Question

A potential energy function for a two-dimensional force is of the form U = 3 x^(4)y - 8 x. Find the force that acts at the point (x, y). (Use x and y as appropriate.)
I know its something about partial derivatives but I'm not sure how to go about it. And the answer has to be in vector components (ihat jhat)

Answer 1

Well, if a force is acting on something and energy is conserved and there is no change in kinetic energy then
Delta U = increase in potential energy from point a to point b = integral Fx dx + integral Fy dy where the integrals are from a to b.
That is the work we would do on the system. The force the system exerts is equal and opposite by Newton #3.
So I have a hunch that
Fx = -dU/dx
and
Fy = -dU/dy
and the vector F = Fx i + Fy j
so here
U = 3 x^(4)y - 8 x
-dU/dx = -12 x^3 y +8
-dU/dy = -3 x^4
and
F = (-12 x^3 y +8)i - 3 x^4 j

Answer 2

Yes! Thank You! I was very close but I messed up on my derivative somehow :)