Newton's third law

Newton's third law says that if object $A$ exerts a force on object $B$, then $B$ exerts a force back on $A$ that is equal in magnitude and opposite in direction. In symbols: $\vec{F}_{B,A} = -\vec{F}_{A,B}$. You can't have one without the other, and the two are always the same size.

Two things about the pair are obvious; one is easy to miss. Obvious: the two forces are equal in magnitude and point in opposite directions. Easy to miss: they act on different objects. One arrow lives on $A$. The other lives on $B$. They never appear together on a single free body diagram.

That last point is what most third-law mistakes get wrong. "Equal and opposite" sounds like a recipe for cancellation, but cancellation requires both forces to act on the same object, and a third-law pair never does. Two balanced forces on a single object can sit at rest because they cancel on that object's diagram. Third-law partners never cancel on a single-object free-body diagram, because they are never on the same diagram in the first place. If you instead draw both interacting objects as one system, the pair becomes internal and does drop out of the system's force sum.

The cleanest way to track which force is paired with which is to name every force by its two endpoints. Read $\vec{F}_{A,B}$ as "force on $A$ by $B$." The first label is the object the force acts on; the second is the object exerting the force. The third-law partner is the same symbol with the subscripts swapped:

$$\vec{F}_{A,B} \quad\text{is paired with}\quad \vec{F}_{B,A}.$$

Same two letters, swapped roles. If you can't write a force in this form, you can't find its partner. The clearest sign that a force is misnamed is that its on-by labels won't swap.

An example. A book sits on a table. The Earth pulls the book down: $\vec{F}_{\text{book, Earth}}$. Swap the labels for its third-law partner: $\vec{F}_{\text{Earth, book}}$, the book pulling the Earth up. Same magnitude, opposite direction, different bodies. The book's weight is paired with the book's pull on Earth, not with anything the table is doing.

A second pair lives in the same setup. The table pushes the book up: $\vec{F}_{\text{book, table}}$. Swap for the partner: $\vec{F}_{\text{table, book}}$, the book pushing the table down. Two pairs, four forces, three bodies. The book's diagram shows two of those forces (weight and the normal force). The Earth's diagram shows one (the book's pull on it). The table's diagram shows one (the book's push on it). No diagram shows both members of the same pair.

Two questions look almost identical, and most third-law mistakes confuse them.

Question A: are these two forces a Newton's third law pair? Question B: do these two forces balance the object they act on? The two questions are about different things. Question A is a relationship across two bodies; Question B is a sum on a single body.

The test for Question A is the on-by swap. Write each force as "force on $X$ by $Y$." If one is "force on $A$ by $B$" and the other is "force on $B$ by $A$" (same two objects, swapped roles), they're a third-law pair. If both forces are on the same object ("force on $A$ by $B$" and "force on $A$ by $C$"), they are not a pair, no matter how perfectly they balance.

The test for Question B is to look at one body's diagram and add up the arrows. If they sum to zero, that body's net force is zero and it does not accelerate. That's Newton's first or second law on that body. The third law says nothing about whether any single body balances. A book at rest on a table has two balanced forces on its diagram (weight down, normal force up). They are equal in magnitude and opposite in direction. They are not a third-law pair, because both act on the book. They are a Newton's-first-law balance: two forces on one body that happen to cancel.

The phrase to remember: partners across two bodies, balance on one body. Same wording, different rules. Never confuse them.

Setup. A 2 kg book sits at rest on a flat table. List the forces, identify the third-law pairs, and explain why the book does not accelerate.

List the forces, one diagram at a time.

The book's diagram: the Earth pulls the book down, $\vec{F}_{\text{book, Earth}}$, of magnitude $mg = 20$ N. The table pushes the book up, $\vec{F}_{\text{book, table}}$, also 20 N.

The table's diagram: the book pushes the table down, $\vec{F}_{\text{table, book}}$, 20 N. The table has other forces on it (its own weight, the floor's normal force), but those are off-topic.

The Earth's diagram: the book pulls the Earth up, $\vec{F}_{\text{Earth, book}}$, 20 N. The Earth barely moves because its mass is enormous, but the force is real.

Find the third-law pairs. Use the on-by swap. $\vec{F}_{\text{book, Earth}}$ pairs with $\vec{F}_{\text{Earth, book}}$: same two objects, swapped roles. $\vec{F}_{\text{book, table}}$ pairs with $\vec{F}_{\text{table, book}}$: same swap on the table-book pair. Two pairs, four forces, three bodies.

Why the book does not accelerate. Look only at the book's diagram. The two forces on the book are equal in magnitude and opposite in direction, so the net force is zero and Newton's second law gives $\vec{a} = \vec{0}$. The book sits at rest. This is balance on a single body. It is not a third-law statement; the two forces on the book happen to balance, but they are not a pair.

The trap to avoid. A common wrong answer is to call $\vec{F}_{\text{book, Earth}}$ and $\vec{F}_{\text{book, table}}$ a third-law pair because they are equal and opposite. They are not. Both act on the book. The on-by swap fails: "force on book by Earth" doesn't have a partner that's "force on book by table." A pair has to swap the two object names.

Newton's third law has a direct consequence for the ropes and strings that show up everywhere in Unit 2. A rope tied at both ends pulls inward on whatever it is tied to, by an amount called the tension. For an ideal string (massless, inextensible), the tension is the same magnitude at every point along the string.

Why? Take any short segment in the middle of an ideal rope. Three things act on it: the rope pulling one way from the left, the rope pulling the other way from the right, and gravity. If the rope is massless, gravity on the segment is zero, so its net force has to be zero by Newton's second law. The two tensions at its ends are then equal in magnitude and opposite in direction. The same argument holds for every segment, so the tension is the same all along. A rope with mass is a separate problem.

An ideal pulley is massless and frictionless. The frictionless part means a rope over the pulley slides freely; the massless part means rotating the pulley costs no force. The result: an ideal pulley redirects the rope without changing its tension. A 5 kg block hanging from one side and a 5 kg block hanging from the other feel the same tension $T$ pulling them upward.

At each end of an ideal string, the rope's pull on the object has a third-law partner: the object's pull on the rope. Both forces are real and both have to appear when you list the third-law pairs in a problem with strings.