1.9 - Newton's Third Law

Introduction

Newton's Third Law of Motion, also known as the Law of Action and Reaction, is one of the fundamental principles of classical mechanics. It describes the nature of forces between interacting objects, stating that forces always occur in pairs. This law is crucial for understanding how objects interact with each other and why isolated forces cannot exist in nature.

Newton's Third Law (Law of Action and Reaction):

For every action, there is an equal and opposite reaction. When one object exerts a force on a second object, the second object simultaneously exerts a force equal in magnitude and opposite in direction on the first object.

In mathematical terms, if object A exerts a force $\mathbf{F}_{AB}$ on object B, then object B exerts a force $\mathbf{F}_{BA}$ on object A such that:

\mathbf{F}_{AB} = -\mathbf{F}_{BA}

This means:

The forces have equal magnitudes: $|\mathbf{F}_{AB}| = |\mathbf{F}_{BA}|$
The forces have opposite directions: $\mathbf{F}_{AB}$ and $\mathbf{F}_{BA}$ point in opposite directions
The forces act on different objects: $\mathbf{F}_{AB}$ acts on object B, while $\mathbf{F}_{BA}$ acts on object A
The forces occur simultaneously: both forces exist at the same instant

Key Concepts

Action-Reaction Pairs

The forces described by Newton's Third Law always come in action-reaction pairs. These are two forces that:

Have the same magnitude
Act in opposite directions
Act on different objects
Are of the same type (both contact forces, both gravitational forces, etc.)

Important Characteristics of Action-Reaction Pairs:

They act on different objects: This is the most crucial point. The action force acts on one object, and the reaction force acts on the other object. They never act on the same object.
They cannot cancel each other: Since they act on different objects, they cannot be added together to give zero net force on a single object. Each force contributes to the motion of its respective object.
They are simultaneous: Both forces exist at the same time. There is no delay between the action and the reaction.
They are always the same type of force: If the action is a gravitational force, the reaction is also a gravitational force. If the action is a contact force, the reaction is also a contact force.

Forces on Different Objects

A common source of confusion is understanding that action-reaction pairs act on different objects. This is essential for correctly applying Newton's laws:

When analyzing the motion of a single object, you only consider forces acting on that object, not forces that the object exerts on other objects.
The reaction force (the force the object exerts on something else) does not appear in the free-body diagram or force analysis for that object.
To find the net force on an object, you sum all forces acting on that object, ignoring forces that the object exerts on other objects.

Examples and Applications

Example 1: A Person Walking

When you walk, you push backward on the ground with your foot. The ground simultaneously pushes forward on your foot (and thus on you). The force you exert on the ground is the action, and the force the ground exerts on you is the reaction. It is this reaction force from the ground that propels you forward.

Action: Your foot pushes backward on the ground ($\mathbf{F}_{\text{foot,ground}}$)
Reaction: The ground pushes forward on your foot ($\mathbf{F}_{\text{ground,foot}} = -\mathbf{F}_{\text{foot,ground}}$)

Example 2: A Book on a Table

A book resting on a table exerts a downward gravitational force on the table. The table simultaneously exerts an upward normal force on the book. These two forces form an action-reaction pair.

Action: Book's weight (gravitational force) on the table ($\mathbf{F}_{\text{book,table}}$)
Reaction: Table's normal force on the book ($\mathbf{F}_{\text{table,book}} = -\mathbf{F}_{\text{book,table}}$)

Note: The book's weight (gravitational force from Earth) and the table's normal force on the book are not an action-reaction pair, even though they are equal and opposite. They both act on the same object (the book), so they cancel to give zero net force on the book, but they are not a Third Law pair.

Example 3: Rocket Propulsion

A rocket engine expels hot gases backward. The gases exert a force on the rocket (pushing it forward), and the rocket simultaneously exerts an equal and opposite force on the gases (pushing them backward). The force on the rocket is what propels it forward.

Action: Rocket pushes gases backward ($\mathbf{F}_{\text{rocket,gases}}$)
Reaction: Gases push rocket forward ($\mathbf{F}_{\text{gases,rocket}} = -\mathbf{F}_{\text{rocket,gases}}$)

Example 4: Two Ice Skaters Pushing Each Other

Two ice skaters push against each other. Skater A pushes on Skater B with force $\mathbf{F}_{AB}$, and Skater B simultaneously pushes on Skater A with force $\mathbf{F}_{BA} = -\mathbf{F}_{AB}$. Both skaters experience acceleration (they move apart), but in opposite directions.

If Skater A has mass $m_A$ and Skater B has mass $m_B$, and the magnitude of the force is $F$, then:

a_A = \frac{F}{m_A} \quad \text{and} \quad a_B = \frac{F}{m_B}

The lighter skater accelerates more than the heavier skater, even though the forces are equal in magnitude.

Example 5: A Ball Hitting a Wall

When a ball hits a wall, the ball exerts a force on the wall, and the wall simultaneously exerts an equal and opposite force on the ball. The force on the ball causes it to bounce back (change direction).

Action: Ball pushes on wall ($\mathbf{F}_{\text{ball,wall}}$)
Reaction: Wall pushes on ball ($\mathbf{F}_{\text{wall,ball}} = -\mathbf{F}_{\text{ball,wall}}$)

Example 6: Gravitational Attraction

The Earth exerts a gravitational force on the Moon, pulling it toward Earth. Simultaneously, the Moon exerts an equal and opposite gravitational force on Earth, pulling Earth toward the Moon. Both objects accelerate toward each other, but because Earth is much more massive, its acceleration is much smaller.

Action: Earth's gravitational force on Moon ($\mathbf{F}_{\text{Earth,Moon}}$)
Reaction: Moon's gravitational force on Earth ($\mathbf{F}_{\text{Moon,Earth}} = -\mathbf{F}_{\text{Earth,Moon}}$)

Both forces have the same magnitude: $F = G\frac{m_E m_M}{r^2}$, where $G$ is the gravitational constant, $m_E$ and $m_M$ are the masses of Earth and Moon, and $r$ is the distance between them.

Common Misconceptions

Important Clarifications:

"Action and reaction cancel each other": This is false. Action-reaction pairs act on different objects, so they cannot cancel each other. They each contribute to the motion of their respective objects. Forces can only cancel if they act on the same object.
"The reaction happens after the action": This is incorrect. Both forces occur simultaneously. There is no time delay between them.
"Equal forces mean equal accelerations": This is false. While the forces are equal, the accelerations depend on the masses. From Newton's Second Law, $a = F/m$, so a lighter object accelerates more than a heavier object when subjected to the same force.
"If forces are equal and opposite, nothing moves": This is incorrect. Equal and opposite forces on different objects cause both objects to accelerate (move). The motion depends on the masses of the objects.
"The stronger object wins": This is a misconception. Both objects experience forces of equal magnitude. The difference in motion comes from differences in mass, not differences in force magnitude.
"Balanced forces are action-reaction pairs": This is incorrect. Balanced forces (forces that cancel) act on the same object. Action-reaction pairs act on different objects and cannot cancel.

Identifying Action-Reaction Pairs

To correctly identify action-reaction pairs, ask these questions:

What two objects are interacting? Action-reaction pairs involve two distinct objects.
What force does object A exert on object B? This is the action force.
What force does object B exert on object A? This is the reaction force.
Do both forces act on different objects? If yes, they form an action-reaction pair. If they act on the same object, they are not a Third Law pair (they might be balanced forces instead).

Example: Identifying Pairs in a Complex Situation

Consider a person standing on a scale in an elevator. Let's identify the action-reaction pairs:

Person and Earth: Earth pulls person down (gravity), person pulls Earth up (gravity). These are equal and opposite, act on different objects, and form an action-reaction pair.
Person and Scale: Person pushes down on scale, scale pushes up on person. Action-reaction pair.
Scale and Elevator Floor: Scale pushes down on floor, floor pushes up on scale. Action-reaction pair.
Elevator and Earth: Earth pulls elevator down (gravity), elevator pulls Earth up (gravity). Action-reaction pair.

Note: The person's weight (Earth's force on person) and the scale's normal force on the person are not an action-reaction pair because they both act on the person. However, they are equal and opposite when the person is at rest or moving with constant velocity, which is why the net force on the person is zero.

Relationship to Other Laws

Connection to Newton's Second Law

Newton's Third Law is essential for applying the Second Law correctly. When two objects interact:

Each object experiences a force from the other object (Third Law pair).
Each object's acceleration is determined by the net force acting on it (Second Law).
The net force on each object is found by summing all forces acting on that specific object, not forces it exerts on other objects.

For example, if object A exerts force $\mathbf{F}_{AB}$ on object B, then:

\mathbf{F}_\text{net on B} = \mathbf{F}_{AB} = m_B \mathbf{a}_B

and

\mathbf{F}_\text{net on A} = \mathbf{F}_{BA} = -\mathbf{F}_{AB} = m_A \mathbf{a}_A

Conservation of Momentum

Newton's Third Law leads directly to the conservation of momentum. For two interacting objects with no external forces:

\mathbf{F}_{AB} = -\mathbf{F}_{BA}

Using Newton's Second Law:

\frac{d\mathbf{p}_B}{dt} = -\frac{d\mathbf{p}_A}{dt}

Rearranging:

\frac{d\mathbf{p}_A}{dt} + \frac{d\mathbf{p}_B}{dt} = \frac{d(\mathbf{p}_A + \mathbf{p}_B)}{dt} = \mathbf{0}

This means the total momentum $\mathbf{p}_A + \mathbf{p}_B$ is constant—momentum is conserved. This is a fundamental principle in physics that arises from Newton's Third Law.

Practical Applications

Understanding Newton's Third Law is crucial in many practical situations:

Rocket propulsion: Rockets work by expelling mass backward, creating a forward reaction force. This is the only way to propel objects in space where there's nothing to push against.
Walking and running: We move forward by pushing backward on the ground. The ground's reaction force propels us forward.
Swimming: Swimmers push water backward with their arms and legs, and the water pushes them forward.
Collision analysis: In collisions, the forces between colliding objects are equal and opposite, which is essential for analyzing momentum conservation.
Structural engineering: Understanding action-reaction pairs is crucial for analyzing forces in structures, bridges, and buildings.
Sports: Athletes use action-reaction principles—for example, a basketball player pushes down on the court to jump up.

Mathematical Formulation

Newton's Third Law can be expressed mathematically as:

Mathematical Statement of Newton's Third Law:

If object A exerts a force $\mathbf{F}_{AB}$ on object B, then object B exerts a force $\mathbf{F}_{BA}$ on object A such that:

\mathbf{F}_{AB} = -\mathbf{F}_{BA}

This is a vector equation, meaning:

Magnitude: $|\mathbf{F}_{AB}| = |\mathbf{F}_{BA}|$ (forces are equal in magnitude)
Direction: $\mathbf{F}_{AB}$ and $\mathbf{F}_{BA}$ point in opposite directions
Simultaneity: Both forces exist at the same instant in time
Different objects: $\mathbf{F}_{AB}$ acts on object B, $\mathbf{F}_{BA}$ acts on object A

Summary

Key Takeaways:

Newton's Third Law states that for every action, there is an equal and opposite reaction. Forces always occur in pairs.
Action-reaction pairs have equal magnitudes, opposite directions, act on different objects, and occur simultaneously.
Action-reaction pairs cannot cancel each other because they act on different objects. Each force contributes to the motion of its respective object.
When analyzing the motion of an object, only consider forces acting on that object, not forces it exerts on other objects.
Equal forces do not mean equal accelerations. The acceleration depends on mass: $a = F/m$, so lighter objects accelerate more than heavier objects for the same force.
Newton's Third Law leads to conservation of momentum for isolated systems.
The Third Law is essential for understanding how objects interact and why isolated forces cannot exist in nature.
Many everyday phenomena—from walking to rocket propulsion—rely on action-reaction force pairs.