Hooke's Law

Derivation

Equation of Motion from Cons of Energy

In the previous chapter Forest_UCM_Energy_Line1D, we saw how the equations of motion could from the requirement that Energy be conserved.

in 1-D

Let consider the case where an object is oscillating about a point of stability

A Taylor expansion of the Potential function U(x) about the equilibrium point is

Further consider the case the the potential is symmetric about the equilibrium point

at the equilibrium point

: Force = 0 at equilibrium

in order to have stable equilibrium

: stable equilibirium

also the odd (2n-1) terms must be zero in order to have stable equilibrium for a symmetric potential (the potential needs to be a max at the end points of the motion).

:

and the leading term is just a constant which can be dropped by redefining the zero point of the potential

This leaves us with

if we make the following definitions

and that the equilibrium point is located at the orgin

Then

Since we began this derivation with the assumption that energy was conserved then the force must be conservative such that

or this 1-D force can be written as

Interpretation (Hooke's law)

Returning back to the conservation of energy equation

Lets consider only the first term in the expansion of the potential U(x)

energy is constant with time
energy is constant with time

A Force exerted by a spring is proportional to the spring displacement from equilibrium and is directed towards restoring the equilibrium condition. (a linear restoring force).

In 1-D this force may be written as

While the above was derived from the assumption of conservation of energy we can apply our two tests for conservative forces as a double check:

1.) The force only depends on position.

2.) The work done is independent of path ( in 1-D and 3-D)