Newton's third law

A force is one object acting on another. The third law says that interaction is never one-sided. When object A pushes or pulls object B, object B pushes or pulls A right back, with the same size and the opposite direction. The two forces appear and disappear together. Neither one is the "real" force and the other an echo.

A naming trick saves a lot of pain. Write every force as "force on [recipient] by [agent]". The third law then guarantees a partner: "force on [agent] by [recipient]". Same size, opposite direction, on the other body. The partner's name is just the original with the two objects swapped.

The figure below shows the gravitational pair between Earth and a book. The familiar view is what you draw on a free-body diagram: gravity on the book, pointing down. The pair view shows the second half of the same interaction, an equal-size upward force on Earth by the book.

Two things to notice. First, the partner is on the other body. Always. The on-by syntax forces this: if you wrote the original as "on book," the partner reads "on Earth." Same body never appears twice. Second, the magnitudes are exactly equal regardless of how the masses compare. Earth is huge and the book is tiny, but the gravity forces in the pair are the same size. The masses change the resulting accelerations, not the forces.

Naming a force in on-by form makes its partner instantly readable. Three steps.

1. Name the force in on-by form. Write it as "force on [A] by [B]." Be specific about both objects and the type of interaction (gravity, contact, friction, tension, electric, magnetic).
2. Swap A and B. "Force on [A] by [B]" becomes "force on [B] by [A]." That sentence names the partner. The two objects literally trade places, which is why partners always live on different bodies.
3. Sanity-check. Same interaction type, equal size, opposite direction. The "different body" part is already baked into the syntax flip, so you don't have to check it separately.

A worked application. Original: "force on the book by Earth," pointing down, magnitude $mg$. Swap: "force on Earth by the book," pointing up, magnitude $mg$. Same interaction (gravity). Equal size. Opposite direction. Done.

Why the naming scheme matters: it forces you to identify both objects in the interaction up front, and it pins down which body the force acts on. A loose label like "the gravity force" hides both. The on-by form also doubles as a free-body diagram filter. Every force on an object's FBD must start with "on [that object]." If you wrote "on Earth" and you're drawing the book's FBD, that force does not belong on the diagram.

Most third-law pairs in AP Physics 1 fall into a small number of recurring shapes. Use this as a lookup while you work.

A small note about ropes. An ideal rope has negligible mass, so the tension on the object end and the tension on the anchor end are equal in size. Each end has its own pair. Treating the whole rope as a single line of force is fine, but the partners of the two end forces still live on different bodies.

Setup. A textbook of mass $m$ rests on a flat table. The force on the book by Earth is $F_g = mg$, pointing down. What is its third-law partner?

Now the question almost every student asks: "Isn't the table's normal force the partner of gravity? They're equal in size, point in opposite directions, and the book is at rest because of them." The answer is no, and the on-by syntax shows why. Both forces start with "on book": force on book by Earth and force on book by table. They act on the same body. Two forces on the same body are never a third-law pair, even when they look identical to one.

What gravity and the normal force do have in common is that they happen to be equal-and-opposite for a book sitting still. That's Newton's first law: the book is in equilibrium, so the net force on it is zero, so the two forces on it cancel. The third law says nothing about cancellation. It only guarantees that any one force has a partner of equal size on the other body.

The partner of the table's normal force on the book, by the swap test, is force on table by book, pointing down into the table. Different pair, different two bodies, different force.

Setup. A horse pulls a cart along a level dirt road. The harness connects them. The force on the cart by the horse, $\vec F_{C,H}$, points forward. By the third law, the partner is the force on the horse by the cart, $\vec F_{H,C}$, equal in size and pointing backward. How can the cart accelerate forward if the two forces are equal and opposite?

This is the classic paradox, and it dissolves the moment you remember which body each force acts on.

Notice what never happened. $F_{C,H}$ and $F_{H,C}$ never appeared in the same FBD. They couldn't have, because they act on different bodies. Cancellation requires forces on the same body. Equal-and-opposite forces in a third-law pair contribute to two different free-body diagrams, so they never get a chance to cancel each other.

Three failure modes show up over and over on AP problems involving Newton's third law. Two are different flavors of confusing which body a force acts on. The third is the classic mass-asymmetry slip.