Question

Why is the gravitational potential energy the integral of the gravitational force? Is this the same for other forces?

Answer 1

Yes, the definition of work (=energy) is\[W := \int\limits_{C} F \, ds\]That is, the line-intergral of the force F along the path C.

Answer 2

Isn't Work= -change in U, so it is the change in energy?

Answer 3

It seems very odd, as \[U(r)=-\int\limits_{reference point (\infty)}^{r}F dr\]

Answer 4

U is supposed to be what now? The potential energy in a conservative force-field?
Ahm.. the thing with Energy is, it's not a "real" physical quantity (sry for that word) but just some number.. hmm..

Well, there's a video of feynman: http://www.youtube.com/watch?v=r_IfV9fkBhk
(if you got Windows and Silverlight installed, you can also google for "project tuva" - there's a better quality of the video) where he explains it quite well..

Or just read through all the articles @ http://en.wikipedia.org/wiki/Forms_of_energy

But yeah.. bottom line is: work is just another word for energy..

Answer 5

U was potential energy, from http://hyperphysics.phy-astr.gsu.edu/hbase/gpot.html#ui

Answer 6

There is a discrepancy with http://en.wikipedia.org/wiki/Potential_energy#Overview
and http://en.wikipedia.org/wiki/Forms_of_energy#Potential_energy
which is correct?

Answer 7

Ahm.. what's the discrepancy? Sry, I'm lazy :P

Answer 8

1) delatw=-deltau
2)w=-deltau

Answer 9

Given that our previous definitions of work and U, PE, were pretty much the same, I'm inclined to think that the 1st is correct.

Answer 10

Well the thing with the deltas in this case is: Work/Energy as I said, are not really a physical quantity, but just ahm.. damn, this is hard to explain (especially since I'm no native speaker :P)

To quote Feynman again: "There is a fact, or if you wish, a law, governing all natural phenomena that are known to date. There is no known exception to this law—it is exact so far as we know. The law is called the conservation of energy. It states that there is a certain quantity, which we call energy, that does not change in manifold changes which nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number and when we finish watching nature go through her tricks and calculate the number again, it is the same."

And as you can see by your definition of potential energy above.. it's no absolute value... Sometimes you say it's negative and becomes 0 at infinity... sometimes you say it's positive and 0 is at the ground (when you have a constant field - near the earth for example, we sometimes say Epot = m*g*h)

I'd suggest, just to watch the video. Feynman is the most ingenious teacher that ever lived, imho ;)

Answer 11

I've seen it before, but this is a wonderful excuse to rewatch!

Answer 12

Sry I can't explain it better, it's rly all the same stuff, it's all Joules, so E, W, U or ΔE - it doesn't rly matter. You can choose your own zero-point and do whatever you want.. as long as you are consistent ;)
There are conventions of course and you should use them. But it doesn't rly matter how big your Energy is.. what matters are local or global minima and maxima and differences in one form of enery etc..
I also rewatch it from time to time.. better than TV :D