# How a rigid body causes a reaction force?

• I
Gold Member
How so? You have to make an assumption about the forces to get momentum conservation. The 1st and 2nd law don't tell anything about the forces!
I naively think that if you have a system of particles Newton's 2nd law says ##\sum_i \dot p_i = F_{tot}## so, assuming the system to be isolated, you get that the total momentum is conserved. In a scattering event this implies ##(m_1v_1 + m_2v_2)_1 = (m_2v_2 + m_2v_2)_2##. Do I have some terrible misconception ?

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But how do you get that ##\vec{F}_{\text{tot}}=0## if you don't assume this through the 3rd Law?

• dRic2
Gold Member
ah I see. My life was a lie up until now.

• PeroK and Nugatory
Even for instantaneous interactions momentum conservation doesn't imply Newton's 3rd law, as soon you have 3 or more interacting bodies.
Can you give an example? I don't see this so easily.
This satisfies momentum conservation, but violates Newton's 3rd law:
F12 = 1N, F21 = -2N,
F13 = 2N, F31 = -2N,
F23 = 2N, F32 = -1N

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hence apply a reaction force?
It is not clear what you mean by "reaction force".

It sounds like you are thinking of Newton's third law as some kind of manifestation of Hooke's law -- that a force exerted on an object causes a deflection, that a deflection causes a stress and that the stress results in a "reaction" force. But Newton's third law does not have that sort of mechanical explanation. It is far more basic.

It is a statement that one object cannot influence another without itself being influenced. This does not happen as a result of a deflection. It happens during a deflection. It happens (for forces at a distance) even without a deflection.

The term "reaction" is unfortunate. It suggests that the force of one object on the other is a cause and the force of the other object on the one is an effect. The truth (at least in the Newtonian model) is that both forces occur together. Neither causes the other.

• Merlin3189, Omega0, Adesh and 2 others
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So, rigid body do deform and hence apply a reaction force?
If I run fast towards a concrete wall and collide with it, I am sure the wall will exert noticeable and painful reactive force on my body without suffering noticeable deformation itself.

When we put sustancial compression force on a bearing steel ball, it does not deform much; actually, it creates noticeable dents on flat surfaces of hardened steel that compress it.
The ball may deform some, but not much even when compressed with 20 tons of force.
At point of violent rupture, the multitude of sparks and speed of flying debris suggest that a huge internal stress had been created at molecular level.

• Merlin3189
Omega0
So, rigid body do deform and hence apply a reaction force?
I meanwhile lost the focus here, so it the problem that you want
to understand Newtons Laws? Or is is the basic conception of a rigid body?
Sorry, but I would love your update, what is the problem?

I meanwhile lost the focus here, so it the problem that you want
to understand Newtons Laws? Or is is the basic conception of a rigid body?
Sorry, but I would love your update, what is the problem?
My problem is how a Newton’s 3rd Law is caused? I mean if I apply a force on some block then how it exerts force on me? For Gravitational force and Electromagnetic forces “action-reaction” is quite obvious but on macroscopic level how can we give an explanation?

By the way, I have received so many great replies.

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This satisfies momentum conservation, but violates Newton's 3rd law:
F12 = 1N, F21 = -2N,
F13 = 2N, F31 = -2N,
F23 = 2N, F32 = -1N
Well, but does it fulfill all the requirements of a proper dynamical law within Galilei-Newton spacetime? I doubt it from using Noether's theorem, which leads to
$$V=\frac{1}{2} \sum_{i \neq j} V_2(|\vec{x}_i-\vec{x}_j|),$$
as long as only two-body forces are in the game.

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For Gravitational force and Electromagnetic forces “action-reaction” is quite obvious but on macroscopic level how can we give an explanation?
In the Newtonian model, forces, masses, displacements, velocities and accelerations all combine linearly. If you can accept Newton's third law at the microscopic level, it is a mathematical fact that Newton's third law at the macroscopic level follows.

The force of one macroscopic object on another is the sum of the forces of all the microscopic components of the one object on all the microscopic components of the other.

• Well, but does it fulfill all the requirements of a proper dynamical law within Galilei-Newton spacetime?
The question was if momentum conservation alone implies Newton's 3rd law. With additional requirements you can obviously deduce Newton's 3rd law.

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That's an interesting question indeed. I.e., you ask for the most general two-body force law if you consider only translation invariance. Of course, using the argument of the Noether theorem you also need to assume that you have a conservative interaction-force law. Then I think also Newton III follows, because the two-body potential is ##V_2(\vec{x}_1,\vec{x}_2)##, where ##\vec{x}_1## and ##\vec{x}_2## are the position vectors of the two particles. Of course you can uniquely rewrite it as a function ##V_2'(\vec{R},\vec{r})## with ##\vec{R}=\vec{r}_1+\vec{r}_2## and ##\vec{r}=\vec{x}_1-\vec{x}_2##. Now to make the Hamiltonian invariant under the translation by an arbitrary vector ##\vec{a}##, ##\vec{x}_1 \rightarrow \vec{x}_1`+\vec{a}## and ##\vec{x}_2 \rightarrow \vec{x}_2+\vec{a}##. This implies that ##V_2'(\vec{R},\vec{r})=\tilde{V}_2(\vec{r})## and thus
$$\vec{F}_{12}=-\vec{\nabla}_1 \tilde{V}_2(\vec{r})=+\vec{\nabla}_2 \tilde{V}_2(\vec{r})=-\vec{F}_{21}.$$

In the Newtonian model, forces, masses, displacements, velocities and accelerations all combine linearly. If you can accept Newton's third law at the microscopic level, it is a mathematical fact that Newton's third law at the macroscopic level follows.

The force of one macroscopic object on another is the sum of the forces of all the microscopic components of the one object on all the microscopic components of the other.
Yeah, I know that the contact force is actually Electromagnetic force between the constituent molecules.

But I think at the time of Newton this fact was quite unknown. So, is there any explanation why Newton’s Third Law occurs/exists purely on the basis of macroscopic level?

The question was if momentum conservation alone implies Newton's 3rd law. With additional requirements you can obviously deduce Newton's 3rd law.
That's an interesting question indeed. I.e., you ask for the most general two-body force law...
Two-body law is an additional restriction, beyond momentum conservation. You could have a three-body force law that satisfies momentum conservation, but not Newton's 3rd law.

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Have a look at this:

https://www.physicsforums.com/threa...y-when-carefully-analysed.979739/post-6263544

and the quoted manuscript about "many-body forces". Also there translation invariance leads to momentum conservation and an extended kind of Newton's 3rd postulate. Of course you are right, all that relies on Hamilton's principle and Noether's theorem and you may find forces that violate the 3rd law and momentum conservation. The only question is whether they describe anything in nature (as far as Newtonian mechanics can be used as an approximation).

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Yeah, I know that the contact force is actually Electromagnetic force between the constituent molecules.

But I think at the time of Newton this fact was quite unknown. So, is there any explanation why Newton’s Third Law occurs/exists purely on the basis of macroscopic level?
If you are digging for some underlying principle by which Newton's Third Law could be deduced from the physics and mathematics known to Newton, I would answer "there is none". It was a testable hypothesis consistent with the available physical evidence. Assuming its truth allowed for additional testable conclusions to be reached. That is pretty much all that one can ask of a physical principle.

It is in the nature of things that first principles cannot be deduced from nothing. They have to be guessed at based on experimental results. That's how science works.

• russ_watters, anorlunda, dRic2 and 4 others
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• Gold Member
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But that's correct. There's no rigid body in nature. It's only an (non-relativistic) approximation for very stiff (elastic) bodies, and indeed the reaction force comes from deformations of the body from its equilibrium state, when no external forces are acting. As stated some times before, the reaction force is electromagnetic and a consequence of the Pauli exclusion principle.

• Gold Member
How the second body exerted force on the first body (if it cannot be deformed)?
I have just dipped into the thread (far too many posts to stagger through them all) and this post demonstrates the problem. There is no such object that cannot be deformed - just give it as high a modulus as you like and you are back in the real world and there is nothing paradoxical at work. Every step function that occurs in scientific theory is a nonsense until we acknowledge that it's a justifiable approximation.

My Grandad, who was in no way a Scientist (I did love him), used to ask the old question "What happens when an unstoppable force meets an immovable object?" It made him feel smart that a teenager couldn't put him straight on the flawed philosophy behind the question. But, again, he had a problem understanding why we couldn't actually hear the Satelloon going overhead in the early 1960s; it "had to have an engine".

• Staff Emeritus
There is no such object that cannot be deformed
If you read the earlier posts, you would see discussions of free electrons, and other subatomic particles. AFAIK, we can't deform those, yet they obey the 3rd law.

Only late in the thread did the OP clarify that he meant only bulk objects in contact. Obviously, the 3rd law is valid for cases other than that one.

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Hm, subatomic particles are described by quantum mechanics rather than classical mechanics. It's a good question, in which sense (other than momentum conservation) the 3rd law is valid in QM, but that's another story...

• I have just dipped into the thread (far too many posts to stagger through them all) and this post demonstrates the problem. There is no such object that cannot be deformed - just give it as high a modulus as you like and you are back in the real world and there is nothing paradoxical at work. Every step function that occurs in scientific theory is a nonsense until we acknowledge that it's a justifiable approximation.

My Grandad, who was in no way a Scientist (I did love him), used to ask the old question "What happens when an unstoppable force meets an immovable object?" It made him feel smart that a teenager couldn't put him straight on the flawed philosophy behind the question. But, again, he had a problem understanding why we couldn't actually hear the Satelloon going overhead in the early 1960s; it "had to have an engine".

• sophiecentaur
Gold Member
If you read the earlier posts, you would see discussions of free electrons, and other subatomic particles. AFAIK, we can't deform those, yet they obey the 3rd law.
I think what we have seen in this thread is a typical PF treatment of a topic. I read the first post and it was very clear to me that the OP was asking about macroscopic structures being 'explained' with a mechanical version of microscopic structures and a possible problem with a too simple model. It's very easy for contributors to introduce ' what they know' about exceptions to an elementary / classical treatment and that can take a thread way off course from where it started. There are always lines of demarkation between mechanical, mathematical and quantum and I really feel it's up to contributors to try to avoid crossing over them unless absolutely necessary. Imo, the OP introduced or implied, perhaps the possibility of a problem in reconciling the mechanical with the mathematical. Once the gloves are off, we could end up talking about the mechanics of inside black holes / degenerate states - you name it. Would that help? It would definitely have given me a problem at A level.
All good fun though.

Let's face it, once we are dealing with fundamental particles, we mostly stick to the conservation laws and don't get too mechanical.

• Merlin3189
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I disagree. One has to clearly state that classical models have their limitations though in some aspects (particularly concerning linear-response theory of electromagnetic interactions, i.e., the usual macroscopic classical elctrodynamics taught in the intro-E&M lecture) they are quite accurate, though sometimes only qualitatively but as effective theories with phenomenological constitutive relations very valuable also quantitatively.

In this case, as usual as soon as properties of matter are concerned, you cannot answer the question without referring to QT. In this case it's also very easy to argue with relativity that there are no strictly rigid bodies in nature though it's a very useful and even quantitatively working non-relativistic model (with the tensor of inertia the "phenomenological constitutive parameters").

As Einstein said: "Make things as simple as possible, but not simpler!"

Gold Member
I disagree.
Which bit are you disagreeing with? Can you be disagreeing that an appropriate depth of treatment should always be used in education?This thread is surely about early steps in the understanding of the mechanical world and not doing the whole lot in one go. That would be too much for almost anybody.
Quoting Einstein (ad hominem) in a vacuum is not really helpful because there would have been a very relevant context to his remark.

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