# How a rigid body causes a reaction force?

• I
The definition of rigid body says it cannot be deformed (theoretically). Now, Newton’s third law is caused (I mean the reaction force is caused ) due to the deformation of the body.

What I have learned is that every body is like a spring, when we push on it we compress it and hence feel a reaction force (equal to action force).

At molecular level, when our hands comes closer to a rigid body the molecules of our hand and the rigid body apply Coulombic force on each other and it’s perfectly understandable how action and reaction is there. But can we explain it without going to molecular level? I mean can we explain it just by using that concept of deformation.

Rigid body doesn’t deform on being applied with a force, then how it pushes back to pusher?

• Frigus

Staff Emeritus
The definition of rigid body says it cannot be deformed (theoretically). Now, Newton’s third law is caused (I mean the reaction force is caused ) due to the deformation of the body
No no no. You get the action-reaction from a simple subatomic particle which does not deform.

No no no. You get the action-reaction from a simple subatomic particle which does not deform.
Means action and reaction can be explained only by that coulombic attraction/repulsion at molecular level?

Staff Emeritus
Means action and reaction can be explained only by that coulombic attraction/repulsion at molecular level?
No, it actually means that momentum is conserved. In this lecture, Professor Susskind does the math to show that the 3rd law is equivalent to conservation of momentum. No need to invoke gravity or Coulomb forces, it works for neutral particles too.

• Frigus
No, it actually means that momentum is conserved. In this lecture, Professor Susskind does the math to show that the 3rd law is equivalent to conservation of momentum. No need to invoke gravity or Coulomb forces, it works for neutral particles too.
So, can you please explain me How reaction force is actually caused?

Staff Emeritus
I don't know how much physics you have already. This is an I level thread.

Have you studied Noether's Theorem yet? If yes, then you can show that position translation symmetry leads to conservation of momentum.

I don't know how much physics you have already. This is an I level thread.

Have you studied Noether's Theorem yet? If yes, then you can show that position translation symmetry leads to conservation of momentum.
Wow! Today I learned a great thing from you, it’s the conservation of momentum that causes the Newton’s third law, not the deformation.

If we ignore the friction, if I push on a rigid body, it will start to move and since the initial momentum of the system (system is me and that rigid body , and I’m assuming everything was at rest initially) was zero therefore the final momentum also needs to be zero and hence I must move opposite to the rigid body and this motion must be caused by something and that something is the reaction force. Am I right and precise ?

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The definition of rigid body says it cannot be deformed (theoretically). Now, Newton’s third law is caused (I mean the reaction force is caused ) due to the deformation of the body.

What I have learned is that every body is like a spring, when we push on it we compress it and hence feel a reaction force (equal to action force).

At molecular level, when our hands comes closer to a rigid body the molecules of our hand and the rigid body apply Coulombic force on each other and it’s perfectly understandable how action and reaction is there. But can we explain it without going to molecular level? I mean can we explain it just by using that concept of deformation.

Rigid body doesn’t deform on being applied with a force, then how it pushes back to pusher?
First, Newton's third law is an incredible insight. When you push or pull something, you don't have a natural sense that it is pushing or pulling you (or at least I don't).

The first thing to realize is that this is a law of nature. It's fundamental to the concept of force. And, it's essentially independent of the mechanism. The third law doesn't come with a defined mechanism that applies in all cases.

Like any law of nature, you can then look at specific physical scenarios and analyse how the law is enforced. A force generated by physically pushing something must, ultimately, have a mechanism at a molecular level. But, then, the electrostatic forces themselves may be explained by a mechanism (quantum mechanical or otherwise).

In short, the third law doesn't come with a explanation of how nature obeys the law in all cases.

• Frigus, SammyS, Merlin3189 and 1 other person
In short, the third law doesn't come with a explanation of how nature obeys the law in all cases.
Yes, that’s a nice explanation. In Electromagnetism Newton’s 3rd law is no surprise but in macroscopic bodies it is a surprise (if we don’t assume conservation law to be more fundamental to Newton’s third law).

Staff Emeritus
I must move opposite to the rigid body and this motion must be caused by something and that something is the reaction force. Am I right and precise ?
Yes.

Have you seen movies of people in zero gravity? For example people on the space station. In zero gravity it is easier to see that if you push an object, that you start moving the opposite direction, thus keeping the system (object and you) with constant momentum.

On Earth, friction often dominates making it hard to see.

• Yes.

Have you seen movies of people in zero gravity? For example people on the space station. In zero gravity it is easier to see that if you push an object, that you start moving the opposite direction, thus keeping the system (object and you) with constant momentum.

On Earth, friction often dominates making it hard to see.
Yes, I have seen them. Thank you for teaching me that.

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In this lecture, Professor Susskind does the math to show that the 3rd law is equivalent to conservation of momentum
I intend to watch the video later, but I am a bit skeptical. If Newton's 3rd laws is equivalent to conservation of momentum then you could derive it starting from Newton's 2nd law (I think). It doesn't seems good because it would make Newton's 3rd law essentially redundant (because it would be a special case of the 2nd law).

An example: consider the scenario when you push something (like forniture) and it doesn't move. How can you "derive" Newton's 3rd law in this scenario using only the second law ? I can't find a way.

Gold Member
If Newton's 3rd laws is equivalent to conservation of momentum then you could derive it starting from Newton's 2nd law (I think).

Starting with Newton's Third Law you can derive conservation of momentum. But they are not equivalent because you cannot derive Newton's Third from conservation of momentum.

And you can't derive it from Newton's 2nd Law.

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What I have learned is that every body is like a spring, when we push on it we compress it and hence feel a reaction force (equal to action force).

This is true in Newtonian physics for the so-called contact forces like the normal force and the friction force. Both objects deform, so there is nothing to distinguish the so-called action from the reaction.

When a book rests on a table top, the book deforms the table top and the table top deforms the book. The book exerts a force on the table top and the table top exerts a force on the book. Either force could be called the action, either could be called the reaction.

In this sense, there really is no such thing as a truly rigid body. We very often model objects as rigid bodies because their deformation is negligibly small compared to other distances involved.

At molecular level, when our hands comes closer to a rigid body the molecules of our hand and the rigid body apply Coulombic force on each other and it’s perfectly understandable how action and reaction is there.

The molecules move closer together or further apart in response to those electromagnetic forces. That is the deformation.

• Frigus
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And you can't derive it from Newton's 2nd Law.

Staff Emeritus
Can't you derive conservation of momentum from the second law when F = 0?
Newton's First Law (not the 3rd law) is the special case of F=ma=0.

• Lnewqban
Gold Member

There's more to the 1st Law. It has, over the centuries, expanded it's limits of validity. At first it was a declaration that a state of uniform motion is equivalent to a state of rest. Since then it's been discovered that that idea is valid for electromagnetic phenomena as well as mechanical. The idea is now called the Principle of Relativity. Stated formally, it's the assertion that all inertial reference frames are equivalent. All that means, though, is that there is no way to distinguish between a state of rest and a state of uniform motion.

Staff Emeritus
There are other expert threads that dig deeply in the nuances of Newton's Laws. This is an "I" level thread. Let's keep it that way.

• hutchphd
Wait a moment! I understood from @anorlunda replies is that Newton’s Third Law is just a weaker form of Conservation of momentum. Although, in many books Conservation of Moemntum is derived from Newton’s Third Law but the reality is the other way round. Reaction force comes into play in order to ensure the momentum is conserved.

P.S. :- It’s now that I have received the notifs and by now @MisterT and @dRic2 have replied and reached a conclusion.

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I intend to watch the video later, but I am a bit skeptical. If Newton's 3rd laws is equivalent to conservation of momentum then you could derive it starting from Newton's 2nd law (I think). It doesn't seems good because it would make Newton's 3rd law essentially redundant (because it would be a special case of the 2nd law).

An example: consider the scenario when you push something (like forniture) and it doesn't move. How can you "derive" Newton's 3rd law in this scenario using only the second law ? I can't find a way.
You can reconstruct Newton's Laws from the fundamental symmetries of Galilei-Newton spacetime using Noether's theorem. This elucidate the issue very much and, even better, it also can be generalized to the special relativsistic case, where Newton's 3rd Law cannot hold, because there are no instantaneous actions at a distance, but momentum conservation of course holds, because Minkowski spacetime is invariant under spatial translations as is Galilei-Newton spacetime. This implies the convenience of field descriptions and locality of interactions in relativistic physics.

• dRic2
... in many books Conservation of Moemntum is derived from Newton’s Third Law but the reality is the other way round...
Newton’s Third Law implies Conservation of Momentum, but not the other way round.

Staff Emeritus
OK I misspoke when I implied that the relationship between the 3rd law and conservation of momentum was bidirectional. My bad for oversimplifying.

But this is still an I level thread. Let's keep the level of replies and the vocabulary at the I level.

The original question had to to with the origin of the reactive forces.

• SammyS
OK I misspoke when I implied that the relationship between the 3rd law and conservation of momentum was bidirectional. My bad for oversimplifying.

But this is still an I level thread. Let's keep the level of replies and the vocabulary at the I level.

The original question had to to with the origin of the reactive forces.
So, can we start once again?

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Newton’s Third Law implies Conservation of Momentum, but not the other way round.
Of course momentum conservation only implies Newton's 3rd law if you have instantaneous actions at a distance (see also my previous posting).

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So, can we start once again?
On a fundamental level all forces are through interactions, and in Newtonian physics you tacitly assume actions at a distance, i.e., forces (often derivable from a gradient, i.e., being conservative forces) are only given by the positions of the particles. It's also almost always a good approximation to assume that there are only pair interactions, i.e., in a many-body system the interaction forces are a some over two-body forces.

The paradigmatic example is Newton's gravitational force. For two bodies you have
$$\vec{F}_{12}=-\frac{G}{|\vec{x}_1-\vec{x}_2|^3} (\vec{x}_1-\vec{x}_2), \quad \vec{F}_{21}=-\vec{F}_{12},$$
i.e., the force goes like the the inverse square of the distance and is directed along the line connecting the bodies, being always attractive.

If you have a closed system, from Newton's 2nd Law you immediately get momentum conservation:
$$m_1 \ddot{\vec{x}_1}=\vec{F}_{12}, \quad m_2 \ddot{\vec{x}}_2=\vec{F}_{21}=-\vec{F}_{12}.$$
Defining the total momentum of the system as
$$\vec{P}=m_1 \dot{\vec{x}}_1 + m_2 \dot{\vec{x}}_2,$$
you indeed find
$$\dot{\vec{P}}=m_1 \ddot{\vec{x}}_1 + m_2 \ddot{\vec{x}}_2 = \vec{F}_{12} + \vec{F}_{21}=0,$$
i.e., the total momentum is conserved.

• Frigus
On a fundamental level all forces are through interactions, and in Newtonian physics you tacitly assume actions at a distance, i.e., forces (often derivable from a gradient, i.e., being conservative forces) are only given by the positions of the particles. It's also almost always a good approximation to assume that there are only pair interactions, i.e., in a many-body system the interaction forces are a some over two-body forces.

The paradigmatic example is Newton's gravitational force. For two bodies you have
$$\vec{F}_{12}=-\frac{G}{|\vec{x}_1-\vec{x}_2|^3} (\vec{x}_1-\vec{x}_2), \quad \vec{F}_{21}=-\vec{F}_{12},$$
i.e., the force goes like the the inverse square of the distance and is directed along the line connecting the bodies, being always attractive.

If you have a closed system, from Newton's 2nd Law you immediately get momentum conservation:
$$m_1 \ddot{\vec{x}_1}=\vec{F}_{12}, \quad m_2 \ddot{\vec{x}}_2=\vec{F}_{21}=-\vec{F}_{12}.$$
Defining the total momentum of the system as
$$\vec{P}=m_1 \dot{\vec{x}}_1 + m_2 \dot{\vec{x}}_2,$$
you indeed find
$$\dot{\vec{P}}=m_1 \ddot{\vec{x}}_1 + m_2 \ddot{\vec{x}}_2 = \vec{F}_{12} + \vec{F}_{21}=0,$$
i.e., the total momentum is conserved.
When you wrote ##F_{21}=-F_{12}## did you use Newton’s Third Law?

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Yes: ##\vec{F}_{12}## is the interaction force on body 1 and ##\vec{F}_{21}## the one on body 2. According to Newton III, ##\vec{F}_{21}=-\vec{F}_{12}##.

Yes: ##\vec{F}_{12}## is the interaction force on body 1 and ##\vec{F}_{21}## the one on body 2. According to Newton III, ##\vec{F}_{21}=-\vec{F}_{12}##.
How the second body exerted force on the first body (if it cannot be deformed)?

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I'm not sure, what you are after here. Any body can be deformed. There is no strictly rigid body in nature, which follows alone from relativity, because the velocity of sound is ##<c## and not ##\infty##.

If you mean contact forces, it's not that easy, because you have to take into account quantum mechanics. In everyday life the interactions among bodies are solely electromagnetic. If two bodies are in contact with each other you have in good approximation electrostatic repulsion and also the Pauli effect due to the indistinguishability of the fermionic electrons.

I'm not sure, what you are after here. Any body can be deformed. There is no strictly rigid body in nature, which follows alone from relativity, because the velocity of sound is ##<c## and not ##\infty##.

If you mean contact forces, it's not that easy, because you have to take into account quantum mechanics. In everyday life the interactions among bodies are solely electromagnetic. If two bodies are in contact with each other you have in good approximation electrostatic repulsion and also the Pauli effect due to the indistinguishability of the fermionic electrons.
So, rigid body do deform and hence apply a reaction force?

Gold Member
So, rigid body do deform and hence apply a reaction force?
Objects are made of atoms and they do deform. Rigid Body is an approximation of reality. This is important to understand. A perfectly rigid body does not exist, so the question "how can a rigid body apply a reaction force if it can't be deformed?" is meaningless. In the approximation of Rigid Bodies, Objects are perfectly rigid and do not deform, yet they apply somehow reaction forces. You don't care how that happens. If you wish to investigate further you have to abandon the hypothesis of rigid bodies and enter in the realm of elasticity. Which is a total different subject.

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• PeroK
Gold Member
Of course momentum conservation only implies Newton's 3rd law if you have instantaneous actions at a distance (see also my previous posting).
But in Newtonian Mechanics, can't you obtain conservation of momentum from the first and the second law ?