Classical Mechanics

Module title: Classical Mechanics

Module code: PHYS2006

Introduction

The aim of this course is to continue with and consolidate the Mechanics studied in PHYS1015, Motion & Relativity. Its ideas link with other courses on oscillations and waves, quantum mechanics and condensed matter. Beginning with a review of Newton's Laws applied to systems of particles, the course moves on to rotational motion, dynamical gravity (Kepler's Laws) and motion in non-inertial reference frames. Systems of coupled oscillators are studied.

Aims and learning outcomes

After studying this course students should be able to:

• discuss the linear motion of systems of particles (eg rocket motion)
• define angular momentum for a particle and a system; define moment of inertia and use it in simple problems; describe how steady precession occurs and work out the precession rate
• demonstrate that a spherically symmetric object acts gravitationally like a point with the same total mass located at the object's centre (providing you are outside the object), solve orbit problems using the conservation of angular momentum and total energy, explain the origin of the Coriolis and centrifugal terms in the equation of motion in a rotating frame and solve problems in rotating frames
• identify normal modes for oscillating systems; find normal modes for systems with many degrees of freedom by applying symmetry arguments and boundary conditions.

Summary of syllabus content

The numbers of lectures indicated for each section are approximate.

• Linear motion of systems of particles [4 lectures] - centre of mass; total external force equals rate of change of total momentum (internal forces cancel); examples (rocket motion.)

• Angular motion [6 lectures] - rotations, infinitesimal rotations, angular velocity vector; angular momentum, torque; angular momentum for a system of particles; internal torques cancel for central internal forces; rigid bodies, rotation about a fixed axis, moment of inertia, parallel and perpendicular axis theorems, inertia tensor mentioned; precession (simple treatment: steady precession rate worked out), gyrocompass described

• Gravitation and Kepler's Laws [6 lectures] – conservative forces; gravity; law of universal gravitation; gravitational attraction of spherically symmetric objects; two-body problem, reduced mass, motion relative to centre of mass; orbits, Kepler's laws; energy considerations, effective potential

• Non-inertial reference frames [4 lectures] - fictitious forces, motion in a frame rotating about a fixed axis, centrifugal and Coriolis terms - apparent gravity, Coriolis deflection, Foucault's pendulum, weather patterns
• Normal Modes [4 lectures] - coupled oscillators, normal modes; boundary conditions and Eigen frequencies;

Summary of teaching and learning methods

The method of teaching is by lectures and problem workshops. The course has 30 lectures and 10 compulsory problem classes. These are held in a lecture theatre, led by a Mayflower Fellow and supervised by a team of assistants. The questions set in the weekly-assessed problem sheets will be discussed there.

Summary of assessment methods

Assessment is by written examination at the end of the course. The paper will have a compulsory section A with between 5 and 10 short questions covering the whole courses and a section B where answers to 2 questions out of 4 will be required. The examination counts 80% of the mark. The problem sheets during the course contribute 20% of the mark.

Special features of module

 N/A

Resources

Fowles and Cassiday's book is full of examples and is the recommended text, although it stops short of discussing one-dimensional crystal models. The treatment in Chow's book parallels the course quite closely and has a modern viewpoint. Kibble or Marion and Thornton cover almost everything, but are mathematically more sophisticated. French and Ebison (and French's book on Vibrations and Waves) have good physical explanations but don't cover all the material. Tipler is very gentle and cover most of the course material.

• T L Chow, Classical Mechanics, John Wiley, 1995.
• G R Fowles & G I Cassiday, Analytical Mechanics 5th ed., Saunders, 1993.
• A P French, Vibrations and Waves, MIT Introductory Physics Series, VNR, 1971.
• A P French & M G Ebison, Introduction to Classical Mechanics, VNR, 1986.
• H E Hall, Solid State Physics, John Wiley, 1974.
• T W B Kibble, Classical Mechanics 2nd edition, McGraw-Hill, 1973.
• J B Marion & S T Thornton, Classical Dynamics of Particles & Systems, 4th ed., Saunders 1995.
• PA Tipler, Physics for Scientists and Engineers   (Vol 1, 5th Edition), Freeman, 2004.

Web pages at: http://www.hep.phys.soton.ac.uk/courses/phys2006