How to check if force is conservative?

[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 ?

 

[Orodruin]

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

 

[hackhard]

no I am nt

 

[Orodruin]

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.

 

[Dale]

F = f(x) i^ + g(y) j^ + h(z) k^

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

 

[HallsofIvy]

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 ?

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 textbook?

 

[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...

 

[jtbell]

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

 

[BvU]

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

 

[Orodruin]

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

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

 

[BvU]

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

 

[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 = F⋅ds = f(x)dx + g(y)dy + h(z)dz

 

[gracy]

This may help.

 

[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?

 

[Orodruin]

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?

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

 

[gracy]

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?

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.

 

[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).

 

[hackhard]

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

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?

 

[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 ...

 

[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

 

[BvU]

More than sufficient. But that way you miss a lot of (non-central) force fields that are conservative...

 

[vanhees71]

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.

Check your claim on the example of the potential in a plane
$$V(x,y)=A \arccos \frac{x}{\sqrt{x^2+y^2}},$$
by evaluating the work done in the corresponding force field for a closed circle around the origin ;-)). It's fun!

 

[hackhard]

please correct me if I am wrong -
a field at a point $\vec{P}$ defined by
$\vec{E}$ = f( P.l , P.m, P.n )$\hat{i}$ + g( P.l , P.m, P.n )$\hat{j}$ + h( P.l , P.m, P.n )$\hat{k}$
where l, m , n are dir cosines of $\vec{P}$
is always conservative if can be proved that ----
there exist a fixed point $\vec{A}$ (fixed in the frame wrt which $\vec{P}$ is defined)
such that for every $\vec{P}$ (ranging over all points in R³), the expression
f( P.l , P.m, P.n )$\hat{i}$ + g( P.l , P.m, P.n )$\hat{j}$ + h( P.l , P.m, P.n )$\hat{k}$
can always be reduced to
w(|$\vec{AP}$|) $\hat{AP}$ (note here w varies only over |$\vec{AP}$|)
can this answer post#1 ? (ive proved something extra but is this proof sufficient?)

 

[ZapperZ]

Oh my god! Why can't you just take the curl of your vector field already?!

This is like watching molasses flow! It is getting to be two pages long, and you still can't seem to get the most obvious indication of a conservative vector.

Zz.

 

[BvU]

Easy now ! We have to wait until grokking is... learning isn't always a linear process that can be accelerated at will . But I do agree, Zz !