How to show that force is conservative?

Question

anonymous · Answer

We can do that if it is possible for it to write in terms of a gradient along with a negative sign.

gorica · Answer

can you do that for $$\vec F=-mg \vec k$$

naveenbabbar · Answer

The work done by a conservative force over a closed loop/path is always zero.

naveenbabbar · Answer

Work done by the force is independent of nature of path followed and depends on initial and final position. the force involved is then conservative.

naveenbabbar · Answer

Example :- Gravitational force, Electrostatic force

gorica · Answer

@naveenbabbar I would appreciate if you could show it on example. Only theory doesn't help me much.

anonymous · Answer

This relation is a simple form of a complicated formula $$F=mM/r ^{2}$$ so we can write,$$F=-gradV=-grad(mM/r ^{1})=mM/r ^{2}$$
where $$V=-mM/r$$ is potential function of the body.

naveenbabbar · Answer

ok wait a minute

anonymous · Answer

Ok!

naveenbabbar · Answer

attachment

naveenbabbar · Answer

Look for the attachment and try to solve this question

gorica · Answer

@Saeeddiscover can you do that using (iii) from this proposition?

gorica · Answer

Can I do that writing like this$$\vec F d \vec r=-dΦ$$ where $$dΦ\neq0$$? If it is =0, will it be conservative force?

anonymous · Answer

it should be, because curl of a gradient is always zero. or$$Curl F=Curl(grad V)=0$$

anonymous · Answer

No it doesn't show the axiom.

gorica · Answer

no, I am not asking you to prove axiom, but if I can show that way that a force is conservative

anonymous · Answer

I strongly recommend looking at some mathematical textbooks such as George B. Arfken's.

anonymous · Answer

Nothing can be adopted if you wish to consider the case $$d \theta=0$$

anonymous · Answer

Any questions!

gorica · Answer

I don't understand it much, but thanks anyway.