EXAM #3 DAY 16: Interacting Objects and Newton’s Third Law: Force Pairs, Free-Body Diagrams, and Multiple Object Systems

Interacting Objects and Force Pairs

Introduction to Multiple Object Systems

In physics, analyzing systems with more than one object requires careful consideration of the forces acting on each object and the interactions between them. This topic is foundational for understanding Newton's Laws, especially when objects interact directly (contact) or indirectly (via strings and pulleys).

• Multiple objects of interest require separate analysis for each object.

• Interactions can be direct (contact forces) or indirect (tension in strings, forces via pulleys).

Mathematical Strategies for Systems of Equations

Solving for Unknowns in Interacting Systems

When analyzing forces in systems with multiple objects, you often encounter systems of equations. These can be solved using several methods:

• Guess and Check: Substitute possible values for variables to see if both equations are satisfied.

• Algebraic Substitution: Solve one equation for a variable and substitute into the other equation.

• Graphical Inspection: Plot both equations and find their intersection point.

Example: Given the equations 150 – y = 10x and y = 5x, solve for x and y.

• Substitute y = 5x into the first equation: 150 – 5x = 10x → 150 = 15x → x = 10

• Then y = 5x = 50

Free-Body Diagrams (FBDs) for Multiple Objects

Drawing and Interpreting FBDs

Free-body diagrams are essential tools for visualizing all the forces acting on each object in a system. Each force is represented as an arrow pointing in the direction of the force, with its tail at the point of application.

• Draw a separate FBD for each object.

• Include all forces: normal, weight, applied, friction, tension, etc.

• Label each force clearly, indicating both the agent and the object acted upon.

Newton’s Third Law and Force Pairs

Definition and Identification of Force Pairs

Newton’s Third Law states that for every action, there is an equal and opposite reaction. This means that forces always come in pairs, known as force pairs or action-reaction pairs.

• Each force in a pair acts on a different object.

• The forces are equal in magnitude and opposite in direction.

• Force pairs are always of the same type (e.g., both are normal forces, both are tension forces).

Example: The force by box A on box B and the force by box B on box A are a force pair.

Practice: Identifying Force Pairs

• Force by hand on box A → Force by box A on hand

• Normal force by floor on box A → Normal force by box A on floor

Analyzing Multiple Interacting Objects

Contact Forces and Internal Forces

When two or more objects are in contact, they exert forces on each other. These internal forces are crucial for understanding the motion of the system.

• Draw FBDs for each object to identify all forces.

• Internal forces (e.g., force by box A on box B) are equal and opposite to the force by box B on box A.

Example Problem: Two Boxes in Contact

Suppose a hand pushes on box A (10 kg), which is in contact with box B (5 kg), with a force of 150 N. If friction is negligible:

• Write Newton’s Second Law for each box:

$150 - F = 10a$ $F = 5a$

• Solve for F and a:

$150 - 5a = 10a \implies 150 = 15a \implies a = 10 \text{ m/s}^2$ $F = 5 \times 10 = 50 \text{ N}$

Newton’s Third Law in Experiments

Types of Interactions

• Tug-o-War: Two objects pull on each other.

• Collision: Two objects crash into each other.

• Moving as One: Two objects are pressed together and move together.

In all cases, the force pairs are equal in magnitude and opposite in direction, as predicted by Newton’s Third Law.

Indirect Interactions: Strings and Pulleys

Ideal Strings and Pulleys

When objects interact via strings and pulleys, the tension force transmits the interaction. For ideal strings and pulleys (massless and frictionless), the tension is nearly the same throughout the string.

• Conditions for equal tension:

  • String mass is negligible compared to the objects.

  • Pulley mass and friction are negligible compared to the tension force.

Example: Pulley System with Two Masses

Consider two masses, M and m, connected by a massless string over a frictionless pulley. The tension in the string is the same on both sides, and both masses have the same magnitude of acceleration.

$M = 1.5 \text{ kg},\ m = 0.5 \text{ kg}$ $T = ?$ $a = ?$

• Draw FBDs for each block.

• Write Newton’s Second Law for each block and solve the system of equations.

Summary Table: Key Concepts for Interacting Objects

Concept	Description	Key Equation
Newton’s Third Law	For every action, there is an equal and opposite reaction.	$F_{A \ on\ B} = -F_{B \ on\ A}$
Free-Body Diagram	Diagram showing all forces acting on an object.	Sum of forces: $\sum F = ma$
System of Equations	Used to solve for unknowns in multi-object systems.	e.g., $F_{net,1} = m_1 a$, $F_{net,2} = m_2 a$
Ideal String/Pulley	Transmits force with negligible mass and friction.	$T_{left} \approx T_{right}$

Key Takeaways

• Always draw separate FBDs for each object in a system.

• Identify and label all force pairs according to Newton’s Third Law.

• Use systems of equations to solve for unknown forces and accelerations.

• For ideal strings and pulleys, tension is nearly the same throughout the string.