If two bodies exert forces on each other, these forces are equal in magnitude and opposite in direction.
This law is often addressed as the action-reaction principle, though the term "principle" might suggest we're dealing with a postulate. As such, it cannot be derived from more basic laws.
Since the "principle" of inertia has nothing dogmatic – it's just a particular case of the $2^{\rm nd}$ law, where the net force is zero – I thought the same could be said about the $3^{\rm rd}$ law.
A recurring example is that of the apple and the Earth.
An apple in free fall accelerates toward the Earth with a free fall acceleration. The force of the apple on the Earth also causes the Earth to accelerate toward the falling apple. By Newton's Third Law, the force of the Earth on the apple is exactly equal and opposite to the force of the apple on the Earth. (…)
The thing is… why does it have to be like this?
This is what I've thought, but I'm not sure if it's a fair way of looking at it, nor if it can be used in general.
Consider the apple-Earth system, with the apple still attached to its tree.
Nothing is moving, and since there are no external forces acting on it (forget the Sun, the Moon, the other planets, etc. for a moment), the net force at any instant should be zero.
Therefore, if it gains some internal forces at some point, like the weight force of the now falling apple, they must balance one another, otherwise an internal non-zero net force would make the whole system accelerate.
So.. qed?
I don't know. Is this approach valid in general? For example, are all the internal forces in a (not-accelerating) cloud balanced?
Best Answer
As you mentioned, the third law is just a postulate in classical mechanics. As a side note, a professor of mine would disagree with you that the first law is without independent content of the second law (although $a=0$ when $F=0$ is a special case of $F=ma$).
This is an extremely insightful observation, but also a reductio ad absurdum. It's true that it would be absurd for a system to spontaneously accelerate itself, but this is not excluded by Newton's first two laws on their own. You have basically rediscovered the concept of the conservation of momentum, which reiterates your point that, if there are no external forces, the center of mass cannot accelerate.
There is also the seemingly obvious but really nontrivial fact that the center of mass of a system obeys $\vec{F}_{\mathrm{tot}}=M_{\mathrm{tot}}a_{\mathrm{cm}}$. This is the fundamental reason why things like tennis balls also follow Newton's second law even though they are really composed of small particles which are the only things guaranteed directly by Newton's laws to obey $F=ma$. This fact is usually just assumed, but would not be generally true if we don't include Newton's third law.
I think the fact that $F=ma$ ought to be "inherited" by big objects as a property gives some kind of elegant intuition for why the third law needs to be included as a postulate.