Newton's Third Law: Action-Reaction and Interacting Objects

Chapter 7: Newton's Third 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. The first two laws define force and its effects, while the third law explains how forces arise from interactions between objects.

• Newton's First Law: An object at rest remains at rest, and an object in motion continues in a straight line with constant velocity, unless acted upon by a net external force.

• Newton's Second Law: The acceleration a of an object of mass m subjected to a net force F_{net} is given by: $\vec{a} = \frac{\vec{F}_{net}}{m}$ where $\vec{F}_{net} = \vec{F}_1 + \vec{F}_2 + \vec{F}_3 + \cdots$ is the vector sum of all forces acting on the object.

Newton's Third Law

Newton's third law describes the nature of forces between interacting objects. It states that forces always occur in pairs, known as action-reaction pairs.

• Newton's Third Law (in Newton's words): "To every action there is always opposed an equal reaction: OR the mutual actions of two bodies upon each other are always equal, and directed to contrary parts."

• Modern Statement: Every force occurs as one member of an action/reaction pair of forces. The two members act on two different objects, are equal in magnitude, and opposite in direction: $\vec{F}_{A\ on\ B} = -\vec{F}_{B\ on\ A}$

• Conservation of Momentum: In an isolated system, the total momentum is conserved. This is a consequence of Newton's third law.

Action/Reaction Force Pairs

Action-reaction pairs can be classified as contact forces or long-range forces.

• Contact Forces: These include normal force, friction, and tension. For example, if block A pushes on block B, block B pushes back on block A with an equal and opposite force.

• Long-Range Forces: These include gravitational forces. For example, the Earth pulls on the Sun, and the Sun pulls on the Earth with equal and opposite forces.

Examples of Action/Reaction Pairs

• Contact Force Example: $\vec{F}_{B\ on\ A}$ (force of B on A) and $\vec{F}_{A\ on\ B}$ (force of A on B) are equal in magnitude and opposite in direction.

• Gravitational Force Example: $\vec{F}_{Earth\ on\ Sun}$ and $\vec{F}_{Sun\ on\ Earth}$ are equal in magnitude and opposite in direction.

Identifying Action/Reaction Pairs

To analyze interacting objects and identify action/reaction pairs, follow these steps:

1. Represent each object: Draw each object as a circle with a name and label.

2. Identify interactions: Draw connecting lines between objects to represent each interaction (contact or long-range force).

3. Identify the system: Determine which objects are part of the system and which are part of the environment.

4. Draw free-body diagrams: For each object in the system, draw a diagram showing all forces acting on that object (not forces exerted by the object).

5. Connect action/reaction pairs: Use dashed lines to connect the two forces in each action/reaction pair, which must be on different free-body diagrams.

Example Problem: Three Blocks on a Frictionless Table

Three blocks with masses 1 kg, 2 kg, and 3 kg are lined up and pushed by a 12 N force applied to the 1 kg block. The acceleration of the system is:

• Formula: $a = \frac{F_A}{m_1 + m_2 + m_3}$ $a = \frac{12\ N}{1\ kg + 2\ kg + 3\ kg} = 2\ m/s^2$

• Finding Forces Between Blocks: Newton's third law is needed to determine the forces that the blocks exert on each other.

Summary Table: Types of Action/Reaction Pairs

Type of Force	Example	Action/Reaction Pair
Contact (Normal, Friction, Tension)	Block A pushes Block B	$\vec{F}_{A\ on\ B}$ and $\vec{F}_{B\ on\ A}$
Long-Range (Gravity)	Earth and Sun	$\vec{F}_{Earth\ on\ Sun}$ and $\vec{F}_{Sun\ on\ Earth}$

Steps for Solving Interacting-Objects Problems

1. Identify the system and environment.

2. Visualize: Draw a pictorial representation and sketch important points in the motion.

3. Draw interaction diagrams: Show all forces acting on each object and connect action/reaction pairs.

4. Draw free-body diagrams: For each object, show only the forces acting on it.

5. Write equations: Use Newton's second law for each object, incorporating information from the free-body diagrams.

6. Equate magnitudes: Set the magnitudes of action/reaction pairs equal and opposite.

7. Include constraints: Add acceleration constraints, friction models, and other relevant information.

8. Solve: Find acceleration, then use kinematics to determine velocities and positions.

9. Check: Verify units, significant figures, and that the answer addresses the question.

Additional info:

• Momentum conservation is a direct consequence of Newton's third law in isolated systems.

• Action/reaction pairs never act on the same object; they always act on different objects.