# Potential Function from a simple conservative force

1. Feb 22, 2010

### K29

1. The problem statement, all variables and given/known data
Find the potential functions for these conservative forces:

(1)F=xi+yj

(2)F=yi+xj

2. Relevant equations
F=-$$\nabla$$V (Force = -del (Pot.energy))

3. The attempt at a solution

So, I'm guessing to get V I just need to integrate F. For the first equation that gets me the right answer hurray!

However I think my method is wrong(or my general understanding the opposite of partial derivatives). Doing the same thing to eqn 2 leaves me with a slight problem.
-($$\int$$y dx +$$\int$$x dy)leaves me with -2xy+C. The correct answer should be -xy+C

(I checked and it is indeed a conservative force)

2. Feb 22, 2010

### ehild

For a conservative force, the work between two points is independent on the connecting path. Start to integrate from point (0,0) to point P(X,Y).
You can follow the line along the x axis from x=0 to x=X and then a straigth line parallel to the y axis from (X,0) to (X,Y). The work done is

$$V(0,0)-V(X,Y)=\int_{(0,0)}^{(X,Y)}{ydx+xdy}=\int_{(0,0)}^{(0,X)}{ydx}+\int_{(X,0)}^{(X,Y)}{xdy}$$

As y=0 along the first line, and x=X along the second one, the first integral is 0, the second one is XY.

ehild

3. Feb 24, 2010

### K29

solved. thanks