Why does work depend on the path for some forces but not for others?

Question

I have learned that work done by conservative forces is independent of the path followed between initial and final position. But that's not the case for non-conservative forces, they depend on the path followed to reach the final point (eg. Friction).

My question is:

1. Why is that so? Why do some forces' work depend on path and others not?

When I hold a thing in my hand and make it follow a short and long random path in different cases and come to some position $x$, I feel I have done a different amount of work in both cases. But gravitation being a conservative force says I have done an equal amount of work in both cases.

2. Where am I mistaken?

3. Also, why do 2 forces exist? Forces are forces, they must have the same nature.

Lastly,

4. Can I state all unidirectional forces are conservative?

5. Are there other classifications of forces?

Answer 1

Why do some forces' work depends on path and others not.

• Nonconservative forces cause energy loss during displacement. For example, friction when an objects moves over a surface converts the stored energy to heat that disappears and is wasted. Therefore the final state depends on how long the path was, because that determines how much energy is lost along the way.

• Conservative forces cause no loss in energy. Therefore, energy associated with such forces can only be converted into other stored forms in the object (kinetic energy) or system (potential energy). In fact the work done by a conservative force is what we describe as potential energy. The word "potential" gives the feeling that it is stored; it is merely a name for the work that the conservative force will do when released. And when released, that potential energy will be work done on the object and it turns into kinetic energy, which is still stored in the body. If you are told what the start and end speeds are, you therefore know that the difference in kinetic energy must be stored. Regardless of the path.

We can consider the conservation in terms of energy like here or entropy and maybe others as well. I personally find the energy approach the most intuitive.

When I hold a thing in my hand and make it follow a short and long random path in different cases and come to the some position 'x' , I feel I have done different amount of work in both cases. But gravitation being a conservative force says I have done equal amount of work in both cases. Where am I mistaken?

Gravity might be a conservative force, but the force you exert on the object is not.

Also why do 2 forces exist? Forces are forces they must be same nature.

Which two are you thinking of?

In any case, yes, forces are "the same thing" so to speak. It doesn't matter what "kind" of force or what created the force - forces are forces and they can be added, for example in Newton's laws, where we don't care about the "type" of force.

Lastly, -can I state all unidirectional forces are conservative?

What do you mean by unidirectional force?

If gravity pulls downwards, so a box slides down an incline, there can still be a friction in only one direction on the incline. The directionality is not a measure of if a force is conservative or not.

Instead think about what kind of energy that force causes. Is it potential or kinetic, then the force is conservative. Is it heat or alike, then not.

-Are there other classifications of forces?

There are many "types" of forces: Electric, magnetic, chemical, gravitational, elastic etc. Those are just names that tell us the origin of them. As stated above, the "type" or origin is of no importance; all forces can cause acceleration in the same manner.

Answer 2

Work is defined as the path differential form associated with a force vector field, i. e., $dW = F_x dx + F_y dy + F_z dz$; the finite work is the integral thereof on a finite line $\gamma$. Once you integrate the variables over, the only variable left is exactly the path you are integrating upon, hence, by definition, integrals of the differential form must indeed be a function of the path, in principle.

One can show that in the very special case of forces derived by a potential function (namely conservative forces) $\textbf{F} = -\textrm{grad}\,V$ the integration over any path happens to not depend on the shape of the path, but only on its initial and end points (because of Stokes theorem on the boundaries of integration).

Forces are forces they must have the same nature.

In the universe, there are four different types of interactions and their form strongly depends on the case at hand and the distribution of masses and charges generating the force.

Answer 3

I look at [non]conservative forces in terms of what they turn energy into. When moving against a force, you do work; when moving with a force, work is done on the object. In a conservative forces, these are both "efficient" energy conversions: 100% of the work you do is turned into usable potential energy, and then 100% of that is turned back into kinetic energy of the object. This is the case with, say, an object falling under gravity in a vacuum: its energy is exactly the same at the bottom as the top.

Friction does not do this. Friction, by definition, converts all of the work done into heat. The force opposing movement in a certain direction $x$ is the sum of the work required to get the potential energy, and the energy that will be lost. In friction there is no potential energy stored, so you are always doing work to move.

The force of your arm on an object is also not conservative. Perhaps, with very efficient muscles, you can lift a weight 100% efficiently. But when you lower it again, the potential energy is not converted back into energy in your arm. In fact, you'll have to start firing your muscle cells to slow it down as it falls, which will waste heat, so you do work even when it's coming down. Again, all the work is turned into heat.

As an example of something in between, an object flying through air under gravity: going up, most of its energy is turned into potential energy, and a bit into turbulence in the air (and eventually heat); and going down, most is turned back into kinetic energy. So perhaps, upon reaching its original height, it has 90% of its original speed. This force was mostly conservative.

Why are there "2 kinds of forces"? In the same sense that there "are two kinds of English sentences": true and false. Conservatism is just a property I can talk about, indicating 0 loss. This is the same as "[perfectly] elastic" and "inelastic".

Why are the 4 fundamental forces conservative? -- because they have nowhere else to put the energy! On a microscale, all energy is "tracked", including heat, as a the kinetic energy of particles creating heat. By definition, energy is the thing that is conserved by all the forces. So we built energy such that the fundamental forces would be conservative, and this determined it. Then, if you choose to ignore certain kinds of energy (such as heat), you can get nonconservative forces.