# Newton's Third Law by David Halliday and Robert Resnick

Figure 5-10 (a) Book B leans against crate C. (b) Forces ${\stackrel{\to }{F}}_{\mathrm{BC}}$ (the force on the book from the crate) and ${\stackrel{\to }{\mathrm{-F}}}_{\mathrm{CB}}$ (the force on the crate from the book) have the same magnitude and are opposite in direction.

Two bodies are said to interact when they push or pull on each other-that is, when a force acts on each body due to the other body. For example, suppose you position a book B so it leans against a crate C (Fig. 5-10a). Then the book and crate interact: There is a horizontal force ${\stackrel{\to }{F}}_{\mathrm{BC}}$ on the book from the crate (or due to the crate) and a horizontal force ${\stackrel{\to }{F}}_{\mathrm{CB}}$ on the crate from the book (or due to the book).This pair of forces is shown in Fig. 5-10b. Newton's third law states that

• Newton's Third Law: When two bodies interact, the forces on the bodies from each other are always equal in magnitude and opposite in direction.

For the book and crate, we can write this law as the scalar relation

FBC = FCB (equal magnitudes)

or as the vector relation

${\stackrel{\to }{F}}_{\mathrm{BC}}$ = ${\stackrel{\to }{\mathrm{-F}}}_{\mathrm{CB}}$ (equal magnitudes and opposite directions), (5-15)

where the minus sign means that these two forces are in opposite directions. We can call the forces between two interacting bodies a third-law force pair. When any two bodies interact in any situation, a third-law force pair is present. The book and crate in Fig. 5-10a are stationary, but the third law would still hold if they were moving and even if they were accelerating.

Figure 5-11 (a) A cantaloupe lies on a table that stands on Earth. (b) The forces on the cantaloupe are ${\stackrel{\to }{F}}_{\mathrm{CT}}$ and ${\stackrel{\to }{F}}_{\mathrm{CE}}$. (c) The third-law force pair for the cantaloupe-Earth interaction. (d) The third-law force pair for the cantaloupe-table interaction.

As another example, let us find the third-law force pairs involving the cantaloupe in Fig. 5-11a, which lies on a table that stands on Earth. The cantaloupe interacts with the table and with Earth (this time, there are three bodies whose interactions we must sort out).

Let's first focus on the forces acting on the cantaloupe (Fig. 5-11b). Force ${\stackrel{\to }{F}}_{\mathrm{CE}}$ is the normal force on the cantaloupe from the table, and force ${\stackrel{\to }{F}}_{\mathrm{CT}}$ is the gravitational force on the cantaloupe due to Earth. Are they a third-law force pair? No, because they are forces on a single body, the cantaloupe, and not on two interacting bodies.

To find a third-law pair, we must focus not on the cantaloupe but on the interaction between the cantaloupe and one other body. In the cantaloupe-Earth interaction (Fig. 5-11c), Earth pulls on the cantaloupe with a gravitational force ${\stackrel{\to }{F}}_{\mathrm{CE}}$ and the cantaloupe pulls on Earth with a gravitational force ${\stackrel{\to }{F}}_{\mathrm{EC}}$. Are these forces a third-law force pair? Yes, because they are forces on two interacting bodies, the force on each due to the other.Thus, by Newton's third law,

${\stackrel{\to }{F}}_{\mathrm{CE}}$ = ${\stackrel{\to }{F}}_{\mathrm{EC}}$ (cantaloupe-Earth interaction).

Next, in the cantaloupe-table interaction, the force on the cantaloupe from the table is ${\stackrel{\to }{F}}_{\mathrm{CT}}$ and, conversely, the force on the table from the cantaloupe is ${\stackrel{\to }{F}}_{\mathrm{TC}}$ (Fig. 5-11d).These forces are also a third-law force pair, and so

${\stackrel{\to }{F}}_{\mathrm{CT}}$ = ${\stackrel{\to }{\mathrm{-F}}}_{\mathrm{TC}}$ (cantaloupe-table interaction).

Checkpoint Suppose that the cantaloupe and table of Fig. 5-11 are in an elevator cab that begins to accelerate upward. (a) Do the magnitudes of ${\stackrel{\to }{F}}_{\mathrm{CT}}$ and ${\stackrel{\to }{F}}_{\mathrm{TC}}$ increase, decrease, or stay the same? (b) Are those two forces still equal in magnitude and opposite in direction? (c) Do the magnitudes of ${\stackrel{\to }{F}}_{\mathrm{CE}}$ and ${\stackrel{\to }{F}}_{\mathrm{EC}}$ increase,decrease,or stay the same? (d) Are those two forces still equal in magnitude and opposite in direction?

Answers: (a) increase; (b) yes; (c) same; (d) yes