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Now I was reading The Feynman Lectures on Physics and found this which I found somewhat peculiar and deep and thus want your assistance here. So here it goes:

The theorem concerning the motion of the center of mass is very interesting, and has played an important part in the development of our understanding of physics. Suppose we assume that Newton’s law is right for the small component parts of a much larger object. Then this theorem shows that Newton’s law is also correct for the larger object, even if we do not study the details of the object, but only the total force acting on it and its mass. In other words, Newton’s law has the peculiar property that if it is right on a certain small scale, then it will be right on a larger scale. If we do not consider a baseball as a tremendously complex thing, made of myriads of interacting particles, but study only the motion of the center of mass and the external forces on the ball, we find $F=ma$, where $F$ is the external force on the baseball, $m$ is its mass, and a is the acceleration of its center of mass. So $F=ma$ is a law which reproduces itself on a larger scale.

Now here, I do understand that the theorem of center of mass reproduces itself on a larger scale and can figure out why it is so, but I fail to understand how this theorem leads to the conclusion that newton's laws of motion also have this peculiar property.

Other than this, I want to know why Newtonian laws have this replicating property. Is it merely an experimental fact which we have observed and encountered every time we use Newtonian mechanics? Or is there something subtle in the laws themselves which grants them this property of replication on larger scales.

PS: I would request you all to avoid use of concepts of quantum mechanics or something advanced as I'm not in a position to understand that all now. I am only familiar with Newton's laws.

I ask for your help in this regard.

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  • $\begingroup$ Correct me if I am wrong: You just want to prove that the net force acting on a body is the same as that acting on the center of mass of the system. $\endgroup$ – SchrodingersCat Dec 22 '17 at 17:24
  • $\begingroup$ Because of linearity: $(M+m)a=Ma+ma$. $\endgroup$ – PM 2Ring Dec 22 '17 at 22:27
  • $\begingroup$ @SchrodingersCat That's not what I want. I know how we can show that the acceleration of center of mass is equivalent to the acceleration of a particle with the total mass of system and acted upon by the resultant force on system. Now, what this shows is that F=ma for center of mass and surely this can be considered as extension of Newton's second law on a larger scale if we introduce this concept of center of mass. Now what about the 1st and the 3rd law. That's precisely what I want. $\endgroup$ – Abhinav Dhawan Dec 23 '17 at 4:52
  • $\begingroup$ And I would like to remark that I really liked the answer of joshphysics where he explained how 1st law is not a consequence of 2nd law, and rather a beautiful assertion about the existence of inertial reference frames. I would like to have similar distinction made here while showing the extension of Newton's 1st law on larger scales. $\endgroup$ – Abhinav Dhawan Dec 23 '17 at 4:57
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To analyze further the example of the baseball, what you are interested in during a game is the mean movement of all particles composing it. Each individual particle may be moving randomly inside the ball at very high speeds but you can not see them directly nor it is relevant to the overall motion of the ball on a human scale.

Now suppose that each individual particle composing the baseball obeys Newton's law. The theorem states then that when considering the whole ball the center of mass also obeys Newton's law, with a mass that is the mass of whole ball, and force that is the total force acting on the ball, that is the sum of all forces acting on every particle.

Here it is the crucial point: from a distance you won't be able to resolve the ball and all you see is a sphere moving. Since reasonably all particles will stay in this sphere uniformly, its central point will also be the center of mass geometrically. But the center of mass will move according to Newton's law and, if the sphere doesn't deformate or tear apart, so will its central point, since they coincide! So the sphere as a whole, for its rigidity, will move according to Newton's law.

The theorem is of course always true, but it may not always be interesting or insightful. For example, take two non interacting balls at rest, and kick one so that it starts moving. The motion of the center of mass is predictable, but it's not really describing anything about the "system" as a whole and most importantly is not directly observable. It is just a property of the two balls combined.

As a less mundane example, what you can infere is, assuming that everything that composes Earth obeys second law, Earth itself is a rigid body obeying the second law. And if every little piece of matter in the solar system obeys this law, you can study and predict the motion of the planets if you know the force that each one exerts on the others.

What you can not infere is, since all ordinary matter appears to be obeying Newton's law, then also atoms or sub-atomical particles obey the same law. The theorem is valid in one direction only.

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I don't know what Faynman had on his thought, but definitely relation between parts of the system and center of the mass is caused by:

  1. Additivity of forces.
  2. Linear relationship between force, mass and acceleration.

Linearity means mass and force depends linearly on each other. As such, sum of masses gives you total mass, and sum of forces gives you total force, and they are still proportional ( when body is rigid).

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It's a consequence of Newton's second and third laws and the superposition of forces. Newton's second law implicitly assumes the superposition of forces. Otherwise, Newton's second is just a definition of net force. That the net force acting on a particle is the sum of the individual forces acting on that particle is a corollary in Newton's Principia. Several physics instructors explicitly treat it as a fourth law of motion; others do so implicitly by teaching about vectors before discussing Newton's laws of motion.

What about a system of particles? The center of mass $X$ of a system of particles is defined as $$M \boldsymbol X = \sum_i m_i \boldsymbol x_i$$ where $M = \sum_i m_i$, $m_i$ and $\boldsymbol x_i$ are the mass and position of particle $i$, and the summations are over all the individual particles that comprise the system.

Assuming that the number of particles and that the mass of each particle remains constant over time enables differentiating twice with respect to time: $$M \ddot{\boldsymbol X} = \sum_i m_i \ddot{\boldsymbol x}_i$$ Newton's second law enables rewriting the right-hand side as the net force acting on the $i^{th}$ particle: $$M \ddot{\boldsymbol X} = \sum_i \ddot{\boldsymbol F}_{\text{net},i}$$ where $F_{\text{net},i}$ is the net force acting on the $i^{th}$ particle. The superposition of forces means this net individual force can be resolved as a sum of external and internal forces: $$F_{\text{net},i} = F_{\text{ext},i} + \sum_{j\ne i} F_{j,i}$$ where $F_{\text{ext},i}$ is the sum of the external forces acting on particle $i$ (forces attributable to the external environment as opposed to interactions amongst the particles that comprise the system). Those inter-particle interactions are captured by the $F_{j,i}$ internal forces. Thus $$M \ddot{\boldsymbol X} = \sum_i \boldsymbol F_{\text{ext},i} + \sum_i \sum_{j\ne i} \boldsymbol F_{j,i}$$ This is where Newton's third law comes into play, which says that $\boldsymbol F_{i,j} = -\boldsymbol F_{j,i}$. This means the second sum on the right ($\sum_i \sum_{j\ne i} F_{j,i}$) is identically zero, leaving only $$M \ddot{\boldsymbol X} = \sum_i \boldsymbol F_{\text{ext},i} \equiv \boldsymbol F_\text{ext,tot}$$

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