Formula f=-dU/dx: Non-Conservative Forces

  • Context: Undergrad 
  • Thread starter Thread starter gracy
  • Start date Start date
  • Tags Tags
    Formula
Click For Summary

Discussion Overview

The discussion revolves around the applicability of the formula ##f = -\frac{dU}{dx}## in the context of conservative and non-conservative forces, particularly when external forces or energy sources are involved. Participants explore the implications of this formula in various physical systems, including spring-mass systems and gravitational fields.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • Some participants suggest that the formula ##f = -\frac{dU}{dx}## is not applicable when non-conservative forces are present, as potential energy is defined only for conservative forces.
  • Others argue that the formula can still apply in certain contexts, such as when forces are position-dependent, but may not hold in open systems or when external forces are involved.
  • A participant questions the context in which the formula is used, indicating that clarity on the definitions of ##f## and ##U## is necessary for proper application.
  • There is a mention of the work-energy theorem, which some participants believe is related but not limited to conservative forces, suggesting that it may apply to other scenarios like braking cars.
  • One participant proposes that the gradient of potential energy gives the force on a single body within a system, rather than the entire system itself.
  • Another participant introduces the idea that friction may be treated as a force with a potential function in certain contexts, although this is noted as potentially controversial or misunderstood.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of the formula ##f = -\frac{dU}{dx}##, with no consensus reached on its limitations or the conditions under which it holds true. The discussion remains unresolved regarding the definitions and contexts of external forces and potential energy.

Contextual Notes

Participants highlight the importance of defining the system boundaries and the nature of forces involved, indicating that the classification of forces as internal or external can affect the validity of the formula. There are also mentions of specific systems, such as spring-mass systems and capacitors, which may influence the discussion.

gracy
Messages
2,486
Reaction score
83
Is it true that we can't use the formula ##f##= ##\frac{-dU}{dx}## when external source of energy or some external force is present?

I think it should rather be "we can't use ##f##= ##\frac{-dU}{dx}## when non conservative force is present" because potential energy is defined only for conservative force .
(U in the above formula is potential energy , right?)
 
Physics news on Phys.org
gracy said:
Is it true that we can't use the formula ff= −dUdx\frac{-dU}{dx} when external source of energy or some external force is present?

If your f is a force then the force can be defined as negative gradient of potential energy - these systems are called conservative as the total energy is conserved
however of dissipative forces are present and one can exactly account for the 'energy dissipated' or energy coming in -i do not think the total energy estimate can not be made in a closed system-what do you think?
 
gracy said:
(U in the above formula is potential energy , right?)

I don't know. Where did you find this formula? In what context is it being used?
 
I don't think it's that simple. The formula looks like the work-energy-theorem which is not limited to internal or conservative forces and potential energy. For example the kinetic energy of a braking car also follows this rule. In some cases this theorem doesn't hold (e.g. for open systems) but I don't know the general conditions for its validity.
 
I believe if the force is a function of position x, then F=-dU/dx applicable, where U is the position-dependent energy. In a spring, F=kx and hence, energy stored in the spring due to its displacement x is U=½kx2. Gravitational potential energy is also a position dependent energy. So, as per my understanding, F=-dU/dx is applicable to any position dependent force and energy system.
 
  • Like
Likes   Reactions: gracy
You did not say what f was. You always can define a function f(x) as the negative derivative of a differentiable function U(x) as f=-dU/dx. What is your problem?
Try to ask a question which has sense.
You asked the same question in an other thread recently, which has been closed .
 
ehild said:
Try to ask a question which has sense.
Sorry :frown:
 
Okay, since you haven't specified what f and U is and what type of system you have in your mind I'm making the assumption that the body on which a force f acts is part of a system with potential energy U.
If by "external force" you mean a force due to some influence outside of the system, then you're right, this relation won't hold. It doesn't matter whether the external force (external to the system) is conservative or not. This formula gives only the relation b/w the potential energy gradient and the force associated with THAT potential only.. Think of a spring- mass system in a gravitational field. Can you use this relation with f as the net force on the mass and U as either the gravitational potential energy? No. The gradient of gravitational potential energy gives the force on the mass due to gravity and the gradient of the spring potential energy gives the force due to the spring. You cannot mix up the two.
This may not be the answer you were looking for, but hope I could make you see it clearly( somewhat).
 
  • Like
Likes   Reactions: gracy, DrewD and cnh1995
gracy said:
Sorry :frown:

Don't be. Since this is Physics Forums and you are posting in the general physics sub, it should be abundantly clear what this formula is despite the fact that you didn't explicitly define the terms. You are asking for a clarification about the formula ##F=-\frac{dU}{dx}## where ##F## is force and ##U## is a scalar potential function (of position; I am not sure if it can be extended to have time dependance, but it doesn't in normal introductory courses). This is essentially the definition of a conservative force. If there is a ##U## such that a force can be written in this way, then ##F## is a conservative force.

(This is a little off topic, but I think there are methods that treat friction as a force with a potential function even though it is not actually conservative. I could be totally wrong about this, but I think I recall one of my profs saying that friction terms are added to a effective potential function for certain situations)

Aniruddha@94 gives a good answer. I assume your confusion is due the the term "external" which is probably not the best term, but it might be easier to grasp than "conservative force" for some people.
 
  • Like
Likes   Reactions: gracy and beamie564
  • #10
DrewD said:
(This is a little off topic, but I think there are methods that treat friction as a force with a potential function even though it is not actually conservative. I could be totally wrong about this, but I think I recall one of my profs saying that friction terms are added to a effective potential function for certain situations)
Wow really?! I didn't know that. Could you please mention some source, it sounds interesting.
 
  • #11
Aniruddha@94 said:
Wow really?! I didn't know that. Could you please mention some source, it sounds interesting.

I'll try to find one, but as I said, I only recall a professor telling me this. I may have just misunderstood what he was saying.
 
  • #12

Is F external to the system? How to know FORCE is external to the system or not?
 
  • #13
When you analyze a problem, you are free to pick the boundaries of the system in any way you like. Having chosen those boundaries, a force is "external" if it acts on an entity inside the system and originates from an entity outside the system.
 
  • Like
Likes   Reactions: gracy
  • #14
gracy said:
Is F external to the system? How to know FORCE is external to the system or not?
Instead of shoving a video, can you explain with your own words what the problem is? What is the set-up? F is a force, acting on what? U is the potential energy of what? What is x?
 
  • #15
Okay, I'll try to explain it. Are your concepts of potential energy, work and forces clear? If not then you should look into them. Whenever I have to understand something related to potential energy, I try to relate the system to a mass-earth system and think of gravitational potential energy ( because gravitation is very intuitive to us), so I'll use that to explain the idea behind the formula.
In the case of capacitors, the parallel plates and the dielectric form the system. There are no frictional forces and no external forces ( we want the force b/w the plates and the dielectric slab). The total potential energy associated with this system is U=(1/2)*CV^2.
Now think about a mass-earth system. The mass is at some height H and the potential energy of the SYSTEM is mgH ( we know mgh is only an approximation, but this will do for this discussion). mgH is the potential energy of the whole system, NOT of a single body. When you change the configuration of the system (I.e, change the height of the mass) then the potential energy of the system also changes, let's say it becomes mgH'. Now, the gradient of this potential energy (gradient is how U changes with distance x) gives us the force exerted ON a body.
The forces in both the bodies are equal in magnitude (Newton's 3rd law), this fits perfectly. Can you see that both the forces ( on mass due to the Earth and vice versa ) come from dU/dx .
Coming back to our capacitor system, the same thing happens. The gradient of U gives the force on the dielectric. (Of course the dielectric is also pulling the plates with an equal force)
 
  • Like
Likes   Reactions: gracy
  • #16
The gradient of Potential energy gives force on just one body and not on entire system. Potential energy gradient gives force on one of the bodies constituting that system. I understand now. Thanks @Aniruddha@94
 

Similar threads

Undergrad System, potential energy, and nonconservative forces
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Principle of Physics: Derivation of -dU/dx=F
  • · Replies 24 ·
Replies
24
Views
3K
Undergrad Conservative forces and internal and external forces
  • · Replies 9 ·
Replies
9
Views
3K
High School When to Use f = -dU/dx with External Energy Sources?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Why is F=dU/dx=0 either side of inflexion point?
  • · Replies 4 ·
Replies
4
Views
11K
High School Work Done By Conservative Forces
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Why are the Navier-Stokes equations inconsistent in this case?
  • · Replies 18 ·
Replies
18
Views
3K
Graduate Conservative forces, Nonconservative forces and Potential Energy
  • · Replies 9 ·
Replies
9
Views
4K
High School Why Do Objects Bounce? Exploring Momentum and Energy Conservation
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Does Compressing a Spring Decrease Its Potential Energy?
  • · Replies 4 ·
Replies
4
Views
2K