Scalar Potential of a One-Dimensional Force

[ourio]

1. The problem statement, all variables and given/known data
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.


2. Relevant equations
How do I find the scalar potential??


3. 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 ..

hopefully that answers your question ..

[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.