# I Can nonconservative forces change either KE or PE?

1. Jan 24, 2017

### fog37

Hello everyone,

conservative forces only depend on position (cannot depend on time), i.e. $F(x)$ and are equal to the spatial derivative of the potential energy function $U(x)$:

$$F(x)= - \partial U(x)/\partial x$$

Conservative forces always have to change the kinetic energy KE and potential energy PE in such a way that the total mechanical energy $ME=KE+PE= constant$.

What about nonconservative forces? They don't maintain the total mechanical energy $(KE+PE)$ constant as they act on an object. But can they still affect the value of $PE$? They shouldn't, since there is no potential energy function associated to nonconservative forces...

thanks.

2. Jan 24, 2017

### Staff: Mentor

You cannot define a (meaningful) potential energy for non-conservative forces. If you could, the force would be conservative. Something that does not exist cannot change its value.

3. Jan 24, 2017

### fog37

I agree.

But when a rocket pushes a spacecraft upward, the spacecraft increases in KE and gains altitude. Can we know say that the thrust force has increased also the potential energy of the rocket?

4. Jan 24, 2017

### jbriggs444

Certainly. A similar example would be a man throwing a ball upward. He succeeds in imparting both kinetic and gravitational potential energy to the ball. Note that the "potential energy" in this case is the potential associated with the force of gravity. It has nothing to do with the "potential" that might or might not even be definable for the force of the man's hand on the ball [or of a rocket motor on a rocket].

5. Jan 24, 2017

### fog37

Ok, but the force exerted by the man's hand seems to be the source of the increase of the potential energy...That is what confuses me since no potential energy can be defined for a nonconservative force...

I think there is no problem for nonconservative force to increase/decrease kinetic energy. that is what the work-kinetic energy theorem says. There is not potential energy change in that theorem.

6. Jan 25, 2017

### A.T.

Moving parallel to the conservative force of gravity will change potential energy. The "cause" for the movement is irrelevant.

You confuse yourself by focusing on irrelevant issues, that are not mentioned in the definitions.

7. Jan 25, 2017

### Stephen Tashi

Suppose an electromagnet is giving metal objects a potential energy due to the magnet's attraction and a man's hand flips a switch and turns the electricity to the electromagnet off. Where did the potential energy go?

In fact, where was the potential energy to begin with? Was it in the magnet? - in the magnetic field? -in the metal objects?

The concept of a potential-something involves a predicted outcome assuming circumstances remain as specified by the type of potential we are defining.

For example, if we have a table top with a constant coefficient of friction then "The work done by friction as we drag a 'unit test weight' from a point (x,y) on the table to the center C of the table" cannot be predicted because the conditions don't contain sufficient information. If we allow ourselves to invent a kind of "path valued" function W(x,y) that represents particular path taken from (x,y) to the center (instead of a single numerical value) we could conceivably define a "frictional potential" P(x,y,W(x,y)) as the work done by dragging a unit test weight from (x,y) to the center of the table along path W(x,y). In that definition, it is understood that nothing happens to disturb the scenario such as a man's hand intervening to press the weight harder against the table.

Potential energy has to do with a predicted outcome given some assumed conditions. If something happens that varies from those conditions then the change in potential energy is similar to what happens where there is a violation of a clause in a legal contract. It is an abstract event that may void the contract or cause some other contingency of the contract to come into effect.

8. Jan 25, 2017

### Staff: Mentor

As you pointed out in the OP the potential ennergy depends only on the position. So the only thing that can affect the PE is the position.

The force does not explicitly increase the PE of the rocket, only the position does. The thrust may implicitly increase the PE, but only through its effect on the position.

9. Jan 26, 2017

### fog37

mmm... I am close to getting it.

Let me ask again: can a nonconservative force change the potential energy of a system?

Also, what is the clear explanation of why conservative forces cannot be functions of time but ONLY functions of position? How would a time dependence of a force be in conflict with conservation of mechanical energy? What principle does it go against?

10. Jan 26, 2017

### jbriggs444

How are you defining the potential energy of the system? Are you including gravitational potential energy? Electrostatic potential energy? Chemical potential energy? Potential energy stored in springs?

A gravitational force cannot change the gravitational potential energy of the system. (*)
Electrostatic repulsion cannot change the electostatic potential energy of a system. (*)

But electrostatic repulsion can change the gravitational potential energy of a system. So can the chemical energy in your muscles. Or the gas in our car's fuel tank.

One way of defining what it means for a force to be conservative is that the integral of work done around a closed loop is zero. If you have a force that changes with time, you could run half of the a loop out while the force has one value and half back in while the force has a different value. The result is non-zero work done, even though any particular snapshot of the force field would be conservative.

An example would be a piston in a cylinder in your engine. During the compression stroke, the gas pressure in the cylinder creates what amounts to a conservative force on the piston. During the power stroke, the gas pressure in the cylinder creates what amounts to a conservative force on the piston. But if you combine the strokes, you get positive work done over the entire [Otto] cycle -- because the force varies over time.

(*) To be picky, external forces can increase the internal potential energy of a system. For instance, the gravitational force from the Sun and moon (e.g. the tides) can influence objects on the Earth, increasing the gravitational potential energy of the Earth/ocean system (e.g. high tide). If one expands the boundaries of of the "system" to include the Sun and moon then their gravitation would be an internal force, part and parcel of the system's total gravitational potential energy. The gravity from the Sun and moon would then not influence the system's gravitational potential energy.

11. Jan 26, 2017

### Stephen Tashi

The work "to go in a closed loop" only refers to the geometric shape of the loop, not some particular timing for how you travel the path around the loop. In a time varying field, "the work to go in a closed loop" is not a uniquely defined quantity since it depends on the trajectory as a function of time, not just the geometric shape of the trajectory. Since "the work to go in a closed loop" in a time varying field is not a uniquely defined quantity, we can't claim it is a quantity always equal to zero.

Suppose the gravity of a planet somehow varied with time and dropped to zero at certain times. If you started out on a journey around a loop, you might luck out and spend most of the time going away from the planet when the field was zero and more of the time coming back toward the planet when the field was "on" and helping you.

12. Jan 26, 2017

### Staff: Mentor

You may want to read about Noether's theorem. It addresses this question and also similar questions about momentum, angular momentum, charge, and other conserved quantities.

13. Jan 26, 2017

### fog37

Thanks. I am reading about Noether's theorem which involves Lagrangians and Hamiltonians and I am new to those concepts. I sort of understand that conservation laws derive from symmetries.
Conservative forces and potential energy depend only on position. Any force that is time-dependent is therefore nonconservative. I am sure there are force that depend on position that are nonconservative forces.

What does the work done by a nonconservative forces do to the system? Does it just affect the kinetic energy of the system? Fundamentally, there are only two types of energies (KE and PE). If nonconservatives force don't mess with PE, they must only affect KE...

Let's move into electromagnetism now. It is well known that the electrostatic field is a conservative force field, hence a potential energy function can be associated to it. The concept of electric potential difference $\Delta V$ derives from the potential energy function. However, in AC circuits and time-changing electronics, we talk about a time varying electric potential difference. Is that concept really correct? It doesn't seem like it would correct since the idea of potential energy and potential must imply no time dependence. We know that a time varying electric field $E(t)$ is not conservative but will still employ the concept of electric potential difference...

14. Jan 27, 2017

### Stephen Tashi

The situation surrounding how "voltage", "emf", "electric potential" etc. are defined is rather embarrassing. (Look at the Wikipedia "talk" pages associated with articles on those and related terms and you will find a variety of contradictory opinions being defended.)

One approach is to say that a "voltage" difference between two locations is determined by calculating the work to move a unit test charge between them while assuming the electric field stays constant in time as you traverse the path between the locations. For voltage differences in a wire, maybe we have to specify that the path is a path that stays within the wire.

With that approach, a voltage difference isn't really a difference computed from a potential function.

15. Jan 27, 2017

### Stephen Tashi

Perhaps those are the only types of energies in courses on mechanics. If "KE" is only defined relative to the property of an object called its mass, then we can ask if there are other properties of an object that can be used to define other non-potential kinds of energy. That would be interesting to discuss, but its a digression.