Newton's Third Law - AP Physics 1: Algebra-Based Study Guide

Getting Started

Every push, pull, or touch in the universe is a two-way street. When you lean against a wall, you feel the wall pushing back on you. This chapter explores the fundamental rule governing these interactions, from the scale of colliding galaxies down to the forces between atoms. Our core question is: What is the universal relationship between the forces that two interacting objects exert on each other?

What You Should Be Able to Do

After working through this section, you should be able to:

• Identify the two objects involved in any single force interaction.

• Represent an interaction as a pair of force vectors, equal in magnitude and opposite in direction, acting on the two separate objects.

• Distinguish between internal and external forces by defining a system boundary.

• Explain why forces internal to a system do not affect the motion of the system's center of mass.

• Describe tension as the force transmitted through a flexible object like a string and state the conditions under which it is constant.

Key Concepts & Mechanisms

System & Preconditions

To properly analyze forces, we must first define the system, which is simply the collection of one or more objects we choose to study. Forces can then be classified based on this definition.

• An internal force is a force that objects within the system exert on each other.

• An external force is a force exerted on an object in the system by something outside the system boundary.

The distinction is critical: as we will see, only external forces can change the overall motion of a system. In many problems, we make simplifying assumptions. A common one is the ideal string model, where a string, rope, or cable is assumed to be massless and unstretchable.

Key Steps / Relations

1. The Interaction Law (Newton's Third Law): The foundational principle of interactions is Newton's Third Law. It states that if object A exerts a force on object B, then object B simultaneously exerts a force on object A. These two forces are always equal in magnitude and opposite in direction. They are often called an action-reaction pair.

2. Mathematical Representation: This relationship is expressed concisely using vector notation. If ${\vec{F}}_{A\text{ on }B}$ is the force object A exerts on object B, and ${\vec{F}}_{B\text{ on }A}$ is the force object B exerts on object A, then:

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

   The negative sign indicates that the force vectors point in opposite directions. Their magnitudes are equal: $|{\vec{F}}_{A\text{ on }B}|=|{\vec{F}}_{B\text{ on }A}|$.

3. Identifying Force Pairs: The most important skill for applying Newton's Third Law is correctly identifying the pairs. The labels are the key: the force "A on B" is paired with the force "B on A". Notice that the forces in a pair always act on different objects. This is why they can never cancel each other out when analyzing the motion of a single object.

4. Internal Forces and System Motion: Consider a system containing both object A and object B. The two forces, ${\vec{F}}_{A\text{ on }B}$ and ${\vec{F}}_{B\text{ on }A}$, are now internal to the system. If we sum all internal forces within any system, the sum is always zero because every force is matched by its equal and opposite partner. Because the net internal force is zero, these forces cannot cause the system's center of mass—the mass-weighted average position of the system—to accelerate. The motion of the system as a whole can only be changed by a net external force.

5. Tension as an Internal Force:Tension (symbol: $T$ or $F_T$; SI unit: newton, N) is the pulling force transmitted axially by means of a string, cable, chain, or similar object. It is the macroscopic result of the intermolecular forces within the object. If you pull on one end of a rope, the first segment of the rope pulls on the second, the second on the third, and so on. Each of these pulls is an action-reaction pair. In an ideal (massless) string, these internal forces transmit perfectly, so the tension has the same value at every point along the string.

Outputs & Effects

• Individual Objects: The acceleration of any single object depends on the vector sum of all forces acting on that object (Newton's Second Law). This sum may include forces that are part of a third-law pair.

• Systems: The acceleration of a system's center of mass depends only on the vector sum of all external forces acting on the system.

Regulation & Limits

• Newton's Third Law is universal; it applies to all known forces, including contact, gravitational, and electromagnetic forces.

• The "action" and "reaction" forces are perfectly simultaneous. There is no time delay.

• The ideal string model is an approximation. Real ropes have mass, which can cause tension to vary from one end to the other, especially if the rope is accelerating.

Key Models & Diagrams

The choice of system is a crucial step in solving dynamics problems. The following matrix shows how defining the system boundary changes the analysis of a person pushing a crate on a frictionless floor.

Key Components & Evidence

• Newton's Third Law: The principle that all forces come in simultaneous, equal-magnitude, opposite-direction pairs acting on interacting objects.

• Force Pair (Action-Reaction Pair): The two forces (${\vec{F}}_{A\text{ on }B}$ and ${\vec{F}}_{B\text{ on }A}$) that constitute a single interaction.

• System: A defined collection of objects whose motion is being analyzed. The choice of system is up to the analyst.

• Internal Force: A force exerted by one part of a defined system on another part. Internal forces always sum to zero.

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

• Free-Body Diagram (FBD): A diagram that isolates a single object (or system) and shows all the external forces acting on it. It is essential for applying Newton's Second Law.

• Tension ($T$): The internal pulling force transmitted through a rope or cable, measured in newtons (N). In an ideal string, its magnitude is constant throughout.

• Center of Mass: The unique point where the weighted relative position of the distributed mass sums to zero. It represents the average position of a system's mass.

• Experimental Evidence: If two spring scales are hooked together and pulled apart, they will always show identical readings, demonstrating the equal-magnitude nature of the force pair.

Skill Snapshots

• Causation

  • The Earth's gravitational pull on a falling apple causes the apple to exert an equal and opposite gravitational pull on the Earth.

  • A bat hitting a baseball causes the baseball to exert an equal and opposite force on the bat, which the batter feels as recoil.

  • The force of a rope pulling a sled forward is caused by the sled pulling backward on the rope with an equal tension force.

• Comparison

  • An action-reaction pair consists of two forces acting on two different objects, whereas balanced forces are two or more forces acting on a single object that sum to zero.

  • Internal forces act between objects within a defined system and sum to zero, while external forces act on the system from the outside and can cause its center of mass to accelerate.

  • An ideal string is a model where tension is uniform and the mass is zero, whereas a real rope has mass, which can cause tension to vary if the rope accelerates.

• Change Over Time (CCOT)

  • Baseline: A cannon and a cannonball are at rest. The total momentum of the cannon-cannonball system is zero, and its center of mass is stationary.

  • Change 1 (The Firing): The explosion exerts a force pushing the cannonball forward and an equal, opposite force pushing the cannon backward. These are internal forces to the system.

  • Change 2 (After Firing): The cannonball moves forward with high velocity and the cannon recoils backward with a smaller velocity.

  • Continuity: Because the explosion was an internal force, the center of mass of the cannon-cannonball system remains in its original position (assuming no external forces like friction).

Common Misconceptions & Clarifications

1. Misconception: The "reaction" force is a result of, and happens after, the "action" force.

   • Clarification: The terms "action" and "reaction" are just labels. The forces are two sides of a single, simultaneous interaction. Neither comes first.

2. Misconception: In an interaction, the more massive or faster-moving object exerts a greater force. (e.g., "A truck hits a car, so the force on the car is bigger.")

   • Clarification: The magnitudes of the forces in an action-reaction pair are always equal. The effects of these forces (the accelerations) are different due to the objects' different masses, according to Newton's Second Law ($a=F/m$). The car experiences a much larger acceleration than the truck.

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

   • Clarification: Forces can only cancel if they act on the same object. Since action-reaction forces act on different objects, they can never cancel. To find an object's acceleration, you sum only the forces acting on that specific object.

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

   • Clarification: This is incorrect. Both forces act on the same object (the book). They are balanced forces, not an action-reaction pair. The reaction to the Earth's gravitational force on the book (${\vec{F}}_{Earth\text{ on }Book}$) is the book's gravitational force on the Earth (${\vec{F}}_{Book\text{ on }Earth}$). The reaction to the table's normal force on the book (${\vec{F}}_{Table\text{ on }Book}$) is the book's contact force on the table (${\vec{F}}_{Book\text{ on }Table}$).

One-Paragraph Summary

Newton's Third Law is a cornerstone of physics, revealing that forces are not isolated events but are components of mutual interactions. It dictates that for any force an object A exerts on an object B, B simultaneously exerts a force of equal magnitude and opposite direction back on A. Crucially, these paired forces act on different objects and thus never cancel one another. By defining a system, we can classify these interactions as internal or external. Internal forces, which always sum to zero, cannot alter the motion of a system's center of mass; this change can only be accomplished by a net external force. This principle explains phenomena from the recoil of a firearm to the constant tension in an idealized rope, providing a powerful tool for analyzing the dynamics of any multi-object system.