How to differentiate b/w a conservative/non-conservative force?

[SciencyBoi]

Homework Statement

How do we determine a specific force mentioned in a question to be conservative or non-conservative?

2. Relevant data
Conservative force is a force whose work done does not depend on the path that is taken while doing it. Examples include electrostatic force, gravitational force. It is because of this property that we are able to define potential energies.
They can be differentiated by evaluating a closed loop and checking if zero.

Non-conservative force are the opposite of conservative forces. Examples include frictional force.

The Attempt at a Solution

I have tried brainstorming about some particular forces like the one shown below;

Here, I cannot determine if the force is conservative or not. The question required application of Work-Energy theorem, wherein, work of F on block was simply given as F.|AB| + mgh (h being the height it rises by)

Which is only possible if F is conservative.

Please shed some photons. I seem to be missing some concept.

 

[andrewkirk]

Wiki gives three alternative definitions of conservative force:

1. It conserves mechanical energy (ie does not produce heat)
2. The work done is path-independent
3. It is part of a force field that has certain mathematical properties.

The definitions are equivalent when the force is part of a field, but not otherwise. In this example, the force F is not part of a field, which means we can't use definition 3, and it is not possible to vary the path, which knocks out definition 2. So we have to use definition 1.

What hypothetical properties would we have to require the block, rope, pulley and curved surface to have in order to satisfy definition 1? (These conditions may not be achievable in practice.)

 

[SciencyBoi]

What hypothetical properties would we have to require the block, rope, pulley and curved surface to have in order to satisfy definition 1? (These conditions may not be achievable in practice.)

I think, there should be no resistive forces, like friction or elasticity in the rope. That should make it a conservative force by definition 1.

But, then another confusion, these three definitions make nearly every force (in an ideal world/scenario) conservative. A world with rigid bodies and no friction would mean conservative forces everywhere. Am I right?

Thanks a lot for your help! I really appreciate it...

 

[andrewkirk]

A world with rigid bodies and no friction would mean conservative forces everywhere. Am I right?

I think problems arise with magnetic fields and time-varying electric fields. See for instance the discussions here and here. Definition 3 cannot be satisfied by a magnetic field, as there is no magnetic potential. However I don't think the lack of conservatism can be put down to heat-generating processes such as friction. I think for magnetic fields, perhaps definition 1 of a conservative force is satisfied but not definition 3. If we restrict ourselves to mechanics - ie excluding electromagnetic forces, and ignore the fact that collisions and rebounds between rigid(-ish) bodies have to be ultimately explained in terms of electrostatic forces (and that even the most elastic collisions always generate some heat), then I think what you wrote works.