Physics Tutorial: Newton's Third Law of Motion

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In this Physics tutorial, you will learn:

  • What are action and reaction?
  • What does Newton's Third Law of Motion says?
  • Why two forces that are involved in an action-reaction process do not cancel each other?
  • What is the condition that two forces obey the action-reaction principle?
  • What are some examples of action-reaction principle?
  • Why not all forces do appear in a force diagram?
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4.7Newton's Third Law of Motion

Introduction

When we punch a wall, our hand gets hurt. Why do you think this occurs?

Why we don't experience the same effect when we push gently an object with a finger? Try to explain.

The Meaning of Action and Reaction

In the Physics tutorial "What Causes the Motion? The Meaning of Force", we gave the following definition of force:

"A force is any interaction that, when unopposed, will change the motion of an object."

Please note the word "interaction" used in the above sentence. By definition, an interaction means "a mutual or reciprocal action." This means that when an action occurs from an object A to another object B, there is a reaction from the object B to the object A as well.

But how such action and reaction are related to each other? We should seek the answer for this question in the simplified definition of Newton's Third Law of Motion, which says:

"For every action, there is an equal size but opposite reaction."

If we consider again the punch-the-wall example mentioned in the Introduction paragraph, we can say that the force we use to punch the wall represents the "action", while the force exerted by the wall on our hand represents the "reaction."

Definition of Newton's Third Law of Motion

We can define the Newton's Third Law of Motion in a more comprehensive manner as follows:

"When one body (object) exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body."

The Newton's Third Law of Motion is also known as the "Action-Reaction Principle."

Mathematically, we can write:

FA-B= - FB-A

Look at the figure below:

Physics Tutorials: This image provides visual information for the physics tutorial Newton's Third Law of Motion

In the figure above, one man A exerting a force FA-B on the cart B is shown. At the same time, the cart B exerts an equal but opposite force - FB-A on the man. The sign minus of the force exerted by the cart on the man stands for the direction; it represents the opposite direction to the positive force exerted by the man on the cart.

The one million dollar question arisen here is: "Since the action force and the reaction that stems from this action are always equal and opposite, why they are not balanced? Why when we push a box, it moves away but we remain at rest?

The answer is: "Because action and reaction forces do not act on the same object and therefore, they cannot be balanced. Remember from the previous tutorial "Newton's Second Law of Motion" that two forces are balanced when they are equal and opposite and they act at the same object, so that the resultant force is zero. In the action-reaction principle (Newton's Third Law of Motion) it is different; there are two forces acting at two different object (FA-B and FB-A). Therefore, they cannot be balanced as the Newton's Second Law of Motion is applied for each object separately. As a result, the accelerations these forces cause depend on the masses of the objects involved, as the forces are equal. In few words, despite the action and reaction forces are equal in magnitude, they cause different effects in the objects they act (more precisely, in their acceleration) because these objects have different masses.

Let's explain this point through a daily life example. Suppose you went for a trip with your friend for a couple of days. You spent ₤200 per person during the trip. Prior to the trip your savings were ₤1000 and those of your friend only ₤200. Therefore, you still have ₤800 left from your savings but your friend doesn't have any money left. As a result, you enjoyed the trip but your friend regrets his decision to come with you because now he has no more money. Thus, although you both spent the same amount of money, these expenditures caused a different effect in your mood. You are happy and your friend is sad.

Likewise, action and reaction cause different effects in the objects movement. Thus, when the action-reaction principle applies, most probably, one of the objects moves more and the other object moves less or it doesn't move at all.

What is the Condition that Two Forces Obey the Action-Reaction Principle?

Not all forces that are equal in magnitude and opposite in direction can qualify to be part of action-reaction principle. The condition is that they must be of the same nature (type). For example, despite the moving force of a car may be equal in magnitude and opposite in direction to the frictional force exerted between the car's tires and the road, these forces do not represent the action-reaction principle because the moving force is caused by the push of the engine to the tire's axis, while frictional force has an intermolecular nature. It depends on the properties of the surfaces in contact and the car's weight.

Another example in this regard is the gravitational force and the normal force acting on an object that rests on a horizontal table. The gravitational force caused by the Earth on the object is a field force as explained in the tutorial "What causes Motion? The Meaning of Force" while Normal Force is a contact force caused by the resistance provided by the ground against deformations when an object pushes it.

But when two people are pulling each other by hands, the forces are of the same nature. Therefore, one of these forces is taken as active (action force) and the other as reactive (reaction force).

Only half of all forces involved in a system is shown in a force diagram. This is because we are interested only in the active forces as they cause motion. Indeed, even if we include the reactive forces in the diagram and the calculations, we would obtain the same result, only that the solution would be longer.

Examples of Action and Reaction Forces

There are many examples of the application of action-reaction principle in daily life. Some of them include:

  1. Gravitational force of the Earth on objects near it - action; gravitational force of objects on the Earth - reaction
  2. Frictional force caused by the surface of the ground to a sliding object - action; frictional force caused by the surface of the sliding object to the ground - reaction
  3. The force of an object pushing down the ground - action; the force exerted by the ground when pushing up on the object - reaction, and so on.

Remark! The weight of an object is caused by the gravitational tug of earth, not the upward push of the table! Therefore, the reaction force that pairs with the object's weight is the tug on earth caused by the object. Usually this reaction force is irrelevant in solving problems but ought to be kept in mind to avoid the common error described at the beginning of the previous paragraph, i.e. taking weight as action and normal force as reaction.

Example 1

Identify as much pairs of action-reaction forces as possible in the figure below.

Physics Tutorials: This image provides visual information for the physics tutorial Newton's Third Law of Motion

Solution 1

We can identify the following pairs of action-reaction forces in the above figure:

  1. Pulling force of the car to the rope - action; pulling force of the rope to the car - reaction
  2. Pulling force of the rope to the cart - action; pulling force of the cart to the rope - reaction
  3. Pulling force of the hanged object to the rope - object; pulling force of the rope to the hanged object - reaction
  4. Frictional force of the ground with the car's tires - action; Frictional force of the car's tires with the ground - reaction
  5. Frictional force of the ground with the cart tires - action; Frictional force of the cart tires with the ground - reaction
  6. Gravitational force caused by the Earth to the car - action; Gravitational force caused by the car to the Earth - reaction
  7. Gravitational force caused by the Earth to the cart - action; Gravitational force caused by the cart to the Earth - reaction
  8. Gravitational force caused by the Earth to the hanged object - action; Gravitational force caused by the hanged object to the Earth - reaction
  9. Pushing force of the car to the ground - action; pushing force of the ground to the car - reaction
  10. Pushing force of the cart to the ground - action; pushing force of the ground to the cart - reaction, etc.

It is not very helpful to draw any force diagram in this case, as it would be very confusing. So, please keep in mind only the force diagram used in Newton's Second Law of Motion.

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