Notes on Newton's Laws of Motion

Chapter 4: Newton's Laws of Motion

Introduction

Description of motion can be achieved through kinematics covering one, two, or three dimensions.
Inquiry into the cause of motion leads to dynamics, the study of the relationship between motion and forces acting upon objects.
The foundational principles of dynamics were first articulated by Sir Isaac Newton, known as Newton's laws of motion.
Newton's laws were deduced from extensive experiments conducted by various scientists rather than being directly derived by him.

What Are Some Properties of a Force?

A force is defined as a push or a pull acting on an object.
A force is an interaction that occurs between two objects, or between an object and its environment.
Forces are classified as vector quantities, which possess both magnitude and direction.
Notation:
- $F$ (force)
- Visual representation includes push (outward) and pull (inward).

Types of Forces

Normal Force

Normal Force (n): This occurs when an object exerts a force on a surface, resulting in the surface exerting a reactive push on the object perpendicular to the surface.
- Notation: $n$ .
- Classification: Contact force.

Friction Force

Friction Force (f): Apart from the normal force, a surface also exerts a friction force parallel to its surface on an object.
- Classification: Contact force.

Tension Force

Tension Force (T): The force experienced by an object when it is being pulled by a rope, cord, or similar medium.
- Classification: Contact force.

Weight

Weight (w): This refers to the gravitational pull exerted on an object by Earth or another massive body, classified as a long-range force.
- Notation: $w$ .
- Classification: Long-range force.

Magnitudes of Common Forces

The SI unit for force is the newton (N).
Examples of force magnitudes include:
- Gravitational force of the Sun on Earth: $3.5 imes 10^{20} ext{ N}$ .
- Weight of a large blue whale: $1.9 imes 10^{6} ext{ N}$ .
- Maximum pulling force of a locomotive: $8.9 imes 10^{5} ext{ N}$ .
- Weight of a 250-lb linebacker: $1.1 imes 10^{3} ext{ N}$ .
- Weight of a medium apple: $2 imes 10^{-6} ext{ N}$ .
- Weight of the smallest insect eggs: $8.2 imes 10^{-8} ext{ N}$ .
- Weight of a very small bacterium: $1 imes 10^{-18} ext{ N}$ .
- Weight of a hydrogen atom: $1.6 imes 10^{-20} ext{ N}$ .
- Weight of an electron: $8.9 imes 10^{-30} ext{ N}$ .
- Gravitational attraction between a proton and electron in a hydrogen atom: $3.6 imes 10^{-40} ext{ N}$ .

Drawing Force Vectors

A spring balance can be used to depict the pull exerted on an object by measuring applied force.
Vectors are drawn to represent forces, where the length represents the magnitude of the force—longer vectors denote greater forces.

Superposition of Forces

When multiple forces act simultaneously on an object, their combined effect can be represented by the vector sum of these forces, also referred to as the net force.

Decomposing a Force into Its Component Vectors

To effectively analyze forces, a system of perpendicular x- and y-axes is utilized.
Components of a force along these axes can be defined as:
- $F_x$ (horizontal component)
- $F_y$ (vertical component)
The components can be calculated using trigonometric relationships.

Notation for the Vector Sum

The resultant or net force acting on an object is termed the vector sum of all forces, mathematically expressed as:
- $F{ ext{net}} = F1 + F2 + F3 + ext{and so on}$ .

Example 4.1: Finding the Net Force

Scenario: Three wrestlers apply horizontal forces on a champion's belt, visualized in Figure 4.7a.
Given magnitudes are:
- $F_1 = 50 ext{ N}$
- $F_2 = 120 ext{ N}$
- $F_3 = 250 ext{ N}$
Components need to be resolved.

Newton's First Law

An object is considered to be in equilibrium when it is either stationary or moving with a constant velocity (steady speed in a straight line).
For equilibrium, it is essential that no external forces act on the object or that multiple forces sum to a net force of zero:
- $ext{Net Force} = 0$ .

Newton's Second Law of Motion

The acceleration (a) of an object can be expressed mathematically as:
Chapter 4: Newton's Laws of Motion
Introduction
- Description of motion can be achieved through kinematics covering one, two, or three dimensions.
- Inquiry into the cause of motion leads to dynamics, the study of the relationship between motion and forces acting upon objects.
- The foundational principles of dynamics were first articulated by Sir Isaac Newton, known as Newton's laws of motion.
- Newton's laws were deduced from extensive experiments conducted by various scientists rather than being directly derived by him.
What Are Some Properties of a Force?
- A force is defined as a push or a pull acting on an object.
- A force is an interaction that occurs between two objects, or between an object and its environment.
- Forces are classified as vector quantities, which possess both magnitude and direction.
- Notation:
 - $F$ (force)
 - Visual representation includes push (outward) and pull (inward).
Types of Forces
Normal Force
- Normal Force (n): This occurs when an object exerts a force on a surface, resulting in the surface exerting a reactive push on the object perpendicular to the surface.
 - Notation: $n$ .
 - Classification: Contact force.
Friction Force
- Friction Force (f): Apart from the normal force, a surface also exerts a friction force parallel to its surface on an object.
 - Classification: Contact force.
Tension Force
- Tension Force (T): The force experienced by an object when it is being pulled by a rope, cord, or similar medium.
 - Classification: Contact force.
Weight
- Weight (w): This refers to the gravitational pull exerted on an object by Earth or another massive body, classified as a long-range force.
 - Notation: $w$ .
 - Classification: Long-range force.
Magnitudes of Common Forces
- The SI unit for force is the newton (N).
- Examples of force magnitudes include:
 - Gravitational force of the Sun on Earth: $3.5 \times 10^{20} \text{ N}$ .
 - Weight of a large blue whale: $1.9 \times 10^{6} \text{ N}$ .
 - Maximum pulling force of a locomotive: $8.9 \times 10^{5} \text{ N}$ .
 - Weight of a 250-lb linebacker: $1.1 \times 10^{3} \text{ N}$ .
 - Weight of a medium apple: $2 \times 10^{-6} \text{ N}$ .
 - Weight of the smallest insect eggs: $8.2 \times 10^{-8} \text{ N}$ .
 - Weight of a very small bacterium: $1 \times 10^{-18} \text{ N}$ .
 - Weight of a hydrogen atom: $1.6 \times 10^{-20} \text{ N}$ .
 - Weight of an electron: $8.9 \times 10^{-30} \text{ N}$ .
 - Gravitational attraction between a proton and electron in a hydrogen atom: $3.6 \times 10^{-40} \text{ N}$ .
Drawing Force Vectors
- A spring balance can be used to depict the pull exerted on an object by measuring applied force.
- Vectors are drawn to represent forces, where the length represents the magnitude of the force—longer vectors denote greater forces.
Superposition of Forces
- When multiple forces act simultaneously on an object, their combined effect can be represented by the vector sum of these forces, also referred to as the net force.
Decomposing a Force into Its Component Vectors
- To effectively analyze forces, a system of perpendicular x- and y-axes is utilized.
- Components of a force along these axes can be defined as:
 - $F_x$ (horizontal component)
 - $F_y$ (vertical component)
- The components can be calculated using trigonometric relationships.
Notation for the Vector Sum
- The resultant or net force acting on an object is termed the vector sum of all forces, mathematically expressed as:
 - $F{\text{net}} = F1 + F2 + F3 + \dots$ .
Example 4.1: Finding the Net Force
- Scenario: Three wrestlers apply horizontal forces on a champion's belt, visualized in Figure 4.7a.
- Given magnitudes are:
 - $F_1 = 50 \text{ N}$
 - $F_2 = 120 \text{ N}$
 - $F_3 = 250 \text{ N}$
- Components need to be resolved.
Newton's First Law
- An object is considered to be in equilibrium when it is either stationary or moving with a constant velocity (steady speed in a straight line).
- For equilibrium, it is essential that no external forces act on the object or that multiple forces sum to a net force of zero:
 - $F_{\text{net}} = 0$ .
Newton's Second Law of Motion
- The acceleration (a) of an object can be expressed mathematically as:
 - $F_{\text{net}} = ma$ (where $m$ is the mass of the object).
- This principle is foundational in the application of force analysis.
Units of Force, Mass, and Acceleration
- SI Units presented in Table 4.2:
 - Force:
 - SI: newton (N)
 - cgs: dyne (dyn)
 - British: pound (lb)
 - Mass:
 - SI: kilogram (kg)
 - cgs: gram (g)
 - British: slug
 - Acceleration:
 - SI: meters per second squared
 - cgs: centimeters per second squared
 - British: feet per second squared
Example 4.4: Determining Acceleration from Force
- Problem: A worker applies a constant horizontal force of $20 \text{ N}$ on a $40 \text{ kg}$ box on a frictionless floor.
- Objective: Find the box's acceleration.
- Solution involves analyzing the external forces acting on the box, revealing net force and subsequent acceleration using the formula:
 - $F_{\text{net}} = ma$ .
Example 4.5: Determining Force from Acceleration
- Problem: A waitress pushes a ketchup bottle ( $m = 0.45 \text{ kg}$ ) along a counter with initial velocity $2.0 \text{ m/s}$ which slows over a distance due to friction.
- Objective: Determine friction force.
- The situation requires understanding the balance between initial momentum and forces acting against the bottle's motion.
Example 4.6: Racing Down the Runway
- Problem Synopsis: A Boeing 737, mass $51,000 \text{ kg}$ , starts from rest and reaches a speed of $70 \text{ m/s}$ after traveling $940 \text{ m}$ under engine thrust.
- Inquiry: Calculate the thrust of each engine generating necessary acceleration.
Mass and Weight
- The weight of an object (within Earth's gravitational influence) is defined as the gravitational force imposed on it.
Newton's Third Law of Motion
- Stated as: If object A applies a force on object B, then object B reciprocally applies a force on object A.
- These forces are equal in magnitude but opposite in direction.
- Equation representation:
 - $F{A \text{ on } B} = - F{B \text{ on } A}$ .
- It is crucial to acknowledge that these forces act on different objects and thus do not cancel each other out.
- Example: A foot kicking a ball causes a reaction force of the ball pushing back on the foot, illustrating action-reaction pairs.
Free-Body Diagrams
- Steps to create a free-body diagram:
 - Identify all acting forces on the object.
 - Establish a coordinate system.
 - Represent the object as a dot at the origin of the coordinate system.
 - Draw relevant vectors for each identified force acting on the object.
 - Illustrate and label the resultant force vector (net force).
This principle is foundational in the application of force analysis.

Units of Force, Mass, and Acceleration

SI Units presented in Table 4.2:
- Force:
- SI: newton (N)
- cgs: dyne (dyn)
- British: pound (lb)
- Mass:
- SI: kilogram (kg)
- cgs: gram (g)
- British: slug
- Acceleration:
- SI: meters per second squared
- cgs: centimeters per second squared
- British: feet per second squared

Example 4.4: Determining Acceleration from Force

Problem: A worker applies a constant horizontal force of $20 ext{ N}$ on a $40 ext{ kg}$ box on a frictionless floor.
Objective: Find the box's acceleration.
Solution involves analyzing the external forces acting on the box, revealing net force and subsequent acceleration using the formula:
- $F_{net} = m imes a$ .

Example 4.5: Determining Force from Acceleration

Problem: A waitress pushes a ketchup bottle ( $m = 0.45 ext{ kg}$ ) along a counter with initial velocity $2.0 ext{ m/s}$ which slows over a distance due to friction.
Objective: Determine friction force.
The situation requires understanding the balance between initial momentum and forces acting against the bottle's motion.

Example 4.6: Racing Down the Runway

Problem Synopsis: A Boeing 737, mass $51,000 ext{ kg}$ , starts from rest and reaches a speed of $70 ext{ m/s}$ after traveling $940 ext{ m}$ under engine thrust.
Inquiry: Calculate the thrust of each engine generating necessary acceleration.

Mass and Weight

The weight of an object (within Earth's gravitational influence) is defined as the gravitational force imposed on it.

Newton's Third Law of Motion

Stated as: If object A applies a force on object B, then object B reciprocally applies a force on object A.
These forces are equal in magnitude but opposite in direction.
Equation representation:
- $F{A ext{ on } B} = - F{B ext{ on } A}$ .
It is crucial to acknowledge that these forces act on different objects and thus do not cancel each other out.
Example: A foot kicking a ball causes a reaction force of the ball pushing back on the foot, illustrating action-reaction pairs.

Free-Body Diagrams

Steps to create a free-body diagram:
- Identify all acting forces on the object.
- Establish a coordinate system.
- Represent the object as a dot at the origin of the coordinate system.
- Draw relevant vectors for each identified force acting on the object.
- Illustrate and label the resultant force vector (net force).