# Hooke's Law

Hooke's Law states that the force exerted by a stretched or compressed spring is a restoring force and is linearly proportional to the distance that the spring is stretched or compressed beyond its relaxed length. The formula for Hooke's law is $\vec F = -k \vec x$ where k is the spring stiffness constant which has units of newtons per meter and $\vec x$ is the displacement from the equilibrium position.

The negative sign here is to indicate that the force is in the direction opposite the displacement. For example, if $\vec x$ is to the right, then $-k \vec x$ is to the left, indicating that the mass is pulled back towards the spring.

The force $\vec F = -k \vec x$ can be interpreted as a vector field depending on the displacement vector $\vec x$. Since this field is the gradient of the function $\frac{1}{2} k|\vec x|^2$, this means that $\vec F$ is a conservative field. As a result, $E = \frac{1}{2} k|\vec x|^2$ is conserved, and is the potential energy stored in the string. Moreover, E depends only on $\vec x$, and not the path that the particle takes to get to $\vec x$, and so it is said to be a path independent quantity.

It should be noted that in reality, Hooke's law is merely an approximation, and no physical spring actually has precisely this behavior. However, for most materials a version of Hooke's law holds for reasonable ranges of $\vec x$. This is called the elastic range of the material.

If a particle moves only under the influence of the force exerted by a spring, then Newton's second law ($\vec F = m \vec a$) implies that its displacement satisfies the second-order differential equation $m\ddot{\vec x} = - k \vec x$. This is solved by $\vec x(t) = \vec x_0 \cos(\omega t)$, where $\omega = \sqrt{k/m}$ is the frequency of oscillation: thus the particle moves in a "sine wave" shape. This is an example of simple harmonic motion.