Newton’s Third Law of Motion and Applications: Forces, Interactions, and Vectors (Chapter 5)

Forces and Interactions

Concept of Interactions

Forces always arise from interactions between two objects. Every force is part of a mutual push or pull, meaning that forces never exist in isolation.

• Pairs of Forces: Every force is part of a pair—when you exert a force on an object, it exerts an equal and opposite force back on you.

• Mutual Push/Pull: For example, pushing against a wall means the wall pushes back on your hand with equal strength in the opposite direction.

• Magnitude Limits: The force you can exert on an object is limited by the force it can exert back. A boxer can hit a punching bag hard, but only a tiny force can be exerted on a tissue paper because it cannot push back much.

• Nature of Forces: There is no strict “giver” or “receiver” in a force interaction; both objects are equal participants. Inanimate objects (like walls or ropes) exert forces just as living things do.

• Force Is Not Possessed: Objects do not possess force; force only exists during active interactions and ceases when the interaction ends.

Examples

• Pushing a wall, hitting a punching bag, or driving a stake with a hammer all involve mutual forces.

Newton’s Third Law of Motion

Statement and Core Principle

Newton’s Third Law states: Whenever one object exerts a force on a second object, the second object simultaneously exerts an equal and opposite force on the first.

• Classic Expression: “To every action there is always an opposed equal reaction.”

• Force Pairs: Forces always occur in pairs (action and reaction), and neither can exist without the other.

Identifying Action–Reaction Pairs

• Identify the two interacting bodies (Object A and Object B):

  • Action: Object A exerts a force on Object B.

  • Reaction: Object B exerts a force on Object A.

• Examples:

  • Walking: Feet push backward on the floor (action); floor pushes forward on feet (reaction).

  • Swimming: Hands push water backward; water pushes swimmer forward.

  • Gravity: Earth pulls object downward; object pulls Earth upward.

Defining Your System

• Internal Forces: Forces within a defined system cancel each other and cannot accelerate the system.

• External Forces: Only forces from outside the system can cause acceleration.

• Cancellation Condition: Equal and opposite forces only cancel if they act on the same object. Action–reaction pairs act on different objects and do not cancel each other on a single object.

Example

• Orange & Apple System: If the system is just the orange, the force from the apple is external and can accelerate the orange. If the system includes both, their mutual forces are internal and cancel; an external force is needed to accelerate both.

Action and Reaction on Different Masses

Equal Forces, Unequal Accelerations

Action and reaction forces are always equal in magnitude and opposite in direction, regardless of the masses involved:

• According to Newton’s Second Law, $a = \frac{F}{m}$, so equal forces produce different accelerations depending on mass.

• Small mass: Large acceleration.

• Large mass: Small acceleration.

Key Examples & Applications

• Falling Boulder & Earth: Both pull on each other with equal force, but the boulder’s acceleration is noticeable while Earth’s is negligible due to its large mass.

• Cannon & Cannonball: The cannonball accelerates rapidly; the cannon recoils slowly. $F_{\text{cannonball}} = -F_{\text{cannon}}$

• Rocket Propulsion: Rockets accelerate by expelling exhaust gases backward; the reaction force pushes the rocket forward. Rockets work even better in space where there is no air resistance.

• Helicopters, Birds, & Airplanes: Helicopter blades push air down (action); air pushes helicopter up (reaction). Birds and planes generate lift by deflecting air downward.

• Car Driving: Tires push backward on the road; the road pushes forward on the tires, accelerating the car.

Bug vs. Bus Problem

• When a bus and a bug collide, the force each exerts on the other is equal and opposite. The bug experiences a huge acceleration (and damage) due to its small mass, while the bus’s acceleration is negligible.

Tug-of-War Insight

• Winning depends on which team pushes harder against the ground (friction), not who pulls harder on the rope. The rope exerts equal and opposite forces on both teams.

Vectors and the Third Law

Vector Components & Resolution

Vectors can be resolved into perpendicular components, typically horizontal and vertical. The parallelogram rule is used for this resolution.

• Vector Resolution: Any vector can be broken into components $V_x$ and $V_y$ such that:

$R = \sqrt{V_x^2 + V_y^2}$

Forces on an Incline

• Flat Surface: The normal force $N$ equals the object’s weight $mg$ ($N = mg$).

• Inclined Surface: Weight $mg$ acts downward and is resolved into:

  • Perpendicular component: $mg_\perp$ (balances normal force $N$; decreases as incline steepens).

  • Parallel component: $mg_\parallel$ (causes acceleration down the slope; increases as incline steepens).

• Vertical Incline ($90^\circ$): Normal force drops to zero ($N = 0$); object is in free fall ($a = g$).

Friction & Equilibrium on an Incline

• Friction ($f$): Opposes the net force down the slope.

• Equilibrium: If friction equals the down-slope force ($f = mg_\parallel$), the object remains at rest or moves at constant velocity ($a = 0$).

Tension Vectors & Suspended Objects

• Angle Dependency: Tension in supporting ropes increases as the angle from vertical increases. As the rope approaches horizontal ($180^\circ$), tension approaches infinity.

Velocity Components (Projectile Motion Preview)

• Horizontal Component ($v_x$): Remains constant (if air resistance is negligible).

• Vertical Component ($v_y$): Changes due to gravity ($g$); decreases upward, increases downward.

Summary of Newton’s Three Laws

First Law (Law of Inertia)

• Principle: An object at rest remains at rest, and an object in motion continues at constant velocity unless acted upon by a net force.

• Key Concept: Inertia is resistance to changes in motion; mass measures inertia.

Second Law (Law of Acceleration)

• Principle: A net force causes acceleration in the direction of the force.

• Formula:

$a = \frac{F_{\text{net}}}{m}$

• Falling Motion:

  • In a vacuum: $a = g$

  • In air: $a < g$ (due to air resistance)

  • Terminal speed: $a = 0$ when air resistance equals gravitational pull

Third Law (Law of Action–Reaction)

• Principle: Whenever one object exerts a force on a second object, the second exerts an equal and opposite force on the first.

• Formula:

$F_{\text{action}} = -F_{\text{reaction}}$

• Key Properties: Forces always occur in pairs, act on different objects, and neither exists without the other.