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I have some trouble understanding Newton's Third Law. I tried understanding it myself (my teacher is an imbecile who accomplishes almost nothing in every lesson), and my failure to grasp the concept cost me dearly on a test. I need some outside help.

Here's what I understand about forces so far. Call me out on it if something's wrong.

So far, I can understand that "equilibrium" of an object is when there is no net force on an object, meaning that it is either at rest or moving at a constant speed, and that the net force acting upon it is zero.

I also understand that the net force is the difference of all of the forces acting on an object. For example when a 10 N eastbound force and a 30 N westbound force act on an object, the difference is 20 N and the object will be accelerated west.

Here's my problem:

For Newton's third law, my textbook describes two demonstrative scenarios: two ice skaters pushing against each other, and a ball on a table on Earth. In scenario 1, I can understand that there are two systems interacting upon each other. What I don't understand is where the points of the Third Law "all forces come in pairs" and "two forces in a pair act on two different object and are equal in magnitude but opposite in direction" are applied here. The second scenario (ball on a table on Earth) I can't understand at all. For the third law to work, there would have to be two exclusive system here, right? What are they? My book points out that the "upward force from the table to the ball" and the "downward force of gravity on the ball" are NOT an interaction pair. Why? It seems like that that would be a perfect example of the Third law.

Thanks in advance for any help.