Physics I: Mechanics – Action-Reaction Pairs and Friction

Action-Reaction Pairs and Friction

Introduction

This study guide covers fundamental concepts in Newtonian mechanics, focusing on action-reaction pairs (Newton's Third Law), friction, free body diagrams, and the analysis of forces in various physical scenarios. These topics are essential for understanding how objects interact and move under the influence of forces.

Free Body Diagrams

Definition and Purpose

• Free body diagrams are graphical representations used to visualize the forces acting on an object.

• Each force is represented as an arrow pointing in the direction of the force, with the length proportional to its magnitude.

• Common forces include gravity (weight), normal force, applied force, and friction.

Example

• A book on a table: Forces acting are the downward weight ($w$), upward normal force ($n$), and any applied force ($F$).

Superposition of Forces

Principle of Superposition

• The superposition principle states that the net force on an object is the vector sum of all individual forces acting on it.

• Acceleration is determined by Newton's Second Law: $\vec{a} = \frac{1}{m} \sum \vec{F}$

Sample Calculation

• If a cart of mass 70 kg is acted upon by two forces, $F_1 = 100$ N at $60^\circ$ and $F_2 = 140$ N at $30^\circ$:

• Calculate components: $a_x = \frac{1}{m} \sum F_x = \frac{171}{70}$ $a_y = \frac{1}{m} \sum F_y = \frac{16}{70}$ $|\vec{a}| = \frac{1}{70} |\vec{F}| = \frac{17}{70}$

Action-Reaction Pairs (Newton's Third Law)

Definition

• Newton's Third Law: For every action, there is an equal and opposite reaction.

• When two objects interact, the force exerted by object A on object B is equal in magnitude and opposite in direction to the force exerted by B on A.

Example

• Pushing a box: The force your hand exerts on the box is matched by the box's force on your hand.

Friction

Types of Friction

• Static friction ($f_s$): Prevents motion between surfaces at rest. Maximum value: $f_{s,\text{max}} = \mu_s N$

• Kinetic friction ($f_k$): Acts when surfaces slide past each other. Value: $f_k = \mu_k N$

• $\mu_s$ and $\mu_k$ are the coefficients of static and kinetic friction, respectively; $N$ is the normal force.

Example

• Pulling a block along a rough surface: To initiate motion, the applied force must exceed the maximum static friction.

Sample Problems

Suspended Traffic Light

• A 10 kg traffic light is suspended by two cables at angles $20^\circ$ and $40^\circ$ with the horizontal.

• Find the tension in each cable using equilibrium conditions:

• Equations: $T_1 = \frac{mg}{\cos \theta_1 + \tan \theta_2} = 86.8\ N$ $T_2 = \frac{mg}{\cos \theta_2 (\tan \theta_1 + \tan \theta_2)} = 106.4\ N$

Horizontal Pull on Frictionless Ice

• Three blocks (10 kg, 20 kg, 10 kg) are pulled with a force of 180 N.

• Find acceleration and tension in ropes:

• Equations: $a = \frac{F}{m_{\text{total}}}$ $T_A = F - m_1 a$ $T_B = m_2 a$

Summary Table: Types of Friction

Type	Symbol	Equation	When Applies
Static Friction	$f_s$	$f_{s,\text{max}} = \mu_s N$	Before motion starts
Kinetic Friction	$f_k$	$f_k = \mu_k N$	During sliding motion

Key Equations

• Newton's Second Law: $\vec{F}_{\text{net}} = m \vec{a}$

• Friction: $f_{s,\text{max}} = \mu_s N$, $f_k = \mu_k N$

• Equilibrium (for suspended objects): $\sum \vec{F} = 0$

Applications

• Analyzing forces in engineering structures (bridges, cables)

• Understanding motion in vehicles and machinery

• Predicting outcomes in sports and everyday activities involving force and friction

Additional info: Some equations and explanations have been expanded for clarity and completeness.