In this explainer, we will learn how to apply Newton’s third law of motion to analyze systems of forces.
Imagine a tennis player swinging a racket and hitting a tennis ball. We know the racket exerts a force on the ball. It is also true, however, that the ball exerts a force on the racket, which is noticeable to the player. In fact, the force the ball exerts on the racket is equal in magnitude and opposite in direction to the force the racket exerts on the ball, as shown below.
This mutual exertion of force is an example of Newton’s third law of motion.
Definition: Newton’s Third Law of Motion
For a pair of interacting objects, Newton’s third law of motion states that the force exerted by the first object on the second is equal in magnitude and opposite in direction to the force exerted by the second object on the first.
Unlike the first two laws of motion, Newton’s third law of motion focuses on pairs of interacting objects. These could be two billiard balls colliding, a rocket ship launching, or even a moon orbiting a planet. Newton’s third law of motion describes the forces involved in these interactions.
Sometimes this law of motion is stated this way: “For every action, there is an equal and opposite reaction.”
Here, “action” and “reaction” refer to the forces that arise whenever two objects interact. Considering the cases listed above, we can say that these forces may be contact forces (as with colliding billiard balls and the rocket and exhaust) or noncontact forces (as in the case of gravitational attraction). The pairs of equal and opposite forces in each scenario are shown below.
Note that what force is considered the “action” and what force is considered the “reaction” is arbitrary. For example, there is no particular reason to call the force on the left billiard ball the “action” force and the force on the right ball the “reaction” force, or the reverse. Likewise for the rocket and exhaust, and in general.
The important point rather is that these forces act on different objects, in opposite directions, with equal strength.
Another pair of interacting objects we might consider is a book at rest on the ground. Two pairs of forces are involved.
First there is the gravitational force on the book due to Earth and the gravitational force of the book on Earth, depicted below. In accordance with Newton’s third law of motion, these forces are equal and opposite.
Likewise, the normal reaction force of the book on Earth and the normal reaction force of Earth on the book are equal and opposite, as follows.
Note that in the context of Newton’s third law, we may use the word “reaction” to mean very different things.
One meaning is that reaction force is one of the forces in a pair of forces of any kind between two interacting objects.
The other meaning is the contact force that acts on solid objects that prevents them from passing through each other, which we often call a normal reaction force.
Newton’s third law of motion holds true even when the masses of two interacting objects are very different. A tennis racket is about 6 times more massive than a tennis ball, but when they collide, the ball exerts just as much force on the racket as the racket does on the ball.
Even though the forces are equal in magnitude, the acceleration experienced by the racket and the ball is not the same. We can see this by considering Newton’s second law of motion,
Given a certain applied force strength, the less mass an object has, the more it will accelerate in response to the force. For a racket colliding with a tennis ball, then, the ball would accelerate roughly 6 times more in magnitude than the racket.
Combining Newton’s third and second laws of motion can lead to surprising results.
Imagine that instead of interacting with a tennis racket, our ball is interacting with Earth through gravity, as follows.
If the ball is tossed in the air, at any given moment Earth pulls on the ball and the ball pulls on Earth—and Newton’s third law of motion states that these pulls are equal in strength.
Since the force on the tennis ball is proportional to its mass, Newton’s second law of motion tells us the ball will accelerate enough for us to notice—we will see it fall back to Earth.
But what about Earth? It has that same force magnitude acting on it, but it is many times more massive than the tennis ball. If we divide the force strength by Earth’s mass, the acceleration we get is so small that, for all practical purposes, it is zero.
Mathematically, however, it is not exactly zero. In this situation, Earth accelerates a very small amount toward the ball in the air!
To see that this really is true, imagine our tennis ball increased in mass until its mass was equal to the mass of Earth, as follows.
These equal masses would attract one another equally and cause equal acceleration in the other. Realizing Earth would accelerate due to the pull of a comparable mass shows that even if that mass was not comparable, both objects, including Earth, would accelerate under the influence of gravity.
To better understand Newton’s third law of motion, let’s now consider some potential descriptions of that law.
Example 1: Newton’s Third Law of Motion
Which of the following statements most correctly describes Newton’s third law of motion?
- When a force is applied to an object, the object exerts an equal-sized force in the opposite direction to the applied force.
- When a force is applied to an object, the object exerts an equal-sized force in the direction of the applied force.
- When a force is applied to an object, the object exerts a force in the opposite direction to the applied force that is proportional to the mass of the object that applies the force.
- When a force is applied to an object, the object exerts a force in the opposite direction to the applied force that is proportional to the mass of the object that the force is applied to.
Let’s recall the definition of Newton’s third law of motion: For a pair of interacting objects, the force exerted by the first object on the second is equal and opposite to the force exerted by the second object on the first.
Options D and C both claim that the reaction force in an action-reaction force pair depends on mass—either the mass of the one object of the mass of the other.
Strictly speaking, the applied force may in some way be related to the mass of the object applying it. For example, if the applied force is gravitational, then that force is proportional to the mass of the object generating it.
Therefore, the force exerted in response, which Newton’s third law of motion states is equal in magnitude to the force applied, may indirectly depend on the mass of the object applying the force.
Generally, though, this is not the case, particularly for applied forces that do not depend on object mass.
We could say then that option C is more likely to be true than option B but that neither one of them is a broadly accurate description of Newton’s third law of motion.
Option B claims that the two forces act in the same direction. We know, though, that in reality the forces in the pair act in opposite directions.
Option A describes equal forces acting in opposite directions without regard to any object properties such as mass. This is the correct answer.
Now let’s look at how Newton’s third law of motion can help us understand a system of interacting objects at rest.
Example 2: Newton’s Third Law of Motion
An object with a weight is attached to a string. The other end of the string is attached to a spring, as shown in the diagram. The spring is stretched until it comes to rest.
- How much vertically downward force does the string apply to the spring?
- How much vertically upward force does the spring apply to the string?
- How much vertically upward force does the string apply to the object?
We can see that the 20 N object is in equilibrium. This means there is a 20 N force acting upward on the weight that counteracts its 20 N downward force. The string supplies this force, as shown below.
If we consider the forces acting on the string (rather than on the weight), we realize there is a downward-pointing 20 N force that must act on the string due to the weight of the object.
Like the object, the string is in equilibrium. Therefore, the spring must exert an upward-pointing 20 N force on the string, as follows.
Finally, considering the forces acting on the spring, we can see the string must be applying a downward-directed 20 N force. Since the spring is at rest, the surface to which it is attached must therefore exert an equal and opposite force on it. The forces acting on the spring appear as shown below.
Using this diagram, we can answer the given question. The downward force the string applies to the spring is 20 N.
We return to the diagram showing the forces that act on the string specifically. That diagram appears as follows.
We see the string being acted on by the spring with a vertically upward force of 20 N.
The force of the string on the object is depicted in the following diagram showing the forces that act on the object only.
In this figure, we see that the string supplies an upward force of 20 N on the object.
Example 3: Newton’s Third Law of Motion
Two balls of equal mass collide head-on, as shown in the diagram. The balls move at speeds and , respectively, where is greater than .
Which of the following diagrams correctly represents the reaction forces that would act when the balls collide, ignoring any objects other than the two balls?
Newton’s third law of motion states that whenever a pair of objects interact, the force from the first object on the second object is equal in magnitude and opposite in direction to the force of the second on the first. This holds true whether or not the objects have the same mass or are moving at the same speed.
We can expect that the force of the gray ball on the green ball will be equal and opposite that of the green ball on the gray ball. Reviewing our answer options, we see only one option (E) claims that the two forces have equal magnitude. This option also shows the forces acting in opposite directions, as we know they should. Our answer then is option E.
Let’s now summarize what we have learned in this explainer.
- Newton’s third law of motion applies to interactions between pairs of objects, rather than individual objects by themselves.
- Given two objects A and B, Newton’s third law states that any force object B exerts on object A is equal in magnitude and opposite in direction to any force object A exerts on object B.
- The forces that act on pairs of objects that interact can both be attractive or both be repulsive.
- The forces that act on pairs of objects that interact can both be contact forces (reaction forces due to a collision or forces between bodies at rest), or they can both be forces that act at any distance (like gravity).