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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

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