Solving Potential Energy: Force & Potential Exam Ques.

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The discussion centers on a sample exam question involving a force defined as F=-c(x-y)^2(î-ĵ). Participants confirm that the force can be shown to be conservative by calculating its curl. The challenge arises in deriving the potential energy function V(x,y) from the force, with the suggestion that using the negative gradient might not yield the correct potential. It is noted that understanding the relationship between work done against a conservative field and potential energy is crucial. The conversation emphasizes the importance of knowing how to derive potential functions from forces for a deeper grasp of the topic.
thenewbosco
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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
 
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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.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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