# Conservative forces, Nonconservative forces and Potential Energy

• fog37

#### fog37

Hello,

I would like to review and validate some concepts that I have been recently thinking about. Hope this is correct and useful to others that need to refresh these concepts.
• Forces can be classified as either conservative or nonconservative. Dissipative forces are always nonconservative forces but not all nonconservative forces are dissipative. For example, the thrust of a rocket is a nonconservative force but not a dissipative force.
• The total work done on a system is the sum of the works done by each force acting on the system: ##W_{total}=W_{conservative}+W_{nonconservative}=\Delta KE##.
• The change in the "mechanical" energy of the system, defined as ##E_{mech}=KE+PE##, is equal to the work done by nonconservative forces: ##W_{nonconservative}=\Delta E_{mech}##. The net work ##W_{conservative} ## done by all the conservative forces never changes the total mechanical energy of the system.
• Once we define the system and its surrounding (everything outside of the system), some of the forces which are external to the system can be conservative and some can be nonconservative. The net force is vector sum of all the conservative and nonconservative forces and is nonconservative if nonzero (adding a conservative force to a nonconservative force produces a nonconservative force).
• If the net external force was either zero or nonzero but conservative, the total "mechanical" energy of the system would remain constant. The total energy of the system (mechanical energy + all other energy forms) is constant only if the net external force is zero.
Let me know if I got things right if you can :)

Here a simple made up problem: a small satellite (weight ##980 N##) is pushed vertically upward a distance of ##6 m## by a thrust force ##F## of magnitude ##2000 N##. The thrust does positive work ##(2000)(6)=12000 J## while gravity does negative work ##(-980)(6)= 5880 J##.
The net work is positive and equal to ##(12000-5880)= 6120 J## and the satellite's kinetic energy increases by that amount.

The thrust force seems responsible for making the satellite gain gravitational potential energy ##PE## by bringing to an altitude of ##6m##. Why can a nonconservative force not change the ##PE## of the satellite? I know ##PE## is not defined for nonconservative forces. If it is not defined, it means that nonconservative forces, like the thrust, cannot increase/decrease ##PE##.
Maybe the correct way to think about it is the following: the thrust force will simply not affect or change the system's potential energy. Although the thrust is a force that brings the satellite to a new position changing the configurational state of the satellite-Earth system, it is not having any role in the potential energy of the system. All we say is that the thrust does positive work which would produce a ##12000K## increase in ##KE## of the satellite. But the conservative gravitational force, as the satellite climbs up, does negative work on the satellite reducing its kinetic energy by ##5880J##. The ##5880 J## are not lost but simply converted from kinetic energy into gravitational potential energy. A dissipative force always does negative work and always reduces the kinetic energy converting it into a different form of energy (thermal energy instead of potential energy) and transfer it to a different body (the entity that produces the dissipative force itself).

Thanks!

• Delta2

Hi. I understand conservative force is the force which can be expressed as ##-\nabla U## where U is potential energy. Non conservative force is the forces otherwise. But I am not familiar with conservative/non conservative work. I will appreciate it if I could learn it in this thread.

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I agree with 4 of your 5 points except the last. If the total external force is not zero then regardless if it is conservative or non conservative i think it changes the total mechanical energy of the system. Your example with the satellite shows that. The satellite gains both kinetic energy and potential energy by the thrust force. The role of the thrust force can be played by any conservative force like for example if we had the satellite upon a huge spring that is initially compressed and then ejects the satellite.

But of course if we include the spring into our system then the total mechanical energy which includes the initial potential energy of the compressed spring is conserved but then the spring force becomes internal to the system, not external...

• fog37
A dissipative force always does negative work and always reduces the kinetic energy ...
This depends on the reference frame. In some frames the same dissipative force can be doing positive work on the object. Only the net work done on both objects by the dissipative force pair is negative in all frames.

... converting it into a different form of energy (thermal energy instead of potential energy) and transfer it to a different body (the entity that produces the dissipative force itself).
Both objects will usually heat up,

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I agree with 4 of your 5 points except the last. If the total external force is not zero then regardless if it is conservative or non conservative i think it changes the total mechanical energy of the system. Your example with the satellite shows that. The satellite gains both kinetic energy and potential energy by the thrust force. The role of the thrust force can be played by any conservative force like for example if we had the satellite upon a huge spring that is initially compressed and then ejects the satellite.

But of course if we include the spring into our system then the total mechanical energy which includes the initial potential energy of the compressed spring is conserved but then the spring force becomes internal to the system, not external...
I would say that if the external force is nonzero, then the system's momentum changes. If the external force does nonzero work (it is possible that the force does zero work), then the system's energy changes. The external work determines if the energy changes and not just the external force alone...

• Delta2
The external work determines if the energy changes and not just the external force alone...
Yes I guess you are right, it might be the case that the external force is perpendicular to the displacement , thus doing zero work.

In essence, a conservative force or conservative force field is an interaction during which the positive work done by the conservative force produces an increase in the ##KE## of the system while reducing the potential energy ##U## stored in the system's configuration (and vice versa).

I would argue that even a nonconservative force can change the configuration of system and its potential energy but only conservative forces, through their conservative work, can be related to that potential energy change. Is that correct?

The potential energy ##U## that a certain system's configuration has does not and should not change with time if the configuration remains in the same identical configuration. For example, if the rock-Earth system mutual distance ##r## does not change, the potential energy ##U_1## of that configuration ##C_1## remains the same. The system's configuration can change if the system change shape but if the system returns to the original configuration ##C_1## it needs to have the same potential energy ##U_1##. This mean that the scalar function ##U## is a unique one-one function with a specific value for every different configuration. This concept requires the work done by a conservative force to be path independent and be an exact differential.

Which configuration has ##U=0##? That is arbitrary. The spatial coordinates ##(x,y,z)## that the function ##U(x,y,z)## depends on should the coordinates of the separation distance vector ##r=(x,y,z)## that separate the two or more entities forming the system.

For example, let's consider a 1D example with two point masses ##m_1## and ##m_2##. The first mass is located at distance ##x_1=2m## from the origin ##O##. The second mass is at ##x_2=4m##. The mutual distance is ##|\Delta x| =| x_2 - x_1|##.

The potential energy function of the two-mass system is $$U= -G \frac {m_1 m_3}{|\Delta x|}$$

Why would the potential energy have a negative value?

Why would the potential energy have a negative value?
Apologies if I'm responding to a rhetorical question.

It doesn't, necessarily. As you point out, the value it takes is physically meaningless. We could add any arbitrary constant (essentially a constant of integration) and no experimental results would change:
Which configuration has ##U=0##? That is arbitrary.
Once we've settled on a constant of integration, the important feature is that as the two masses get closer (smaller ##\Delta x##) an attractive force will make them move faster (greater kinetic energy) and, accordingly, have a lower potential energy.

But ##\frac{1}{\Delta x}## gets larger as ##\Delta x## gets smaller. We need something that does the opposite. So a minus sign is appropriate. Not because it makes potential energy either positive or negative. But because it makes the change in potential energy go in the right direction compared to the change in configuration.

• jim mcnamara and fog37
Apologies if I'm responding to a rhetorical question.

It doesn't, necessarily. As you point out, the value it takes is physically meaningless. We could add any arbitrary constant (essentially a constant of integration) and no experimental results would change:

Once we've settled on a constant of integration, the important feature is that as the two masses get closer (smaller ##\Delta x##) an attractive force will make them move faster (greater kinetic energy) and, accordingly, have a lower potential energy.

But ##\frac{1}{\Delta x}## gets larger as ##\Delta x## gets smaller. We need something that does the opposite. So a minus sign is appropriate. Not because it makes potential energy either positive or negative. But because it makes the change in potential energy go in the right direction compared to the change in configuration.
Thank you jbriggs444.

I got it, thanks: ##U## is negative only if we set ##U=0## when the masses are infinitely far apart. The closer the two point masses are, the smaller (and more negative) their potential energy is.
I did mention that which configuration assume ##U=0## is arbitrary but relativity seems to require ##U=0## only when the point masses are indeed infinitely far apart (see Sherwood's article).

What would you fix/correct about my above statement "...I would argue that even a nonconservative force can change the configuration of system and its potential energy but only conservative forces, through their conservative work, can be related to that potential energy change. Is that correct?..." regarding nonconservative forces being able to change the system configuration hence its ##U## but that change being only trackable using the work done by conservative forces?

Is it somewhat correct?

I got it, thanks: ##U## is negative only if we set ##U=0## when the masses are infinitely far apart. The closer the two point masses are, the smaller (and more negative) their potential energy is.
Nicely put.
What would you fix/correct about my above statement "...I would argue that even a nonconservative force can change the configuration of system and its potential energy but only conservative forces, through their conservative work, can be related to that potential energy change. Is that correct?..." regarding nonconservative forces being able to change the system configuration hence its ##U## but that change being only trackable using the work done by conservative forces?

Is it somewhat correct?
I am not terribly comfortable with broad pronouncements about classes of forces. You formulate a perfectly reasonable sounding statement with one context in mind and then find yourself trying to apply it in another context where it blows up in your face.

Yes, non-conservative forces can change the configuration of the system. Here I have in mind the experimenter reaching into a system with his hands. He lifts a toy cart to the top of the track, stretches a rubber band or throws a ball at a rod sitting on an air table.

And yes, only for a conservative force, are you going to be able to define a potential.

However, if you start with a non-conservative force and find a subset of the configuration space within which that force turns out to be conservative and if you can constrain your system so that it stays within that subset of the configuration space -- then the "non-conservative" force might be conservative enough to have a potential. For instance, a mild steel bar that you never bend past its elastic limit.

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• fog37