Force to lift a chain: Conservative or not?

[AJKing]

Homework Statement

You are lifting a chain straight up at a constant velocity v_0. The chain has a linear mass density λ. What is the force required to lift the chain as a function of height?

The Attempt at a Solution

U = mgh = λygh

The height in the potential energy is the same as the potential energy at the center of mass

h = y/2

U = λgy^2/2

This is a conservative potential energy in one dimension

F = -∂U/∂y= -λgy

Is this correct?

Can conservative forces consider objects of non-constant mass as I've done?

 

[Simon Bridge]

Is the chain initially lying on the ground then.

Per your question:
By definition of a conservative force, the work should be zero on a closed path right?
So check.

 

[AJKing]

Is the chain initially lying on the ground then.

Per your question:
By definition of a conservative force, the work should be zero on a closed path right?
So check.

Yes, it's on the ground.

Right, the work done on the object should be zero about a closed path. Regardless of the fact that my work is not zero.
The work done on this chain will certainly be zero about any closed path.
The change in mass does not matter.

Then I suppose my force is conservative and that my equation is correct in the case that the chain doesn't leave the surface. As soon as it does, I've got to rewrite to:

F = -λLg

which is constant.

Is that all correct?

 

[Simon Bridge]

The applied force here is not really from a field though is it?

Technically it is gravity that is the conservative force - since it can be described as the gradient of a potential function. Your applied force only exists at a point - it (or rather, whatever is applying the force) is the thing doing the work.

But your math looks fine from here.

To check this sort of thing all you need is to check the reasoning - to lift at constant speed, the applied force has to be equal to gravity. Presumably it was briefly larger than gravity at some earlier stage in order to accelerate to the constant speed.

This is one would modify the model to allow for finite sized links in the chain.

 

[HalfLostAndSearching]

I would respond by saying that the force required to lift a chain is a conservative force. This is because the potential energy of the chain, as given by the equation U = λgy^2/2, is only dependent on the height of the chain and not on the path it takes to reach that height. This is a characteristic of conservative forces.

Regarding your question about non-constant mass, I would say that in this scenario, the chain's linear mass density is considered a constant value and does not affect the conservative nature of the force required to lift it. However, if the mass of the chain was changing as it was being lifted, then the force required would not be conservative.