L20: Interacting Objects, Contact Forces, and Applications of Newton's Third Law

Section 5.7 Interacting Objects

Newton's Third Law of Motion

Newton's third law is fundamental in understanding how objects interact through forces. It states that every force is part of an action/reaction pair, and these forces always act on different objects.

• Action/Reaction Pair: For every force exerted by object A on object B, there is an equal and opposite force exerted by object B on object A.

• Direction and Magnitude: The two forces in an action/reaction pair point in opposite directions and are equal in magnitude.

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

• Application: This law applies to all interactions, including contact forces and forces at a distance (such as gravity).

Objects in Contact

Analyzing Forces Between Contacting Objects

When two objects are in contact, their motions and the forces acting on them are linked. Analyzing such systems requires considering both objects together.

• Free-Body Diagrams: Draw a separate free-body diagram for each object to identify all forces acting on it.

• Linked Accelerations: If objects are in contact and move together, they share the same acceleration: $a_A = a_B = a$

• Interdependence: The forces on one object depend on the forces on the other due to Newton's third law.

• Key Point: You cannot solve for the motion of one block without considering the motion of the other block.

Tactics Box 5.2: Working with Objects in Contact

To analyze systems of contacting objects, follow these steps:

1. Draw each object separately and prepare a force identification diagram for each.

2. Draw a separate free-body diagram for each object.

3. Write Newton’s second law in component form for each object: $\sum F_x = m a_x$ $\sum F_y = m a_y$

Action/Reaction Pairs in Contact

• Identify the action/reaction pairs of forces. For example, if object A exerts a force on object B, object B exerts an equal and opposite force on object A.

• Newton’s third law allows you to equate the magnitudes of the two forces in each action/reaction pair.

• Objects in contact will have the same acceleration if they move together.

Worked Example: Block 5.10

Horizontal Forces on Sliding Blocks

Consider two blocks, A and B, sliding to the right on a frictionless surface, with a hand H slowing them. Box A has a larger mass than Box B. Only horizontal forces are considered.

• Force Relationships: $F_{\text{on H}} = F_{\text{on B}} = F_{A \text{ on B}} = F_{B \text{ on A}}$ $F_{\text{on H}} > F_{A \text{ on B}} = F_{B \text{ on A}}$ $F_{\text{on H}} < F_{A \text{ on B}} = F_{B \text{ on A}}$

• Key Point: The forces between the blocks are equal in magnitude and opposite in direction, as per Newton's third law.

• Example Application: If the hand applies a force to slow the blocks, the force transmitted between the blocks depends on their masses and the applied force.

Worked Example: Block 5.11

Identifying Action/Reaction Pairs

Given two blocks connected by a string on a frictionless surface, with a force applied and friction present, identify which pair of forces is an action/reaction pair.

• Correct Pair: The friction force acting on A and the friction force acting on the surface (not shown) are an action/reaction pair.

• Key Point: Action/reaction pairs always act on different objects.

• Example: The tension in the string acting on block A and the tension acting on block B are an action/reaction pair.

Worked Example: Example 5.18 Pushing Two Blocks

Calculating Contact Force Between Blocks

A 5.0 kg block A is pushed with a 3.0 N force against a 10 kg block B; both move together on a frictionless surface. Find the force block A exerts on block B.

• Step 1: Draw free-body diagrams for both blocks, showing all forces (applied force, contact force, normal force, weight).

• Step 2: Write Newton’s second law for each block: For block A: $F_{\text{applied}} - F_{A \text{ on B}} = m_A a$ For block B: $F_{A \text{ on B}} = m_B a$

• Step 3: Since both blocks move together, $a$ is the same for both.

• Step 4: Solve for $F_{A \text{ on B}}$: $F_{A \text{ on B}} = \frac{m_B}{m_A + m_B} F_{\text{applied}}$

• Numerical Example: $F_{A \text{ on B}} = \frac{10}{5 + 10} \times 3.0 = 2.0 \text{ N}$

• Key Point: The contact force accelerates only block B, while the applied force accelerates both blocks.

Summary Table: Steps for Analyzing Interacting Objects

Step	Description
1	Draw each object separately
2	Prepare force identification diagram for each object
3	Draw free-body diagram for each object
4	Write Newton’s second law for each object
5	Identify action/reaction pairs
6	Equate magnitudes of action/reaction forces
7	Set accelerations equal if objects move together

Additional info:

• These notes cover the analysis of systems with multiple interacting objects, focusing on Newton's third law and its application to contact forces.

• Examples illustrate how to set up and solve problems involving blocks in contact, including the use of free-body diagrams and Newton's second law.

• Key concepts include the identification of action/reaction pairs, the importance of considering all objects in the system, and the mathematical relationships between forces and accelerations.