An ideal spring exerts a force that tends to return it to its equilibrium length. For displacements within its elastic range, the restoring force is proportional to displacement:

$$F=-kx.$$

Here $x$ is displacement from equilibrium and $k>0$ is the spring constant.

The minus sign

If the spring is stretched so that $x>0$, the force points toward negative $x$. If it is compressed so that $x<0$, the force points toward positive $x$.

The minus sign therefore expresses the restoring direction rather than a negative force magnitude.

Stiffness

The spring constant has units

$$\mathrm{N/m}.$$

A larger $k$ means a larger restoring force for the same displacement and therefore a stiffer spring.

For example, if

$$k=200,\mathrm{N/m}$$

and the spring is stretched by

$$x=0.05,\mathrm m,$$

then

$$F=-10,\mathrm N.$$

Elastic range

Hooke's law is an ideal linear model. Real springs obey it only over a range where deformation remains approximately proportional to force and the material returns to its original shape.

Because the restoring force is proportional to displacement, combining Hooke's law with Newton's second law produces the equation of simple harmonic motion.