Physics Department, College of Education University of Sulaimani

Subject: Analytical Mechanics Course Book – Year 2

Lecturer's name: Halo Anwar Abdulkhalaq Academic Year: 2016 - 2017

1. Course name 2. Lecturer in charge 3. Department/ College 4. Contact 5. Time (in hours) per week 6. Office hours 7. Course code 8. Teacher's academic profile 9. Keywords

Analytical Mechanics Halo Anwar Abdulkhalaq Physics/ College of education e-mail: [email protected] Theory: 3 4 hours https://sites.google.com/a/univsul.edu.iq/halo-anwar/

Newton's Laws, Vector Analysis, Gravity, Kepler's laws, Lagrangian Mechanics.

10. Course overview: The general structure of this course is centred around the Newtonian mechanics with different approach than that taken in the first year's mechanics course. The approaches used in this course are more analytical, complex and sophisticated but at the same time more useful especially in the complex systems. The Lagrangian and Hainiltonian are one of the methods which we want to reach for dealing with the complex problems in dynamical properties of systems in physics. So that the overview of the course will be as follow. A brief introduction to dimensional analysis and vector algebra; concepts of velocity and acceleration will be given in chapter one. Chapter two will include Newton's laws of motion; motion in one dimension. Inertial frames of reference. The Harmonic motion will be explained in chapter three. While, in chapter four Motion of a particle in three dimensions, potential energy and conservative forces will be studied. In chapter five motion in a noninertial frame of reference and fictitious forces will be analysed. Chapter six will be devoted to Gravitation, central forces, conic sections and Orbits, orbital energy, criteria for stable orbits and Rutherford scattering. In chapter seven there will be expanded discussion about Many particle systems; three-body problem, conservation laws and collisions and rocket motion. Chapter eight will include; rotation of a body about a fixed axis, laminar motion and moments of inertia. In chapter nine rotation of a body in three dimensions, motion of tops and gyroscopes will be discussed.

Finally, Lagrangian and Hamiltonian mechanics will be studied in detail in chapter ten.

11. Course objective: The main objective of this course is to take students to higher level of calculations and analysis in physics. In this course students will learn some analytical techniques to deal with complex problems in physics. Understanding the dynamical properties of any system in physics and formulating the physical phenomena occurring in a system are the main goals of this course. One of the most important techniques in physics to deal with complex systems is Lagrangian and Hamiltonian techniques. Learning these techniques students will have better understanding of the problems and also will have more problem solving skills and abilities in physics. 12. Student's obligation Students are supposed to attend all lectures, and participate in exams and quizzes. Also students should return assignments and exercises on time, this allows them to not get their marks reduced. These obligations are students main responsibilities, in abnormal situations students can notify the lecturer and the department so that exceptional decisions can be considered. 13. Forms of teaching The lecture notes of the course will be given on the day of the lecture, that is by writing the lecture notes on a white board and students are responsible of writing down the notes. Later on the lecture notes may be uploaded to my university webpage. Some lectures require to be presented by a projector, so beside writing down the lecture notes on the board there will be slides presented for more graphical details.

14. Assessment scheme The assessment of this course will be divided into some different categories of assessments , for example there will be two main exams that takes most of the final marks. There might be coursework during the course, this consists of some exercises that students should read and search outside

the lecture notes to be able to solve them and return them on time. Finally, there can be weekly quizzes that is at the start of each lecture student should be ready to answer questions about previous lecture. Two main exams : 40% Quizzes & courseworks 10% Final exam 50% 15. Student learning outcome: At the end of this course, students should have learned the following topics and be able to use the knowledge in the future studies: 1. Understand the vector algebra and be able to use it in other fields of physics. 2. Understand Newton's laws of motion in one and multi-dimensions, also distinguish between inertial and non-inertial frame of references. Then learn the laws of harmonic motion and non-linear oscillatory motion. 3. Learn how the fiction forces work. 4. Understand the Newtonian gravitation, Kepler's laws and central forces in physics. 5. Understand many particle systems, conservation laws and rocket motion. 6. Understand rotation of a body about a fixed axis, laminar motion and moment off inertia. 7. Understand Lagrangian and Hamiltonian mechanics and be able to use it for solving problems in complex systems. 16. Course Reading List and References‌: [1] Grant R. Fowles, and George L. Cassiday, "Analytical Mechanics", Seventh Edition, Thomson Learning Inc., (2005). [2] Dr. Omed Gh. Abdullha lecture notes, Physics dep. School of Science, University of Sulaimani. [3] Louis N. Hand, and Janet D. Finch, "Analytical Mechanics", Cambridge University Press, (1998). [4] Leonard Susskind lectures: https://www.youtube.com/watch?v=ApUFtLCrU90.

17. The Topics: 1. Vector analysis

Lecturer's name Week(1,2,3)

2. Newton's laws of motion; motion in one dimension. Inertial frames of reference.

Week(4,5,6)

3. General motion of a particle in three dimension

Week(7,8)

4. Non-inertial reference systems

Week(9,10)

5. Gravitation, Kepler's laws and central forces.

Week(11,12,13,14)

6. Dynamics of systems of particles, Centre of mass, reduced mass, rocket motion

Week(15,16)

7. Mechanics of rigid bodies; planar motion, moment of inertia. and Motion of Rigid Bodies in Three Week(17,18,19,20) Dimensions

8. Lagrangian Mechanics; Hamiltonian principle, generalized coordinates, Lagrangian equations of motion, Hamiltonian equations.

Week(21,22,23,24)

18. Practical Topics (If there is any) Non

19. Examinations: Generally, the types of questions in this course will be solving problem-like questions, below given a sample of the type of questions that might come up in exams: Q1/ The orbit of a particle moving in a central field is a circle passing through the origin (r=ro cos𝜃). Show that the force law is inverse‐fifth power. Solution:

Q2/ A honeybee hones in on its hive in a spiral path in such a way that the radial distance decreases at a constant rate, r=b-ct, while the angular speed increases at a constant rate, 𝜃 = 𝑘𝑡. Find the speed as a function of time. Solution: We have 𝑟 = 𝑏 − 𝑐𝑡 → 𝑟 = −𝑐 𝑎𝑛𝑑 𝑟 = 0. Thus from 𝑣 = 𝑟 𝒆𝑟 + 𝑟𝜃𝒆𝜃 → 𝑣 = −𝑐𝒆𝑟 + 𝑏 − 𝑐𝑡 𝑘𝑡𝒆𝜃 So → 𝑣 = [𝑐 2 + 𝑏 − 𝑐𝑡 2 𝑘 2 𝑡 2 ]1/2

Q3/A small ball is fastened to a long rubber band and twirled around in such away that the ball moves in an elliptical path given by the equation where b and w are constants. Find the speed of the ball as a function of t. In particular, find v at t =0 and at t = π/2w, at which times the ball is, respectively, at its minimum and maximum distances from the origin.

Solution:

Q4/ A block of mass M has a string of mass m attached to it. A force F is applied to the string, and it pulls the block up a frictionless plane that is inclined at an angle 𝜃 to the horizontal. Find the force that the string exerts on the block. solution:

21. Peer review: 1. Hazhar A. Rasull 2. Rawand Hamamin