# Scalar Potential of a One-Dimensional Force

ourio

## Homework Statement

A particle of mass m is subject to the one dimensional force F = x-x^3. Determine whether or not this force is conservative. If it is: a) write the scalar potential and find the turning points, b) write the kinetic energy and show that the sum of the kinetic and potential energy is independent of position.

## Homework Equations

How do I find the scalar potential??

## The Attempt at a Solution

In order to be conservative, I know that the curl must equal zero. Taking the curl:
$$\nabla$$ x F = 0

thebigstar25
scalar potential is just a term used when you have a conservative vector field, so if you have a conservative vector field, then you can write this vector as minus the gradient of a scalar potential ..

and in your question, if you proved that the force vector is conservative then you can say that this force vector = - gradient of V (V is the scalar potential)..
so to find this scalar potential just do the reverse operation, V = - integral the force vector dx ..

chrisk
Another way to approach the problem is to use Stoke's Theorem relating the line integral to the surface integral. The curl of F is substituted into the surface integral expression and since curl F = 0 then the line integral equals zero. So, the work done by F around any closed path is zero.