**Hooke's law** is a law which states that the force, *F*, required to bend a spring (or some other elastic object) is directly proportional to the distance *X* by some constant *k*, known as the stiffness constant.

- $ F=kX $

This is sometimes written as

- $ F=-kX $

In this case, *F* is equal to the force with which the spring pushes back.

By integrating with respect to *x*, we can find the work needed to compress or stretch a spring a given distance and the potential energy stored in said spring.

- $ W = E_p = \frac{1}{2} k x^2 $

Hooke's law is only an approximation, as all materials will deform past a certain point (called the elastic limit). In fact, many objects deviate from Hooke's law well before their elastic limits. However, for most cases, Hooke's law is fairly accurate.

## Harmonic motionEdit

If a weight is attached to a spring and the spring is stretched or compressed released, the motion can be described as

- $ x=A \cos ( \sqrt{ \tfrac{k}{m}} t ) $

where *A* is the amplitude, or how far the spring is stretched, *k* is the stiffness constant, and *m* is the mass of the weight. By taking the first and second derivatives, the speed and acceleration can be found.

The total energy of the system is equal to

- $ E = \frac{1}{2} k A^2 $