Spring Force Formula: F = −kx, Hooke's Law with Examples

The spring force formula is $F = -kx$ : the force a spring exerts equals its spring constant k multiplied by how far it is stretched or squashed, x. A spring with k = 200 N/m stretched by 5.0 cm pulls back with 10 N.

The minus sign says the force points opposite to the displacement. Stretch the spring and it pulls back; compress it and it pushes out. This is Hooke's law, valid as long as the spring is not pushed past its elastic limit. Along with tension and friction, it is one of the contact forces covered under the laws of motion. Enter positive values in the calculator below, since it handles sizes only, and add the direction yourself.

Quick facts

Formula
$F = -kx$
Size only
$|F| = kx$
SI unit of force
newton (N)
Unit of k
N/m, dimensions [M¹ T⁻²]
Stored energy
$U = \frac{1}{2}kx^2$
Valid range
Up to the elastic limit of the spring

Formula for spring force: what k and x mean

Symbols in Hooke's law
Symbol	Meaning	Unit
$F$	Restoring force from the spring	N
$k$	Spring constant: force per metre of stretch	N/m
$x$	Change in length from the natural, unloaded length	m

x is the change in length, not the total length. A 20 cm spring stretched to 26 cm has x = 6 cm = 0.06 m. A large k means a stiff spring, like a car's suspension; a small k means a soft one, like the spring inside a ballpoint pen.

Worked examples on the force of a spring

Example 1: direct use

A spring with k = 200 N/m is stretched by 5.0 cm. Find the force it exerts.

$x = 0.050\ \text{m}$ , so $F = kx = 200 \times 0.050 = 10\ \text{N}$ , directed back toward the natural length.

Example 2: finding k from a hanging mass

A 250 g mass hung from a spring stretches it by 4.9 cm. Taking g = 9.8 m/s², find the spring constant.

At rest the spring force balances the weight: $kx = mg$ .

$k = \frac{mg}{x} = \frac{0.250 \times 9.8}{0.049} = \frac{2.45}{0.049} = 50\ \text{N/m}$

Example 3: compression

A 1200 N/m spring in a toy launcher is pushed in by 3.0 cm. How hard does it push back?

$F = 1200 \times 0.030 = 36\ \text{N, outward}$

Springs in series and in parallel

Two springs side by side share the load, so they act as a stiffer spring: $k_p = k_1 + k_2$ . Two springs end to end each carry the full load and both stretch, so they act as a softer spring: $\frac{1}{k_s} = \frac{1}{k_1} + \frac{1}{k_2}$ .

For two identical 300 N/m springs, parallel gives 600 N/m and series gives 150 N/m. Cutting one spring in half also doubles its constant, since each half stretches only half as much under the same force.

Graph of the spring force formula

Plot force against extension. Below the elastic limit the plotted points form a straight line from (0, 0), and k is its slope. The area under the line is the work done in stretching the spring, $\frac{1}{2}kx^2$ , stored as elastic potential energy; see the potential energy of a spring page. Past the elastic limit the line curves, the spring stays longer after unloading, and Hooke's law no longer applies.

Units and dimensions of the spring constant

Since $k = \frac{F}{x}$ , its unit is N/m. In dimensions, [M L T⁻²] ÷ [L] = [M¹ L⁰ T⁻²], a result derived in full on the spring constant dimensions page. Note that the spring force itself has the ordinary dimensions of force, [M¹ L¹ T⁻²], and is measured in newtons.

A quick unit check catches many slips. Multiply k in N/m by x in m and the metres cancel, leaving newtons. If you are left with N cm/m or N/m², a length was not converted or the formula was rearranged the wrong way.

Where students go wrong with springs

Using the total length instead of the extension.
Leaving x in centimetres while k is in N/m.
Dropping the minus sign in a vector answer, or keeping it when only the size is asked.
Applying Hooke's law to a spring that has been overstretched.

When a mass bounces on a spring, the spring force changes every instant, so use energy or simple harmonic motion, not the constant acceleration equations.

Spring forces feed into F = ma like any other force, as the force formula explains with more cases.

Frequently asked questions

What is the formula of spring force?

Spring force is F = −kx, where k is the spring constant in N/m and x is the extension or compression in metres. The minus sign shows it opposes the displacement.

What is the unit of the spring constant?

Newton per metre, N/m. Its dimensional formula is [M¹ T⁻²].

What does the minus sign in Hooke's law mean?

The spring always pulls or pushes back toward its natural length, opposite to the displacement x. That is why it is called a restoring force.

Does Hooke's law always work?

Only up to the elastic limit. Beyond that the spring deforms permanently and the force is no longer proportional to the extension.