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 <math> \vec F = -k \vec x </math> where <math>k</math> is the spring stiffness constant which has units of Newtons per meter and <math> \vec x </math> 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 <math>\vec x</math> is to the right, then <math>-k \vec x</math> is to the left, indicating that the mass is pulled back towards the spring. This signifies that it is a restoring force; that it acts to return the mass to equilibrium.

The force <math> \vec F = -k \vec x</math> can be interpreted as a vector field depending on the displacement vector <math>\vec x</math>. Since this field is the gradient of the function <math>\frac{1}{2} k|\vec x|^2</math>, this means that <math>\vec F</math> is a conservative field. As a result, <math>E = \frac{1}{2} k|\vec x|^2</math> is conserved, and is the potential energy stored in the string. Moreover, <math>E</math> depends only on <math>\vec x</math>, and not the path that the particle takes to get to <math>\vec x</math>, 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 <math>\vec x</math>. 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 (<math>\vec F = m \vec a</math>) implies that its displacement satisfies the second-order differential equation

<math>m\ddot{\vec x} = - k \vec x</math>.

This is solved by:

<math>\vec x(t) = \vec x_0 \cos(\omega t + \phi)</math>

where <math>\phi</math> is a phase shift and <math>\omega</math> is the angular frequency and is:

<math>\omega = \sqrt{\frac{k}{m}}</math>

Thus the particle moves in a sinusoidal manner. This is an example of simple harmonic motion.