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If one defines force as the time derivative of momentum, i.e. by

$$ \vec{F}= \frac{d}{dt} \vec{p} $$

how can this include static forces? Is there a generally accepted way to argue in detail how to get from this to the static case? If not, what different solutions are discussed?

Edit: I shoud add, that by static forces I mean forces involved in problems where bodies don't move.

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There is no such thing as static and dynamic force. Could you clarify what you mean? I suppose you're talking about multiple forces acting on the same system such that in total they produce zero net force (e.g. gravity vs. reaction from the ground acting on the chair you're sitting on)? – Marek Jul 31 '11 at 17:04
What you are missing here from Newton's laws is the sum of forces. – ja72 Aug 1 '11 at 12:44
up vote 3 down vote accepted

I don't exactly know what you mean by static forces. But I am going to take a wild guess here and assume that by that you mean forces involved in problems where bodies don't move. I think you assumed that Newton's second law quantifies a force. This is actually wrong. First of all realize that a force is an interaction and it still acts whether the body on which it acts moves or not. Newton's second law quantifies the total effect of all such forces on a body of mass $m$ and not the force itself. For example the Newton's law of Gravitation tells you that the force between two masses is:

$$\vec{F} = G\frac{Mm}{r^2}$$

Now this is practically useless unless you specifies what a force does on a body. That's where Newton's second law comes in. So along with Newton's second law, you have a complete theory of (classical) gravitation.

Also the $\vec{F}$ in Newton's second law is the total force acting on a body having momentum $\vec{p}$. So when bodies don't move the net forces on them is zero. But that does not mean that you can not have forces acting on it.

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Yes I want to take Newton's second law as a definition of Force. I came across this question when studying the Karlsruhe physics course, where force is defined in this way. If I am not mistaken this goes back to Ernst Mach and is used as definition of force in some textbooks as well, for example in the german books by Demtröder or Fließbach. However my problem is that I don't see how one can take this as a definition and also to be able to describe static situations with this concept. – martin Aug 1 '11 at 7:54
@martin Ernst Mach defined a force in an entirely different way. According to Mach when two bodies interact (assuming they are the only bodies which do so) the following equation holds: $m_1/m_2 = -a_2/a_1$. Thus it is meaningful to define a force as $m\vec{a}$. But note that this form of the law isn't as useful as Newton's second law as you consider only two particles. And clearly a system containing just two objects won't be static. So Mach's definition (as its normally given) can't be practically used to solve such problems. – Bernhard Heijstek Aug 1 '11 at 12:21

By "static forces" I think you mean forces within a physical system in static equilibrium.

You get static equilibrium out of the kinetic effects of force by noting the sum of the forces at every point is zero and therefore so is the sum of their kinetic effects.

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