# B How to check if force is conservative?

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1. Mar 8, 2016

### hackhard

given a force
F = f(x) i^ + g(y) j^ + h(z) k^ (x,y,z,are coordinates of body)
how can i prove or disprove that the force is conservative ?

2. Mar 8, 2016

### Orodruin

Staff Emeritus
Are you aware of the requirements for a vector field to have a scalar potential?

3. Mar 8, 2016

no im nt

4. Mar 8, 2016

### Orodruin

Staff Emeritus
Then my suggestion is to go to your standard textbook in vector analysis and review this. Once you have done that, you should be able to figure out whether the field you quoted is conservative or not.

5. Mar 8, 2016

### Staff: Mentor

Please use LaTeX for equations. This is very confusing as written.

6. Mar 8, 2016

### HallsofIvy

Knowing the definition of "conservative force" would be a good start! Unfortunately, you appear to be saying that you do not know that. Can you look it up in your text book?

7. Mar 8, 2016

### BvU

I suppose you googled conservative force ?

A conservative force is a force with the property that the work done in moving a particle between two points is independent of the taken path.[1] Equivalently, if a particle travels in a closed loop, the net work done (the sum of the force acting along the path multiplied by the distance travelled) by a conservative force is zero.

So integrate $\int \vec F \cdot d\vec s$ over a closed loop and prove that that is 0 independent of the loop chosen....

8. Mar 8, 2016

### Staff: Mentor

Or use a simpler method which is indicated on the Wiki page that you linked to.

9. Mar 8, 2016

### BvU

That requires reading and digesting a substantial part of the lemma !

10. Mar 8, 2016

### Orodruin

Staff Emeritus
... or just knowing about the curl theorem ...

11. Mar 8, 2016

### BvU

We are just giving this away ! I'd like to see hack post some work !

12. Mar 8, 2016

### gleem

Another way to look at a conservative force is to see if the work done in the field by moving an object depends only on the initial a final positions of the object . In other words is the incremental work done by the force over an incremental distance an exact differential. i.e.can you show that there is a function W such that dW = Fds = f(x)dx + g(y)dy + h(z)dz

13. Mar 8, 2016

### gracy

14. Mar 9, 2016

### hackhard

if it is known that total work done by a force on a body in moving it in a closed loop back to initial position is zero
is the force always conservative?

15. Mar 9, 2016

### Orodruin

Staff Emeritus
That is the definition of a conservative force ... or rather one of three equivalent definitions.

16. Mar 9, 2016

### gracy

It could happen that the total work is still zero even when the force is non conservative. In a non-conservative field not all closed paths have zero total work. But yes, in case of conservative forces all closed paths have zero total work.

Last edited: Mar 9, 2016
17. Mar 9, 2016

### BvU

Re #14: Isn't that more a follow-up question (*)? How are you doing with the original exercise ?

(*) And not a very clear one. Do you mean "in one specific closed loop" (then: no) or "in any possible loop" (then yes).

18. Mar 9, 2016

### hackhard

so ill have to net work zero for all possible closed loops. is that correct?
and will i have to prove this for all loops through each point in space?

19. Mar 9, 2016

### BvU

That could take a while, isn't it ? . Since you are clearly (refer to post #11) smart enough to do a lot of thinking before doing a lot of working, perhaps you realize that the force function you start with is pretty general, so (#8, #10) could lead you to a few simple conditions for f, g and h that are needed to make this force field conservative ...

20. Mar 9, 2016

### hackhard

if i prove for a force -
mag of field vector at any point depends only on the mag of displacement vector from a fixed point O
dir of field vector is always parallel or antiparallel to that displacement vector
will it be sufficient to prove field is conservative