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 where is the spring stiffness constant which has units of Newtons per meter and 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 is to the right, then 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 can be interpreted as a vector field depending on the displacement vector . Since this field is the gradient of the function , this means that is a conservative field. As a result, is conserved, and is the potential energy stored in the string. Moreover, depends only on , and not the path that the particle takes to get to , 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 . 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 () implies that its displacement satisfies the second-order differential equation
This is solved by:
where is a phase shift and is the angular frequency and is:
Thus the particle moves in a sinusoidal manner. This is an example of simple harmonic motion.