Changes

'''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 k is the spring stiffness constant which has units of newtons per [[meter]] and <math> \vec x </math> is the displacement from the equilibrium position.

'''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.

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.

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.