Course: Classical Mechanics

Topic outline

General

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- An- Najah National University
  
  Faculty of Science
  
  Physics Department
  
  Classical Mechanics
  
  Course Name: Classical Mechanics
  
  Course Number: 22352
  
  Textbook: Grant R. Fowles, George L. Cassiday, “Analytical Mechanics”, 7th ed. , Thomson Brooks/Cole, 2005
  
  Prerequisite: Physics 22353 Mathematical Physics or concurrent and DEQ .
- Course Objectives:
  
  To demonstrate knowledge and understanding of the following fundamental concepts in :
  
  Newtonian mechanics in one dimension .
  
  Oscillations .
  
  General motion of a particle in three dimensions .
  
  Newton’s law of motion in non-inertial frames of reference .
  
  Particle motion under central forces .
  
  The dynamics of systems of particles .
  
  Lagrangian and Hamiltonian formulation of mechanics .
  
  To apply the familiar techniques, based on Newton’s laws, to systems in a variety of coordinate systems and references frames
  
  To develop the Lagrangian and Hamiltonian formulations of mechanics which are important in the study of quantum mechanics
  
  To develop an understanding of Classical Mechanics of particles and to develop your math skills as applied to physics.
- Students Learning Outcomes:
  By the end of this course, students are expected to be able to :
  
  Demonstrate knowledge of core principles in mechanics;
  
  Understand the principles of Newtonian mechanics and have a working knowledge of its application;
  
  Understand and answer problems on damped and forced oscillatory systems, and simple coupled systems ;
  
  Demonstrate a working knowledge of classical mechanics and its application to standard problems such as central forces;
  
  Understand and apply Lagrange’s equations to simple physical systems;
  
  Solve dynamical problems involving classical particles by using the Lagrangian and Hamiltonian formulation .
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Topic 1
Topic One :Fundamental Concepts:Vectors (Chapter One)
- Introduction:
  
  In this topic we will cover Vector properties , Scalar & Vector product, Transformation matrix, Derivative of a vector, Position vector of a particle: velocity & acceleration in rectangular coordinates, Velocity & Acceleration in plane polar ,cylindrical & spherical coordinates .
- Objectives :
  
  Review the basic properties of vectors,dot product,cross product,scalar triple product and vector triple vector.
  
  Describe the Cartesian(Rectangular) coordinates for a point in space.
  
  Resolve a vector into its rectangular components.
  
  Describe the polar coordinates.
  
  Calculate the position vector of a particle ,velocity and acceleration in rectangular, spherical and cylindrical coordinates.
- Plan:
  
  First :Study the power point presentation "Chapter one.ppt" and "Review .ppt"
- File
  
  Chapter one File
- File
  
  Review File
- Second :Do the Assignment 1.
  
  Do the Homeworks(1-4).
- Assignment
  
  Assignment 1
- Assignment
  
  Hw#1 Assignment
- Assignment
  
  Hw#2 Assignment
- Assignment
  
  Hw#3 Assignment
- Assignment
  
  Hw#4 Assignment
- Third :Animation for illustration .
- 1. Animation describe Cylindrical coordinates
- File
  
  Cylindrical coordinates File
- 2. Animation about a rotating wheel on a rotating mount
- File
  
  A rotating wheel on a rotating mount File
- 3.Animation about Rotated axes File
- File
  
  Rotated axes File
Topic 2

Not available
Topic 3
Topic Two :Newtonian Mechanics in One Dimension (Chapter Two)
- Introduction :
  
  In this topic we will cover Newton’s laws and inertial systems, Simple applications of Newton’s laws, Constant applied force, Position-dependent forces(Conservative, Potential energy), Time-dependent forces,Velocity-dependent forces.
- Objectives :
  
  Study the motion of the objects under constant applied force F = constant .
  
  Study the time-dependent applied force F = F(t).
  
  Study the velocity-dependent applied force F = F (v) : Fluid resistance.
  
  Study the position-dependent applied force F = F(x): the concepts of kinetic and potential energy.
- Plan:
- First :Study the power point presentation "Chapter two.ppt"
- File
  
  Chapter two File
- Second :Do the Assignment 2.
  
  Do the homeworks (5-7).
- Assignment
  
  Assignment 2
- Assignment
  
  HW#5 Assignment
- Assignment
  
  HW#6 Assignment
- Assignment
  
  HW#7 Assignment
- Third :Animation for illustration .
- 1.Animation which show you a plumb bob at an angle in an accelerating frame of reference
- File
  
  A plumb bob at an angle in an accelerating frame of reference File
- 2. Animation about mass spring in which you can try by your self
- File
  
  mass spring lab File
Topic 4
Topic Three :Oscillations (Chapter Three)
- Introduction :
  
  In this topic we will cover Linear restoring force: Harmonic motion, Damped harmonic motion(Critical damping, underdamping or overdamping), Forced harmonic motion .
- Objectives :
  
  Describe the general characteristics of simple harmonic motion. (e.g.:mass-spring system,simple pendulum).
  
  Apply energy principles to the simple harmonic oscillator,noting that to total energy is conserved if one assumes there are no nonconservative forces acting on the system.
  
  Derive a qualitative description of damped oscillations(overdamped, underdamped,critically damped)and forced oscillations .
- Plan:
- First:Study the power point presentation "Chapter three.ppt
- File
  
  Chapter three File
- Second :Do the Assignment 3.
- Do the Homeworks(8-9).
- Assignment
  
  Assignment 3
- Assignment
  
  Hw#8 Assignment
- Assignment
  
  Hw#9 Assignment
- Third :Animation for illustration .
- 1.Animation which describe displacement versus time for the underdamped harmonic oscillator
- File
  
  displacement_versus_time_for_the_underdamped_harmonic_oscillator File
- 2.The vertical case for the harmonic oscillator in the following animation
- File
  
  The vertical case for the harmonic oscillator File
- 3.The following animation about the critically damped harmonic oscillator
- File
  
  The critically damped harmonic oscillator File
- 4. In the following video you can study Critically damping, overdamped, underdamped oscillations
Topic 5

Not available
Topic 6
Topic Four :General Motion of a Particle in Three Dimensions (Chapter Four)
- Introduction:
  
  General principles, Potential energy function in three-dimensional motion: The Del operator, Projectile motion, The harmonic oscillator in two and three dimensions, Motion of charged particles electric and magnectic fields, Constrained motion of a particle .
- Objectives :
  
  Discuss the motion of a particle in 3-dimensions.
  
  Study the potential energy function in 3-dimensions.
  
  Solve the projectile problem under the effect of gravity only.
  
  Solve the projectile problem under the effect of both gravitational field and the linear air resistance.
  
  Derive the equation for an ellipse.
  
  Study the harmonic oscillator in two & three dimensions.
  
  Study the motion of a charged particle in an electromagnetic field.
  
  Discuss the constrained motion of a particle.
- Plan:
- First:Study the power point presentation "Chapter four.ppt
- File
  
  chapter four File
- Second :Do the Assignment 4.
- Assignment
  
  Assignment 4
- Do the Homeworks(10-11).
- Assignment
  
  Hw#10 Assignment
- Assignment
  
  Hw#11 Assignment
- Third :Animation for illustration .
- 1.The helical path of a particle moving in a magnetic field
- 2.The potential energy function
- 3.Force field gradient of potential energy function
- 4.A particle sliding on a smooth sphere
Topic 7
Topic Five :Noninertial Reference Systems (Chapter Five)
- Introduction :
  
  In this topic we will cover accelerated coordinate systems and inertial forces, Rotating coordinate systems, Dynamics of a particle in a rotating coordinate system, Effect of earth’s rotation, The Foucault Pendulum
- Objectives :
  
  Analyze the motion of the object in the accelerated reference frames ( Noninertial frame ).
  
  Discuss the Pendulum in an accelerated reference frame.
  
  Derive the velocity due to rotation of the primed coordinate system.
  
  Derive the accelerated of the object in the noninertial frame relative to the inertial frame ( fixed frame ).
  
  Derive the Transverse acceleration , Coriolis acceleration & Centripetal acceleration.
  
  Study the Dynamics of a particle in a rotating coordinate system.
  
  Derive the Transverse force , Coriolis force & Centrifugal force.
  
  Study the effects of the Earth's rotation on the particle ( Plumb Bob ) which is at rest on the surface of the earth ( Static Effects ).
  
  Discuss examples relating to the Earth's Ratation.
  
  Study the Dynamic Effects ( motion of a projectile ) ( e.g. : a falling object ).
- Plan:
- First:Study the power point presentation "Chapter five.ppt
- File
  
  Chapter five File
- Second :Do the Assignment 5.
- Assignment
  
  Assignment 5
- Assignment
  
  Hw#14 Assignment
- Assignment
  
  Hw#15 Assignment
- Third :Animation for illustration .
- 1.A point on a rolling disc .
- 2.Pendulum suspended in an accelerating railroad car as seen by the noninertial observer and the inertial observer.
- File
  
  A plumb bob at an angle in an accelerating frame of reference File
- 3.The angular velocity vector of a rotating coordinate system.
- File
  
  The angular velocity vector of a rotating coordinate system File
- 4.Rotating coordinates fixed to a rolling wheel.
- File
  
  Rotating coordinates fixed to a rolling wheel File
Topic 8
Topic Six :Lagrangian Mechanics ( Chapter Ten)
- Introduction :
  
  In this topicn we will cover Hamilton’s variational principle, Generalized coordinates, Lagrange’s equations of motion for conservative systems, Generalized momentaIgnorable coordinates, Forces of constraint: Lagrange multipliers, Generalized forces , Hamilton’s equations .
- Objectives :
  
  1- Discuss Hamilton's variational principle
  
  2- Derive Lagrange's equations of motion for conservative systems
  
  3- Define the generalized momenta & ignorable coordinates
  
  4- Discuss the forces of constraints: Lagrange Multipliers
  
  5- Develop the Lagrangian formulations of mechanics
  
  6- Understand and apply Lagrange's equations to simple physical systems
  
  7- Develop the Hamiltonian formulations of mechanics
  
  8- Derive Hamilton's equations
- Plan:
- First :Study the power point presentation "Chapter ten.ppt"
- File
  
  Chapter ten.ppt File
- Second :Do the Assignment 6.
- Assignment
  
  Assignment 6
- Assignment
  
  Hw#12 Assignment
- Assignment
  
  Hw#13 Assignment
- Third :Animation for illustration .
- Atwood's Machine
- Double Atwood Machine
- 3.Video Lectures:
Topic 9
Topic Seven :Gravitation and Central Forces (Chapter Six)
- Introduction:
  
  In this topic we will cover Gravitational force between a uniform sphere and a particle, Kepler’s Laws of planetary motion :The law of Ellipses(Kepler’s first law),The law of areas(Kepler’s second law),The harmonic law(Kepler’s third law),Potential energy in a gravitational field and in a general central field, Energy equation .of an orbit in a central field, Orbital energies in an inverse-square field, Effective potential, Stability .
- Objectives :
  
  Discuss Kepler's laws of planetary motion.
  
  Derive the differential equation for the orbit.
  
  Derive the equation for an ellipse.
  
  Derive energy equation of an orbit in a central field.
  
  Derive effective potential for the Inverse-Square law of force & limits of the radial motion.
  
  Discuss the nearly circular orbits in central fields: Stability.
- Plan:
- First :Study the power point presentation "Chapter six.ppt"
- File
  
  Chapter six File
- Second :Do the Assignment 7.
- Assignment
  
  Assignment 7
- Third :Animation for illustration .
- 1.Animation of the Kepler's second law for a planetary orbit
- 2.Animation of the Kepler's laws.
- File
  
  kepler's laws File
- 3.Video Lecture:
Topic 10
Topic eight :Dynamics of System of Particles (Chapter seven)
- Introduction :
  
  In this topic we will cover Center of mass and linear momentum of a system, Angular momentum and kinetic energy of a system, Motion of two interacting bodies, Collisions, Oblique collision and scattering, Motion of a body with variable mass: rocket motion .
- Objectives :
- Plan:
- First :Study the power point presentation "Chapter seven.ppt"
Topic 11

Not available
Topic 12
Topic 13
Topic 14
Topic 15
Topic 16

Topic outline

General

To demonstrate knowledge and understanding of the following fundamental concepts in :

Newtonian mechanics in one dimension .

Oscillations .

General motion of a particle in three dimensions .

Newton’s law of motion in non-inertial frames of reference .

Particle motion under central forces .

The dynamics of systems of particles .

Lagrangian and Hamiltonian formulation of mechanics .

To apply the familiar techniques, based on Newton’s laws, to systems in a variety of coordinate systems and references frames

To develop the Lagrangian and Hamiltonian formulations of mechanics which are important in the study of quantum mechanics

To develop an understanding of Classical Mechanics of particles and to develop your math skills as applied to physics.

Demonstrate knowledge of core principles in mechanics;

Understand the principles of Newtonian mechanics and have a working knowledge of its application;

Understand and answer problems on damped and forced oscillatory systems, and simple coupled systems ;

Demonstrate a working knowledge of classical mechanics and its application to standard problems such as central forces;

Understand and apply Lagrange’s equations to simple physical systems;

Solve dynamical problems involving classical particles by using the Lagrangian and Hamiltonian formulation .

Topic 1

Topic One :Fundamental Concepts:Vectors (Chapter One)

Introduction:

First :Study the power point presentation "Chapter one.ppt" and "Review .ppt"

1. Animation describe Cylindrical coordinates

2. Animation about a rotating wheel on a rotating mount

3.Animation about Rotated axes File

Topic 2

Topic 3

Topic Two :Newtonian Mechanics in One Dimension (Chapter Two)

Introduction :

In this topic we will cover Newton’s laws and inertial systems, Simple applications of Newton’s laws, Constant applied force, Position-dependent forces(Conservative, Potential energy), Time-dependent forces,Velocity-dependent forces.

First :Study the power point presentation "Chapter two.ppt"

1.Animation which show you a plumb bob at an angle in an accelerating frame of reference

2. Animation about mass spring in which you can try by your self

Topic 4

Topic Three :Oscillations (Chapter Three)

Introduction :

In this topic we will cover Linear restoring force: Harmonic motion, Damped harmonic motion(Critical damping, underdamping or overdamping), Forced harmonic motion .

Describe the general characteristics of simple harmonic motion. (e.g.:mass-spring system,simple pendulum).

Apply energy principles to the simple harmonic oscillator,noting that to total energy is conserved if one assumes there are no nonconservative forces acting on the system.

Derive a qualitative description of damped oscillations(overdamped, underdamped,critically damped)and forced oscillations .

First:Study the power point presentation "Chapter three.ppt

1.Animation which describe displacement versus time for the underdamped harmonic oscillator

2.The vertical case for the harmonic oscillator in the following animation

3.The following animation about the critically damped harmonic oscillator

4. In the following video you can study Critically damping, overdamped, underdamped oscillations

Topic 5

Topic 6

Topic Four :General Motion of a Particle in Three Dimensions (Chapter Four)

Introduction:

General principles, Potential energy function in three-dimensional motion: The Del operator, Projectile motion, The harmonic oscillator in two and three dimensions, Motion of charged particles electric and magnectic fields, Constrained motion of a particle .

First:Study the power point presentation "Chapter four.ppt

1.The helical path of a particle moving in a magnetic field

2.The potential energy function

3.Force field gradient of potential energy function

4.A particle sliding on a smooth sphere

Topic 7

Topic Five :Noninertial Reference Systems (Chapter Five)

Introduction :

In this topic we will cover accelerated coordinate systems and inertial forces, Rotating coordinate systems, Dynamics of a particle in a rotating coordinate system, Effect of earth’s rotation, The Foucault Pendulum

First:Study the power point presentation "Chapter five.ppt

1.A point on a rolling disc .

2.Pendulum suspended in an accelerating railroad car as seen by the noninertial observer and the inertial observer.

3.The angular velocity vector of a rotating coordinate system.

4.Rotating coordinates fixed to a rolling wheel.

Topic 8

Topic Six :Lagrangian Mechanics ( Chapter Ten)

In this topicn we will cover Hamilton’s variational principle, Generalized coordinates, Lagrange’s equations of motion for conservative systems, Generalized momentaIgnorable coordinates, Forces of constraint: Lagrange multipliers, Generalized forces , Hamilton’s equations .

1- Discuss Hamilton's variational principle

2- Derive Lagrange's equations of motion for conservative systems

3- Define the generalized momenta & ignorable coordinates

4- Discuss the forces of constraints: Lagrange Multipliers

5- Develop the Lagrangian formulations of mechanics

6- Understand and apply Lagrange's equations to simple physical systems

7- Develop the Hamiltonian formulations of mechanics

8- Derive Hamilton's equations

First :Study the power point presentation "Chapter ten.ppt"

3.Video Lectures:

Topic 9

Topic Seven :Gravitation and Central Forces (Chapter Six)

**First :Study the power point presentation "Chapter one.ppt" and "Review .ppt"**