Solving Potential Energy: Force & Potential Exam Ques.

[thenewbosco]

Just studying for an exam and the following question appeared on the sample exam:
Given the force: $$F=-c(x-y)^2(\hat{i}-\hat{j})$$ where i and j are the unit vectors.
a) Show the force is conservative
b) Show the potential energy is given by $$V(x,y) = \frac{c}{3(x-y)^3)}$$ assuming V(0,0) =0.


So for a) it is simple to show using the curl of F. but for b i am not sure how to get the potential energy function given the force. $$F=-\nabla V$$ will give the force easily if i have the potential function, but I am not sure how to go the other direction. perhaps for the exam to show it i could just take the negative gradient, (which appears to be wrong for the potential given in this question) but i would like to just know how to go the other way for knowledge. thanks

 

[tim_lou]

if a field is conservative, then the work done against the field on a object from point a to point b is conserved and equals to the change in potential. Well, basically, if you know the work done alone a path, you know the potential function.