Newton's Third Law of Motion

In Newton's first law we asked what happens when the net force on a body is zero and in second law we asked what happens if there is nonzero net force on a body. Newton's third law simply allows us to identify what forces act on a body.

A force can not appear alone. A force is an interaction between two bodies. It's an interaction which is not possible with a single body. When you kick a ball, the ball gets the force but only the ball is not involved; it's the interaction between you and the ball, that is you are required to exert force on the ball. You can not think about moving a ball by the ball itself!

A force always comes in a pair. If you kick a ball, the ball also kicks back on you with the same magnitude of force in opposite direction. If you pull a door, the door also pulls back on you. The result of any force (action) is the opposite force (reaction) of the same magnitude.

Figure 1 You exert a force on a wall.
Figure 2 A book on a table.

In Figure 1 you exert a force on a wall and the wall exerts the same magnitude of force on you in opposite direction. In Figure 2 a book is on a table. The weight of the book exerts downward force on the table and the table exerts the same magnitude of force on the book in upward direction. Both forces balance each other and the book stays on the table.

\[\begin{align*} {{\vec{F}}_{\text{wall on you}}}&=-{{\vec{F}}_{\text{you on wall}}} \\ \text{or,}\quad {{\vec{F}}_{\text{wall on you}}}+{{\vec{F}}_{\text{you on wall}}}&=0 \end{align*}\]

If you summarize the similar observations, you end up with a Newton's third law of motion which can be stated as

NEWTON'S THIRD LAW: If a body $A$ exerts a force on body $B$ (action), the body $B$ exerts the same magnitude of force on body $A$ (reaction) in opposite direction.

If body $A$ exerts force $\vec F_\text{AB}$ on body $B$, the body $B$ exerts the same magnitude of force $\vec F_\text{BA}$ on body $A$ but in opposite direction. The subscript $\text{AB}$ means the force of body $A$ on body $B$ and similarly $\text{BA}$ means the force of body $B$ on body $A$.

We know that every force in nature has a pair. And this pair includes two forces having the same magnitude but opposite direction. We call that pair action-reaction pair. If you apply a force on a body, you do action on the body and the body applies the same magnitude of force back on you as a reaction.

The statement of Newton's third law above is the modern statement of what Newton originally stated based on action and reaction forces as "for every action there is equal and opposite reaction". Here equal and opposite forces means equal magnitudes of forces interacting in opposite direction. The modern statement includes both action and reaction forces called action-reaction pair or interaction-pair or third-law pair.

Newton's third law always involves two forces and the most important thing to note is that these two forces never act on the same body. Two forces acting on the same body do not form an action-reaction pair. In Figure 2 a book rests on a table. The weight of the book acts on the book vertically downwards and the normal force of table acts on the book vertically upwards. In this case both of these forces act on the same body which is the book, so these two forces do not form an action-reaction pair.

The only action-reaction pair in Figure 2 is the force book exerts on the table due to its weight and the normal force the table exerts on the book. Here the forces are not applied on a single body but applied on the table by the book as an action and on the book by the table as a reaction. These forces act on each other forming an action-reaction pair. You should be careful that forces equal in magnitude and opposite in direction do not always form an action-reaction pair. So, in Figure 2 $m \vec g$ and $\vec n$ are equal in magnitude and opposite in direction but do not form action-reaction pair.

The book exerts force on the table and the table exerts the same magnitude of force on the book in opposite direction. It is not possible to exert force on a body without the body exerting force back in opposite direction.The action and reaction forces are not always contact forces. Newton's third law is also applies for long-range forces such as gravitational force.

Newton's third law involves a pair of forces acting on different bodies but Newton's first and second law involve forces acting on a single body. In Newton's third law, it is very important to determine the correct action-reaction pair. Newton's third law helps us identify the forces acting on a body and Newton's first and second law focus on all forces acting on a body to determine the net force.

Pushing a Car

When you push a stuck car on a way to your adventure, you exert force on the car and the car exerts the same magnitude of force on you in opposite direction. When the car moves, the car still exerts the same magnitude of force back on you. How is this possible? If both forces are equal in magnitude and opposite in direction, how the car can move forward? The answer lies to the correct interaction pair or action-reaction pair.

Remember that these two forces do not act on the same object. One force acts on the car (you exert on car) and the other force acts on you (car exerts on you). If we go to the atomic level, while you exert force on the car, the atoms of your hand and the car interact like very tiny springs and there is always the same magnitude of force at both ends of a compressed or stretched spring.

In Figure 2 the net force on the book is zero but Newton's third law is equally valid even if the net force is not zero. For example, you are pushing to move the car. You exert force on the car and the car exerts force of equal magnitude back on you in opposite direction. Once your force is greater than the force car exerts on you, the car moves. Now your force is greater and car can not provide enough force back on you and to provide that force the car moves. Newton's third law applies whether two bodies are stationary or moving (moving with constant velocity or accelerating).

Mechanics
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