Newton's Third Law: Concepts, Applications, and Free-Body Diagrams

Module 09: Newton’s 3rd Law

Introduction to Newton’s Laws

Newton’s laws of motion are fundamental principles that describe the relationship between the motion of an object and the forces acting upon it. In this module, we focus on Newton’s third law, its algebraic formulation, and its applications in everyday life and physics problems.

Algebraic Formulation of Newton’s Laws

• Newton’s First Law (Law of Inertia): An object remains at rest or in uniform motion unless acted upon by a net external force.

• Newton’s Second Law: The net force on an object is equal to the mass of the object multiplied by its acceleration.

Key Equations:

• Net force as a sum of all forces: $\vec{F}_{\text{net}} = \sum_i \vec{F}_i$

• If $\vec{F}_{\text{net}} = 0$, then velocity is constant: $\frac{d\vec{v}}{dt} = 0$

• Newton’s second law: $\vec{F}_{\text{net}} = m\vec{a}$

Newton’s Third Law: Action-Reaction Principle

Newton’s third law states that for every action, there is an equal and opposite reaction. This law describes the mutual interactions between two objects.

• Formal Statement: For every force exerted by object A on object B, there is a force of equal magnitude but opposite direction exerted by object B on object A.

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

• Action-Reaction Pair: The two forces act on different objects and are always equal in magnitude and opposite in direction.

Microscopic Origin: Forces Mediated by Exchange Particles

At the microscopic level, fundamental forces (such as electromagnetic and strong nuclear forces) are mediated by exchange particles (e.g., photons, gluons). These particles transfer momentum between objects, resulting in repulsion or attraction.

• Example: Two people on skateboards throwing a ball to each other experience mutual repulsion due to the exchange of momentum.

• Application: This is a physical illustration of Newton’s third law at the particle level.

Newton’s Third Law in Everyday Life

Newton’s third law is evident in many daily activities where motion is generated by pushing against the environment.

• Running: A runner pushes against the ground with their foot; the ground pushes back with an equal and opposite force, propelling the runner forward.

• Skiing: A skier pushes against the snow with their poles; the snow pushes back, allowing movement.

• Equations: $\vec{F}_{\text{ground on us}} = -\vec{F}_{\text{us on ground}}$ $\vec{F}_{\text{ground on us}} = -\frac{1}{m}\vec{F}_{\text{us on ground}}$ (for acceleration context)

Identifying Action-Reaction Pairs

While Newton’s third law seems straightforward, it is important to correctly identify which forces form action-reaction pairs. Only forces between two different objects qualify.

• Key Points:

  • Every force occurs as one member of an action/reaction pair.

  • The two members act on two different objects.

  • The two members are equal in magnitude and opposite in direction.

• Example: The normal force exerted by a table on a box and the force exerted by the box on the table are an action-reaction pair.

• Non-Example: Internal forces within a single object do not form action-reaction pairs.

Free-Body Diagrams and Action-Reaction Pairs

Free-body diagrams are used to represent all the forces acting on a single object. Action-reaction pairs are not shown together in a single free-body diagram because they act on different objects.

• Example: For a box on a table, the free-body diagram for the box includes its weight and the normal force from the table.

• Stack of Books: Each book experiences contact forces from the book above and below, as well as its own weight. The table only "feels" the total weight of the stack.

Applications: Connected Objects and Accelerating Systems

When analyzing systems of connected objects (such as train cars or objects connected by strings), Newton’s third law helps set up equations for each object.

• Example: In a train with multiple cars, each car experiences forces from the adjacent cars. The net force on each car determines its acceleration.

• Equations for Each Car: $F - F' = m a$ (where $F$ and $F'$ are forces from adjacent cars)

Gravitational Force and Weight

The gravitational force acting on an object is called its weight. The direction of gravity is usually defined as "downward."

• Equation: $F_{\text{gravity}} = \text{weight} = m g$

• $F = m(-g) = -mg$ (if upward is positive y-direction)

• Units: Mass is measured in kilograms (kg), and force in Newtons (N).

Units for Mass and Force: SI vs. Imperial

SI units are standard in physics, but imperial units are sometimes used. It is important to distinguish between mass and weight in different unit systems.

• SI Units: 1 kg is the base unit of mass; 1 N is the unit of force.

• Imperial Units: 1 lb (pound) is a unit of force, not mass. 1 lb = 0.45359237 kg.

• Conversion: Always convert imperial units to SI units for consistency in calculations.

• Note: The acceleration due to gravity ($g$) varies by location (e.g., Earth vs. Moon).

Summary Table: Action-Reaction Pairs

The following table summarizes examples and non-examples of action-reaction pairs:

Situation	Action-Reaction Pair?	Explanation
Box on Table	Yes	Normal force by table on box and force by box on table
Internal forces within a single object	No	Forces act within the same object, not between two objects
Gravitational force between Earth and astronaut	Yes	Earth pulls on astronaut; astronaut pulls on Earth with equal and opposite force
Contact force between stacked books	Yes	Each book exerts a force on the one above/below

Key Takeaways

• Newton’s third law applies to all interactions between two objects.

• Action-reaction pairs are always equal in magnitude and opposite in direction, but act on different objects.

• Free-body diagrams help analyze forces on individual objects, but do not show both members of an action-reaction pair together.

• Understanding units and conversions is essential for accurate calculations.

Additional info: Some context and examples were expanded for clarity and completeness, including the microscopic origin of forces and the distinction between SI and imperial units.