Newton's Third Law and Applications

Action-reaction pairs and force interactions

Newton's Third Law and Applications

Newton's Third Law

Statement: For every action, there is an equal and opposite reaction.

More precisely: When object A exerts a force on object B, object B exerts an equal magnitude force on object A in the opposite direction.

$\vec{F}_{A \text{ on } B} = -\vec{F}_{B \text{ on } A}$

Or more compactly: $\vec{F}_{AB} = -\vec{F}_{BA}$

Key Characteristics of Action-Reaction Pairs

Equal magnitude: $|F_{AB}| = |F_{BA}|$
Opposite directions: One points one way, the other points the opposite way
Same type of force: If one is gravitational, both are gravitational
Different objects: Each force acts on a different object
Simultaneous: Both forces exist at the same time

Identifying Action-Reaction Pairs

Template

"Object A exerts force on object B" ⟺ "Object B exerts force on object A"

Examples

Example 1: Book on table

Action: Earth pulls down on book (weight)
Reaction: Book pulls up on Earth

NOT action-reaction:

Weight (Earth on book) and Normal force (table on book) - these act on the same object!

Example 2: Hammer hits nail

Action: Hammer exerts force on nail (to the right)
Reaction: Nail exerts force on hammer (to the left)

Example 3: Rocket propulsion

Action: Rocket pushes gas backward
Reaction: Gas pushes rocket forward

Common Mistakes

❌ Wrong: Normal force and weight are action-reaction pairs ✓ Correct: They act on the same object, so they can't be action-reaction

❌ Wrong: Action happens first, then reaction ✓ Correct: Both happen simultaneously

❌ Wrong: The heavier object exerts more force ✓ Correct: Forces in a pair are always equal magnitude

Why Don't Action-Reaction Forces Cancel?

Action-reaction forces act on different objects, so they don't cancel!

Example: Push a wall

You push wall to the right with force $F$
Wall pushes you to the left with force $F$
Net force on you: $F$ to the left (you accelerate backward)
Net force on wall: $F$ to the right (but wall doesn't move—it's also attached to Earth!)

Forces only cancel if they act on the same object.

Applications of Newton's Third Law

Walking

Your foot pushes backward on ground
Ground pushes forward on your foot (you accelerate forward)
Without friction, you can't push on ground → can't walk (like ice skating!)

Swimming

You push water backward with your hands
Water pushes you forward
More water displaced → greater forward force

Rocket Propulsion

Engine expels gas backward (action)
Gas pushes rocket forward (reaction)
Works in space (doesn't need air to "push against")

Tension in Ropes

If rope has negligible mass:

Tension is the same throughout the rope
Forces at both ends of rope segment are equal and opposite (third law)

Normal Force

Object pushes down on surface
Surface pushes up on object (normal force)
Perpendicular to surface

Internal vs. External Forces

Internal Forces

Forces between objects within a system
Come in action-reaction pairs
Cancel out when considering the system as a whole
Example: Tension between two connected blocks

External Forces

Forces from outside the system
Don't have reaction partners within the system
Cause acceleration of the system
Example: Friction from ground on a car

For a system: Only external forces affect the motion of the center of mass.

Problem-Solving with Third Law

Identify the two objects in the interaction
Name both forces:
- Force object A exerts on object B
- Force object B exerts on object A
Remember: Equal magnitude, opposite direction
Draw separate FBDs for each object if needed
Apply Second Law to each object separately

Connected Objects

When objects are connected (ropes, contact, etc.):

Draw separate FBDs for each object
Apply Newton's Second Law to each object
Use Third Law to relate interaction forces
Solve the system of equations

Example: Two blocks connected by rope

Tension force on block 1: $T$ (pulls block 1)
Tension force on block 2: $T$ (pulls block 2)
If rope is massless: same tension throughout

Misconceptions Clarified

Q: If forces are always equal and opposite, how does anything accelerate?

A: Action-reaction forces act on different objects! The net force on each object determines its acceleration.

Q: When a horse pulls a cart, doesn't the cart pull back equally hard on the horse? How does the cart move?

A: Yes, cart pulls on horse (backward) and horse pulls on cart (forward) with equal forces. But these act on different objects!

For the cart to accelerate: Forward force from horse > Backward friction on cart
The horse accelerates forward because: Ground pushes horse forward > Cart pulls horse backward

Q: If I push a wall and it pushes back equally hard, why do I move but the wall doesn't?

A: The wall is attached to the Earth (huge mass!). The force accelerates you: $a = F/m$ . Same force on Earth: $a = F/M_{Earth} \approx 0$ (negligible).

📚 Practice Problems

1Problem 1medium

❓ Question:

A 1000 kg car pulls a 500 kg trailer with an acceleration of 2.0 m/s². Assume no friction. (a) What is the net force on the car-trailer system? (b) What force does the car's engine produce? (c) What is the tension in the connection between the car and trailer?

💡 Show Solution

Solution:

Given: m_car = 1000 kg, m_trailer = 500 kg, a = 2.0 m/s²

(a) Net force on system: Total mass = 1000 + 500 = 1500 kg F_net = m_total × a = 1500 × 2.0 = 3000 N

(b) Engine force: Since there's no friction, the engine force equals the net force: F_engine = 3000 N

(c) Tension in connection: Consider just the trailer (free body diagram): F_net on trailer = m_trailer × a T = 500 × 2.0 = 1000 N

Check: Consider the car alone: F_engine - T = m_car × a 3000 - T = 1000 × 2.0 T = 3000 - 2000 = 1000 N ✓

By Newton's 3rd Law, the trailer pulls back on the car with 1000 N.

2Problem 2easy

❓ Question:

You push on a wall with a force of $50$ N. What is the force the wall exerts on you?

💡 Show Solution

Given:

Force you exert on wall: $F_{you \text{ on wall}} = 50$ N

Find: Force wall exerts on you

Apply Newton's Third Law:

Every action has an equal and opposite reaction. When you push on the wall, the wall pushes back on you.

$F_{wall \text{ on you}} = -F_{you \text{ on wall}}$

Magnitude: $|F_{wall \text{ on you}}| = 50$ N

Direction: Opposite to the force you exerted

Answer: The wall exerts a force of 50 N on you, directed away from the wall (pushing you backward).

Key point: The forces are equal in magnitude but opposite in direction. They act on different objects (you vs. wall), so they don't cancel.

3Problem 3medium

❓ Question:

💡 Show Solution

Solution:

Given: m_car = 1000 kg, m_trailer = 500 kg, a = 2.0 m/s²

(a) Net force on system: Total mass = 1000 + 500 = 1500 kg F_net = m_total × a = 1500 × 2.0 = 3000 N

(b) Engine force: Since there's no friction, the engine force equals the net force: F_engine = 3000 N

(c) Tension in connection: Consider just the trailer (free body diagram): F_net on trailer = m_trailer × a T = 500 × 2.0 = 1000 N

Check: Consider the car alone: F_engine - T = m_car × a 3000 - T = 1000 × 2.0 T = 3000 - 2000 = 1000 N ✓

By Newton's 3rd Law, the trailer pulls back on the car with 1000 N.

4Problem 4medium

❓ Question:

You push on a wall with a force of 100 N. (a) What force does the wall exert on you? (b) If you and the wall don't move, explain why the forces are balanced. (c) Why don't the action-reaction pair cancel each other out?

💡 Show Solution

Solution:

(a) Force from wall: By Newton's 3rd Law, the wall pushes back on you with 100 N in the opposite direction.

(b) Why balanced (no movement): The forces ARE balanced, but we must consider all forces:

Force you exert on wall: 100 N (action)
Force wall exerts on you: 100 N (reaction)

However, these act on DIFFERENT objects:

On YOU: Wall pushes you backward (100 N). But friction from ground pushes you forward (100 N). Net force on you = 0.
On WALL: You push forward (100 N). But the wall's foundation pushes it backward (100 N). Net force on wall = 0.

Act on different objects
Cannot be added together because they're not acting on the same object
Only forces on the SAME object can cancel

The 100 N on you and 100 N on wall are separate - they don't cancel because they're on different objects!

5Problem 5medium

❓ Question:

💡 Show Solution

Solution:

(a) Force from wall: By Newton's 3rd Law, the wall pushes back on you with 100 N in the opposite direction.

(b) Why balanced (no movement): The forces ARE balanced, but we must consider all forces:

Force you exert on wall: 100 N (action)
Force wall exerts on you: 100 N (reaction)

However, these act on DIFFERENT objects:

On YOU: Wall pushes you backward (100 N). But friction from ground pushes you forward (100 N). Net force on you = 0.
On WALL: You push forward (100 N). But the wall's foundation pushes it backward (100 N). Net force on wall = 0.

Act on different objects
Cannot be added together because they're not acting on the same object
Only forces on the SAME object can cancel

The 100 N on you and 100 N on wall are separate - they don't cancel because they're on different objects!

6Problem 6medium

❓ Question:

A $60$ kg astronaut floating in space pushes a $120$ kg satellite with a force of $240$ N. What is the acceleration of: (a) the satellite, (b) the astronaut?

💡 Show Solution

Given:

Astronaut mass: $m_A = 60$ kg
Satellite mass: $m_S = 120$ kg
Force astronaut exerts on satellite: $F_{A \text{ on } S} = 240$ N

Part (a): Acceleration of satellite

Apply Newton's Second Law to the satellite: $a_S = \frac{F_{A \text{ on } S}}{m_S} = \frac{240}{120} = 2 \text{ m/s}^2$

Direction: In the direction the astronaut pushed

Part (b): Acceleration of astronaut

By Newton's Third Law: $F_{S \text{ on } A} = -F_{A \text{ on } S}$

Magnitude: $|F_{S \text{ on } A}| = 240$ N

Direction: Opposite (satellite pushes back on astronaut)

Apply Newton's Second Law to the astronaut: $a_A = \frac{F_{S \text{ on } A}}{m_A} = \frac{240}{60} = 4 \text{ m/s}^2$

Direction: Opposite to the satellite's acceleration (astronaut moves backward)

Answers:

(a) Satellite acceleration: 2 m/s² (forward)
(b) Astronaut acceleration: 4 m/s² (backward)

Key insight: Even though the forces are equal, the accelerations are different because the masses are different! The less massive astronaut accelerates more ( $a \propto 1/m$ ).

7Problem 7hard

❓ Question:

A $5$ kg book rests on a table. Identify all the forces on the book and state whether any pairs are action-reaction pairs according to Newton's Third Law.

💡 Show Solution

Forces acting on the book:

Weight ( $\vec{W}$ or $\vec{F}_g$ ): Earth pulls down on book
- Magnitude: $W = mg = (5)(9.8) = 49$ N
- Direction: Downward
Normal force ( $\vec{N}$ or $\vec{F}_N$ ): Table pushes up on book
- Magnitude: $N = 49$ N (since book is at rest)
- Direction: Upward

Are weight and normal force action-reaction pairs?

NO! They are not action-reaction pairs because:

Both forces act on the same object (the book)
Action-reaction pairs must act on different objects
They are different types of forces (gravitational vs. contact)

What ARE the action-reaction pairs?

Pair 1: Gravitational forces

Action: Earth pulls down on book (weight) = 49 N downward
Reaction: Book pulls up on Earth = 49 N upward

Pair 2: Contact forces

Action: Book pushes down on table = 49 N downward
Reaction: Table pushes up on book (normal force) = 49 N upward

Summary:

Weight and normal force act on the book → they balance (net force = 0, so $a = 0$ )
The reaction to weight acts on Earth
The reaction to normal force acts on the table

Key insight: Just because two forces are equal and opposite doesn't make them an action-reaction pair! They must act on different objects and be the same type of force.

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