# Work done by a non-conservative force and change in potential energy

I know that the work done by a non-conservative force is equal to the change in total mechanical energy (from Work-Energy Theorem). But I read in a place that "Non-conservative forces don't affect PE". So I am confused. How does the work done by a non-conservative force affect the potential energy?

• As far as I know it is not reasonable to introduce a "potential energy" for non-conservative forces. Could you clearify what this is supposed to be? Maybe I just missunderstood. – Hagadol Sep 21 '15 at 14:05
• Sorry, this is what I mean - WE.T states that Wc + wnc = !KE (! = change), Wc = -U, Wnc = !KE + U So is this U playing any role or is it just to show that the magnitude of Wnc is equal to (!KE + U)? Why do we say that PE is not defined for NC forces? – Shodai Sep 21 '15 at 16:23
• Let me explain:The work done by non-conservative forces is dependent on the path taken. Hence the work done by non-conservative forces to move an object from the initial point to a particular final point is different for different paths taken and hence the potential energy of the body at the particular final point is different for different paths taken. Hence it is not wise to define the potential energy of a body at a particular point for non-conservative forces since it will not be useful to us because of its different values depending on the path taken. – MrAP Oct 16 '16 at 15:01

## 2 Answers

Your question seems to arise from a problem in which there is both a conservative and a non-conservative force. When you say "PE" you must be referring to the PE of the conservative force (by definition there is no PE of a non-conservative force).

The work done by the conservative force does not depend on the path. Therefore you can define the potential as

$$\phi(x_0) - \phi(x) \equiv W_{x_0\to x}$$

Notice that:

• The potential is defined up to a global offset: you can arbitrarily choose the value $\phi(x_0)$ but afterwards any value of $\phi(x)$ is defined.
• This is a well posed definition just because $W_{x_0\to x}$ is a well defined quantity (depends only on $x_0$ and $x$, by definition of conservative force). This is not the case for the non conservative force.
• Why is PE not defined for NC, can you please tell me? – Shodai Sep 21 '15 at 16:21
• I hope that now it is clear – dPol Sep 23 '15 at 8:32
• I see, yes. Potential energy cannot be defined for any point because if I take the body from one point to another, then work done is different, and so one potential energy cannot be given to that point. Is that right? And so when we say that Wnc = Change in KE + U, then it just represents the magnitude of Wnc right? Meaning changing Wnc will only change KE, and not U? Thanks! – Shodai Sep 23 '15 at 15:26

In Newtonian Mechanics, the work-energy theorem states that net work = change in KE (not in PE).

Let's consider an example. Imagine a block of wood falling down on a rough ramp. The non-conservative force here is friction, and the conservative force is gravity. On a frictionless surface, gravity converts PE into KE while conserving the total energy. However, in the rough ramp that we are considering, the non-conservative friction does work on the block and thus decreases KE to 0 eventually(dissipation of energy).

The interesting point about work-energy theorem is that it doesn't rely on the force being conservative, while conservation of energy clearly does. So there is really no need to think about the concept of PE while using work energy theorem because PE is defined only for conservative force fields.

• Yes I get that the WE.T talks about net change in KE not PE but what I'm saying is - Wc + Wnc = Change in KE. >Wnc = Change in KE + U. Thus non conservative forces have a value equal to that of (change in KE + U). My question is, can this U arise due to just non-conservative forces? What is meant when we say that non-conservative forces do not have PE defined? – Shodai Sep 21 '15 at 16:20
• Take a conservative force field like the gravitational force field, the key to defining PE is that the work done from one point to another is PATH-INDEPENDENT. PE cannot be defined for friction, because if you travel a curved path instead of a straight path, the rough surface does more negative work to the moving body! Thus you cannot define a PE for every point on the surface. The equality that you brought up is a consequence of mixing up non conservative and conservative forces in one system. If a system is influenced exclusively by nonconservative work, the U term would not exist at all – Zhengyan Shi Sep 21 '15 at 16:53
• I still don't get it, why can NC forces not cause a change in PE? – Shodai Sep 22 '15 at 14:57
• Since fundamentally PE is a scalar field defined by conservative work. If you are doing non conservative work, it does not affect the PE. Going back to the ramp example, How much PE you lose is determined by how much work Gravity has done on you, not how much work friction has done. – Zhengyan Shi Sep 22 '15 at 15:06

## protected by Qmechanic♦Jan 6 '17 at 9:37

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