Newton’s Third Law - Mcat

In the classes I teach, a disconcerting number of students don’t really understand Newton’s third law of motion and get confused by the statement:

"For every action, there is an equal and opposite reaction."

In fact, the law would be more accurately expressed this way:

"For every force, there is an equal and opposite force."

Or, in mathematical terms:

"F1 on 2 = -F2 on 1"

For a book lying on a table (i.e., stationary), it is commonly thought that Newton’s 3rd Law is when the Normal Force and the weight of the book (m*g) cancel each other out and therefore the book goes nowhere. Actually, these two forces are not related by the Third Law. Given a (gravitational) force between the Earth and the book, the Third Law implies a force between the book and Earth, not the not the book and the table. Consider the math:

FG = G(MEarth)(Mbook)/ r^2

You usually learn that acceleration due to gravity, g, is

g = GMEarth/r^2

and the force due to gravity, aka weight, is mg. Newton’s 3rd Law states that the book must exert the same amount of force on the Earth… and it does!

acceleration (a) of due to the book = G(Mbook)/r^2.

Since in each case you’re using masses of the book and earth for accelerations, the forces come out the same when you multiply by the mass of the other body. Forces are masses * accelerations, which means you will arrive at the same force whether or not you’re using g or the acceleration due to the book. We’ve gotten used to just thinking about acceleration as g (which is immensely useful, no doubt), but it puts up blinders when thinking about forces, most notably weight.

Using the gravitational equation, you can figure out the force on the book due to the earth, as well as the force on the Earth due to the book. Each time, you will find the magnitude to be the same (although different directions). Using mg incorporates the mass of the book (or whatever) into the gravitational equation. Because of this, whenever you figure out the forces on the book or on the Earth, you end up using all the same variables. Try it out! This is what Newton meant by the Third Law.

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