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I can sort of see why Newton's third law makes sense, since if I sit on an office chair and push at my desk, I move away from it. Apparently this happens because when I push the desk, the desk pushes me back with an equal and opposite force. I've been told that Newton came up with his law through similar experiences. But why must Newton's third law be true in the sense he defined force in his second law? While I can believe that when I push something, I'm also being pushed back, but why is it that the product of mass and acceleration must be the same for both of us (except for the sign)? I've asked this before and I've been shown that it works because of conservation of momentum in collisions. But this seems circular, as usually I have seen conservation of momentum in collisions derived using Newton's third law.

EDIT: I'm aware that there are other ways to produce conservation of momentum as a whole, such as Noethers theorem, but I'm more interested in knowing how the third law was thought of in Newton's time.

S. Rotos
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If two bodies are interacting, you must write the Newton equations for each body. The interaction force in one equation is the opposite to that in the second one. Those forces act on different bodies and crete different accelerations. It is not "the same force" - the forces are "external" for each body.

Now, if there is no other external forces but their interaction, the center of mass stays at rest, it is an experimental fact. Otherwise the couple of bodies, they say, is affected with another exterlal force.

Vladimir Kalitvianski
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While I can believe that when I push something, I'm also being pushed back, but why is it that the product of mass and acceleration must be the same for both of us (except for the sign)?

You can consider those two laws unrelated.

  • Newton's 3rd law is observed when you do experiments and momentum conservation considerations like you describe.

  • Newton's 2nd law can be observed by simply pushing with different forces at different objects. You will quickly realize that a doubled pushing force causes doubled acceleration. You will see it again and again in each experiment you do. So you start realizing that this "force being proportional to acceleration" is a general pattern - a law - of this world. When you then perform the same experiments on different masses, you will reach the same conclusion for force and mass.

Newton's 2nd law is thus directly created from empirical evidence. The only thing the 2nd and 3rd law have in common, is that we are talking about the same push, the same force.

Steeven
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At a first glance Newton's third law appears to be true but there are cases where it doesn't hold. One very simple example is that of Lorentz's forces acting on a charged particle due to the other.

Keep in mind that Newton's third law talks about forces only but nothing about fields. Hence a much more generalized way would be to talk about conservation of momentum (which holds true for both forces and fields).

A simple google search will tell you things in much more details!

Debobrata
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