Newton's Third Law - AP Physics C: Mechanics Study Guide

Getting Started

Every push, pull, or gravitational tug in the universe is part of a mutual interaction between two objects. When you push on a wall, the wall simultaneously pushes back on you. This chapter explores the fundamental law governing these interactions, answering the core question: How do we mathematically describe the symmetric, paired forces that arise when objects or systems influence one another?

What You Should Be Able to Do

By the end of this chapter, you will be able to:

• Identify and represent Newton's third law force pairs on separate free-body diagrams for any two interacting objects.

• Decompose the net force on a system of particles into the vector sum of external forces and internal forces.

• Prove that the vector sum of all internal forces within a system is zero and, as a consequence, that the system's center of mass acceleration is determined solely by the net external force.

• Model tension as a continuous, internal force transmitted through a flexible medium like a cable, relating it to the external forces applied to the system.

Key Concepts & Mechanisms

System & Preconditions

To analyze forces, we must first define the system: a collection of objects whose motion we intend to study. The boundary of the system is a conceptual surface that separates it from its environment. Forces are then categorized as either internal (exerted by one part of the system on another) or external (exerted by an agent in the environment on a part of the system). Our analysis assumes we are working in an inertial reference frame—a non-accelerating frame of reference where Newton's laws are valid. We often idealize objects as point particles or rigid bodies, and connectors like strings as massless and inextensible.

Key Steps / Relations

The dynamics of interacting objects are governed by a causal chain rooted in Newton's third law.

1. Identify the Interaction: For any two objects, A and B, that interact, there exists a pair of forces. Object A exerts a force on B, denoted ${\vec{F}}_{A\text{ on }B}$, and object B simultaneously exerts a force on A, denoted ${\vec{F}}_{B\text{ on }A}$.

2. State the Governing Law (Newton's Third Law): These two forces are always equal in magnitude and opposite in direction. This relationship is expressed as a vector equation:

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

   This pair of forces is often called an "action-reaction pair."

3. Sum Forces on a System: Consider a system of $N$ particles. The total force on the $i$-th particle is the vector sum of the net external force on it, ${\vec{F}}_{i,ext}$, and the sum of all internal forces exerted on it by other particles within the system:

   $${\vec{F}}_i=m_i{\vec{a}}_i={\vec{F}}_{i,ext}+\sum _{j\ne i}^N{\vec{F}}_{j\text{ on }i}$$

4. The Consequence of Internal Force Cancellation: To find the equation of motion for the entire system, we sum the forces on all particles:

   $${\vec{F}}_{net,sys}=\sum _{i=1}^N{\vec{F}}_i=\sum _{i=1}^N{\vec{F}}_{i,ext}+\sum _{i=1}^N\sum _{j\ne i}^N{\vec{F}}_{j\text{ on }i}$$

   The double summation term represents the sum of all internal forces. Due to Newton's third law, for every force ${\vec{F}}_{j\text{ on }i}$ in the sum, there is a corresponding force ${\vec{F}}_{i\text{ on }j}=-{\vec{F}}_{j\text{ on }i}$. The sum therefore consists of pairs of equal and opposite vectors, and the entire double summation evaluates to zero.

   $$\sum _{i=1}^N\sum _{j\ne i}^N{\vec{F}}_{j\text{ on }i}=0$$

Outputs & Effects

The cancellation of internal forces is a profound result. It simplifies the system's equation of motion to:

$${\vec{F}}_{net,ext}=\sum _{i=1}^N{m}_i{\vec{a}}_i$$

where ${\vec{F}}_{net,ext}$ is the vector sum of all external forces acting on the system. By defining the system's center of mass position${\vec{r}}_{cm}=\frac{1}{M_{tot}}\sum m_i{\vec{r}}_i$, where $M_{tot}=\sum m_i$, we can differentiate twice with respect to time to find the center of mass acceleration, ${\vec{a}}_{cm}$. This yields the fundamental equation for system motion:

$${\vec{F}}_{net,ext}=M_{tot}{\vec{a}}_{cm}=M_{tot}\frac{d^2{\vec{r}}_{cm}}{d{t}^2}$$

This shows that the motion of a system's center of mass is determined only by external forces. Internal forces, like those in an explosion or the tension within a rope connecting two blocks, can change the configuration and internal energy of the system, but they cannot alter the trajectory of its center of mass.

Regulation & Limits

Newton's third law is universally applicable within the domain of classical mechanics and inertial reference frames. The key to its application is the careful definition of the system boundary. The concept of tension as a uniform internal force is an idealization; in a real, massive rope that is accelerating, tension will vary along its length. However, for an idealized massless string, the net force on any segment must be zero, which implies the tension is constant throughout.

Key Models & Diagrams

The process of analyzing interacting systems using Newton's second and third laws can be visualized as follows:

Key Components & Evidence

• Force ($\vec{F}$): A vector quantity describing an interaction that causes a change in an object's motion. Its SI unit is the Newton (N), where 1 N = 1 kg·m/s².

• Newton's Third Law: The principle that for every action, there is an equal and opposite reaction: ${\vec{F}}_{A\text{ on }B}=-{\vec{F}}_{B\text{ on }A}$. This is a law of interaction, not of equilibrium.

• Action-Reaction Pair: The two forces that constitute a single interaction. Crucially, they act on different objects and are of the same type (e.g., both gravitational or both contact forces).

• System: A defined collection of objects. The choice of system boundary is a strategic step in problem-solving.

• Internal Force: A force exerted by one object inside the system on another object inside the system. Internal forces always sum to zero.

• External Force: A force exerted on an object within the system by an agent outside the system. Only external forces can accelerate the system's center of mass.

• Center of Mass (${\vec{r}}_{cm}$): The mass-weighted average position of a system, defined by the integral ${\vec{r}}_{cm}=\frac{1}{M}\int \vec{r}dm$ for a continuous body or the sum ${\vec{r}}_{cm}=\frac{1}{M}\sum m_i{\vec{r}}_i$ for discrete particles.

• Center of Mass Acceleration (${\vec{a}}_{cm}$): The second time derivative of the center of mass position, ${\vec{a}}_{cm}=d^2{\vec{r}}_{cm}/d{t}^2$. It directly relates the net external force to the system's total mass.

• Tension ($\vec{T}$): The macroscopic pulling force transmitted through a continuous medium like a string or cable. It is the net result of countless intermolecular action-reaction pairs within the material.

• Free-Body Diagram (FBD): A diagram that isolates a single object (or system) and represents all external forces acting on it as vectors originating from the object.

Skill Snapshots

Causation

• Driver: Object A exerts a force on object B. → Change: Object B simultaneously exerts a force of equal magnitude and opposite direction on object A.

• Driver: The sum of internal forces within a system is calculated. → Change: The sum is always zero, meaning internal interactions cannot cause the system's center of mass to accelerate.

• Driver: An external force pulls on one end of an ideal, massless string. → Change: A uniform tension force is established throughout the string, transmitting the external force to an object attached at the other end.

Comparison

• An action-reaction pair consists of two forces acting on two different objects, whereas two balanced forces in an equilibrium problem (e.g., normal force and gravity) act on the same object.

• Internal forces can change the relative positions and velocities of a system's components (e.g., in an explosion), but only external forces can change the velocity of the system's center of mass.

• The gravitational force the Earth exerts on you is equal in magnitude to the gravitational force you exert on the Earth, despite the vast difference in your masses and the resulting accelerations.

Change, Continuity, and Conservation

• Baseline: A system of two blocks connected by a string moves with a constant center of mass velocity ${\vec{v}}_{cm}$.

• Change: An external frictional force is applied to one block. The center of mass now accelerates (${\vec{a}}_{cm}\ne 0$) because a net external force exists.

• Change: The string connecting the blocks is cut. The blocks move independently, but as long as no new external forces act, the center of mass continues to move with the same acceleration it had just before the string was cut.

• Continuity: Throughout the entire process, for every contact force and every gravitational force between any two particles, Newton's third law holds perfectly.

Common Misconceptions & Clarifications

1. Misconception: Action-reaction forces cancel each other out.

   • Clarification: Forces can only cancel if they act on the same object. An action-reaction pair, by definition, acts on different objects and therefore cannot cancel. They are balanced in magnitude but produce distinct effects (accelerations) on their respective objects.

2. Misconception: The "action" force occurs first, and the "reaction" force follows.

   • Clarification: The terms "action" and "reaction" are historical and arbitrary. The forces are two simultaneous aspects of a single interaction. Neither force precedes the other.

3. Misconception: In a collision, the object with more mass or speed exerts a greater force.

   • Clarification: The magnitudes of the interaction forces are always identical. When a truck collides with a bicycle, the force the truck exerts on the bicycle is exactly equal to the force the bicycle exerts on the truck. The dramatic difference in outcomes is due to their different masses ($a=F/m$).

4. Misconception: For a book on a table, the downward force of gravity and the upward normal force are an action-reaction pair.

   • Clarification: This is incorrect because both forces act on the same object: the book. The reaction to the Earth's gravitational force on the book is the book's gravitational force on the Earth. The reaction to the table's normal force on the book is the book's contact force pushing down on the table.

One-Paragraph Summary

Newton's third law provides a complete description of mechanical interactions, stating that forces always occur in symmetric pairs, equal in magnitude and opposite in direction (${\vec{F}}_{A\text{ on }B}=-{\vec{F}}_{B\text{ on }A}$). This principle has a profound consequence when analyzing systems of multiple objects: all internal forces, which exist as action-reaction pairs within the system, sum vectorially to zero. This simplifies the dynamics of a complex system to a single differential equation, $M_{tot}{\ddot{\vec{r}}}_{cm}={\vec{F}}_{net,ext}$, revealing that only the net external force can alter the motion of the system's center of mass. This concept explains how tension transmits force through a cable and forms the essential foundation for the law of conservation of momentum. The predictive power of mechanics relies on correctly identifying these force pairs and distinguishing internal from external influences.