Force fields, vectors, and work (mostly just confused by notation)

[joriarty]

Homework Statement

Consider a force field F = c(iy - jx). From the force field calculate the work required to move a particle from the origin to the point 2i + 4j without acceleration along the two different paths:

• From the origin to 2i then to 2i + 4j
• From the origin to 4j then to 2i + 4j

Comment whether the force field is conservative or not

2. The attempt at a solution

I'm just slightly confused by the notation used in the question. I know that i and j are just the unit vectors in the x and y directions, but what is c in the force field expression? And if i and j are already noted in this expression, why are x and y used as well?

Assuming c is just some arbitrary constant, then is the work done simply 6c J for both paths? My logic for this is that for the first path the work done is 2c J along the x-axis and then 4c J along the y-axis (and the other way around for the second path).

Thus the force field is conservative (work done is independent of the path taken).

Is my logic correct, or am I missing something? Thank you

 

[radou]

There are conditions for the conservativeness of a force field, you should check one of these (formally) before concluding if it is conservative. If c is an arbitrary constant, x and y are probably variables, so F=F(x, y).

 

[joriarty]

There are conditions for the conservativeness of a force field, you should check one of these (formally) before concluding if it is conservative. If c is an arbitrary constant, x and y are probably variables, so F=F(x, y).

I don't understand - how can I check these conditions formally? Have I not already done so by showing that the work done is the same for both paths in the first part of the question? Note that I am asked to comment on whether or not the force field is conservative, which implies that there are no additional calculations required.

 

[radou]

Well, then that's it - you have shown that the work is independent of the path taken.

You could also calculate ∇ x F, which equals 0 for a conservative force field, but since you're not asked to..

 

[joriarty]

OK, thanks! :)