# Changes

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 : $\vec x(t) = \vec x_0 \cos(\omega t+ \phi)$,  where $\phi$ is a phase shift and $\omega$ is the angular frequency and is: $\omega = \sqrt{\frac{k/}{m}}$ is the frequency of oscillation: thus  Thus the particle moves in a "[[sine|sine wavesinusoidal]]" shapemanner. This is an example of simple harmonic motion.