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thenewbosco

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Just studying for an exam and the following question appeared on the sample exam:

Given the force: [tex]F=-c(x-y)^2(\hat{i}-\hat{j})[/tex] where i and j are the unit vectors.

a) Show the force is conservative

b) Show the potential energy is given by [tex]V(x,y) = \frac{c}{3(x-y)^3)}[/tex] 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. [tex]F=-\nabla V[/tex] 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

Given the force: [tex]F=-c(x-y)^2(\hat{i}-\hat{j})[/tex] where i and j are the unit vectors.

a) Show the force is conservative

b) Show the potential energy is given by [tex]V(x,y) = \frac{c}{3(x-y)^3)}[/tex] 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. [tex]F=-\nabla V[/tex] 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

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