# Conservative Forces

The basic definition of a conservative force is that

A force is conservative if the work it does on a particle moving between any two points is idepedent of the path ae by the particle.

Furthermore, the work done by a conservative force exerted on a particle moveing through any closted path is zero.

That means the work done by a conservative force as an object moves from A to B is just the negative or opposite of the work done as an object moves from B to A.

For a conservative force, we can always write a potential energy U. Then the work done by the conservative force is

Wc = Ui - Uf

or

Wc = - U = - ( Uf - Ui )

We have already seen this with the force of gravity.

In talking about work done by a varying force, we also saw this for the elastic force of a spring. The force exerted by a spring is given by

F = - k x

We found that the work done by a spring as it acts on an object that moves from initial position xi to final position xf is

Ws = (1/2) k xi2 - (1/2) k xf2

If we define the spring potential energy by

Us = (1/2) k x2

then we can write

Ws = Usi - Usf

Ws = - Us = - (Usf - Usi)

The force of gravity and the force exerted by a spring are examples of conservative forces. For any conservative force, we can write a Potential Energy and the work done by that conservative force is equal to

Wg = - U = - (Uf - Ui)