant k k = F / x = 10 N / 0.04 m = 250 N/m.
250 N/m
The unit algebra Stiffness is force per unit of stretch. 1 lbf/in is 175.1 N/m; 1 N/mm is 1,000 N/m.
N / m
Stiffness in other units
250 N/m 2.5 N/cm 0.25 N/mm 1.428 lbf/in
Energy stored ½kx²: the area under the force-extension line. Half of F × x, because the force grows from zero as the spring stretches.
0.2 J
Period with the mass above on it 2π√(m/k) with m = 1 kg: how fast that mass would bounce on this spring.
0.3974 s
Hanging stretch per kg g / k at g = 9.807 m/s²: how far each kilogram hung on it pulls it down.
3.923 cm (0.03923 m)
Force is proportional to extension; stored energy grows with its square
ExtensionForceEnergy stored
1 cm (0.01 m)2.5 N0.0125 J
2 cm (0.02 m)5 N0.05 J
4 cm (0.04 m)10 N0.2 J
6 cm (0.06 m)15 N0.45 J
8 cm (0.08 m)20 N0.8 J
  • The force here is the one you apply to the spring. The spring pulls back with an equal and opposite force, which is why the law is often written F = −kx.
  • Hooke's law holds only within the elastic limit. Stretched further, a spring deforms permanently and the line bends.
  • Results are shown to four significant figures. Round to the precision of your least precise measurement.

How the Simple Harmonic Motion Calculator works

k = F / x. Hooke's law says the force needed to stretch or compress a spring is proportional to how far it moves: F = kx. The constant k, the spring constant or stiffness, is that force divided by the distance, in newtons per meter.

Worked example, the one above: a 10 N force stretches a spring 4 cm. Convert the stretch to meters first, 0.04 m, then k = 10 / 0.04 = 250 N/m. Every centimeter of stretch takes 2.5 N. Turned around, the same spring under 15 N stretches x = F / k = 15 / 250 = 0.06 m, and stretched 8 cm it pulls back with F = kx = 250 × 0.08 = 20 N.

Measuring k in practice. Hang the spring, add known masses, and record the extension each time. The force is each mass times g: 0.5 kg weighs 4.903 N under standard gravity. Plot force against extension: within the elastic limit the points fall on a straight line through the origin, and its slope is k. Using the slope rather than a single reading averages out measurement error and shows where the line starts to curve, which is where Hooke's law stops applying.

Units. N/m is the SI unit. Springs are often specified in N/mm (multiply by 1,000 for N/m) or, in the US, in pounds per inch: 1 lbf/in is 175.1 N/m. Getting the stretch in centimeters or millimeters into meters before dividing is the step most often missed, and it is out by a factor of 100 or 1,000.

The minus sign. You will often see F = −kx. That is the force the spring exerts, which always points back toward its natural length: stretch it to the right and it pulls left. The calculator uses the force you apply, which points the same way as the stretch, so k comes out positive when F and x have the same sign.

Energy stored. The work to stretch a spring is ½kx², the area of the triangle under the force-extension line. At 4 cm on a 250 N/m spring that is ½ × 250 × 0.04² = 0.2 J. Double the stretch and the energy quadruples, which the table under the result shows.

Springs together. Two springs side by side (in parallel) share the load, so their stiffnesses add: k = k1 + k2. End to end (in series) each carries the whole force and the stretches add, so 1/k = 1/k1 + 1/k2. Two identical springs in series are half as stiff.

Hooke's law is a linear approximation that real springs obey only within their elastic limit. Past it they yield and do not return to their original length, and no single k describes them.

Simple Harmonic Motion Calculator FAQ

How do you calculate a spring constant?

Divide the force by the extension it causes: k = F / x. A 10 N force that stretches a spring 4 cm (0.04 m) gives k = 10 / 0.04 = 250 N/m. Keep the extension in meters to get N/m.

What are the units of the spring constant?

Newtons per meter (N/m) in SI. Engineering catalogs often use N/mm, which is 1,000 N/m, and US ones pounds-force per inch: 1 lbf/in = 175.1 N/m.

Why is Hooke's law written F = −kx?

The minus sign describes the spring's own force, which always opposes the displacement and pulls back toward the natural length. The force you apply to stretch it is +kx. Both describe the same stiffness k.

What happens to the spring constant when springs are combined?

In parallel the stiffnesses add: two 250 N/m springs side by side make 500 N/m. In series the reciprocals add: the same two end to end make 125 N/m.

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