| CODE | PHY2110 | ||||||
| TITLE | Relativistic Mechanics and Waves | ||||||
| UM LEVEL | 02 - Years 2, 3 in Modular Undergraduate Course | ||||||
| MQF LEVEL | Not Applicable | ||||||
| ECTS CREDITS | 4 | ||||||
| DEPARTMENT | Physics | ||||||
| DESCRIPTION | Study-unit work: Six worksheets will be handed over during the lecturing period concerning waves and a selection of question taken from the book Special Relativity by A.P. French will be assigned. There will also be some questions embedded in the notes. Model answers for this work will be made available. Aims: The aim of this study-unit is two fold: Part of it is meant to build on the study-unit PHY1160 Waves and Optics furthering on the nature of vibrations and waves, while the other is intended as an introduction to the theory of special relativity. Since the study-unit can be naturally considered as made up two parts, each part will be described separately: Waves: Damped and forced simple harmonic motion will be reviewed. This will include the procedure of how to derive the governing differential equations for a given system as well finding the analytical solutions in the case that the forcing term is a non harmonic function but is either a general periodic or a non periodic function using Fourier analyses. Together with this, the way in which simple harmonic motions can superimpose or be couple with other oscillators and the link between forced and damped simple harmonic motion and electric circuitry will be discussed. This part is also intended to illustrate the difference between linear and nonlinear oscillators and the way in which nonlinear oscillators can be treated. Special relativity: This part explains the reasons for the need to depart from classical mechanics when high speeds or energies are involved and focuses on how these situations are treated mathematically. The notion of Lorentz transformations for coordinates, velocities and accelerations are illustrated together with their applications. This part will include the relativistic Doppler effect. In addition, it is shown how the principles of conservation of energy and momentum can be generalized for high speeds and energy, and these are then applied to various situations. Learning outcomes: At the end of the study-unit, successful students will be able to: Waves: • derive as well as understand the differential equations concerning both coupled as well as isolated system that consist of damped and forced simple harmonic motion, together with the various ways in which they can be solved; • understand the relation between the governing differential equation for mechanical and electrical systems and be able to transpose the methods used for the mechanical systems to the electrical systems; • understand what is meant by the principle of superposition and be able to apply it to solve problems; • understand the difference between linear and nonlinear systems and how we can use a phase plot; • linearize a nonlinear autonomous system around an equilibrium point and determine the nature and type of stability of the equilibrium point; • understanding the basic characteristics of a chaotic system and how Poincaré plots are used to analyze such systems; • understand what is meant by a Fourier series and be able to derive the cosine and sine and the complex Fourier series of periodic functions; • understand what it meant by Fourier transform and be able to derive the Fourier transform of a given function; • be aware of the link between the Fourier transform and the Fourier series; • understand what is meant by and be able to draw amplitude and phase spectra; • understand what is meant by the band width theorem and its implications; • understand amplitude modulation through the use of Fourier transformations; • understand what is meant by the Dirac delta function and be able to use it to solve problems; • understand how is it possible to obtain analytical solution to forced and damped simple harmonic oscillators when the forcing term is a periodic function; • understand how is it possible to obtain analytical solution to forced and damped simple harmonic oscillators when the forcing term is a non-periodic function through the use of a convolution integral; Relativistic mechanics: • state and explain what are reference frames and the reasons that brought to the break down of classical mechanics; • understand and be able to explain the definition of simultaneity and the method used to synchronies clocks as well as the implications these have on our measurements; • understand what the Lorentz transformations for coordinates, speeds and accelerations are and be able to reproduce their derivations. It will also be expected that the student will be able to use these equations to solve problems; • understand what is an event and be able to draw (Minkowski) space time diagrams and use them to solve problems; • understand and be able to explain what is meant by rest frames, and proper and nonproper measurements • understand and explain what is meant by length contraction and time dilation, as well as reproducing their derivation; • understand and explain what is meant by causality; • understand the notion of the space time interval and their invariance and be able to use it to solve problems; • understand and be able to distinguish between what is meant by measuring and looking at objects; • understand, explain and be able to derive the Doppler effect and be able to use the equations to solve problems; • understand how the principle of conservation of energy and momentum can be extended to the relativistic case and the able to use the equations to solve problems on photon absorption or emission, scattering of particles and/or photons and creation of particles. Study-unit content: Waves: Review of forced and damped simple harmonic motion The governing differential equations; the general case of damped simple harmonic motion including heavy; critical and light damping; damped simple harmonic motion with a harmonic forcing and resonance; Superposition of oscillations Rotating vector description of simple harmonic motion; the superposition of n harmonic oscillators; Electrical systems The analogy between electric circuits and mechanical systems; use of techniques from simple harmonic analysis on electric circuits; Introduction to non linear vibrations The differences between linear and non linear vibrations; stable and unstable equilibriums; the phase space; Autonomous systems The definition of autonomous systems; the differential equation; the phase plane analysis; Stability analyses of equilibrium points Eigenvectors and Eigenvalues; stability and the nature of solution curves for the linear systems; Taylor expansion; linearization techniques; stability and the nature of solution curves for the non linear systems; Chaotic systems; Limit cycles; chaotic systems; the Poincaré map; Coupled oscillators Two coupled pendulums; normal coordinates; coupled oscillations of a loaded string; the wave equation for a string; Fourier series Definition; the derivation of Fourier series as a result of inner products; odd and even functions; the complex Fourier series; the line Fourier spectrum; solution of equations of forced and damped simple harmonic oscillators with periodic forcing terms; The Fourier Transform Definition; the continuous Fourier spectrum; the Bandwidth Theorem; Forced response of damped simple harmonic oscillator The delta function; the response to an impulse force; the response of the system to any force – Convolution; Amplitude Modulation Linearity of Fourier transforms; the scaling, time shift, the frequency shift property of Fourier transforms; the Fourier transforms of derivatives; definition of amplitude modulation; Special Relativity: Frames of reference and the principle of relativity Definition of inertial and non inertial frames of reference; the origins of the principle of relativity; transformation laws between inertial frames according to Galileo and Newton; The limitations of classical mechanics The constancy of the velocity of light; the wave and particle theory of light; the Michelson-Morley experiment; the maximum speed of massive object; photons emitted by objects moving at high speed; the energy momentum relation of photons; the energy momentum relation for massive objects; Introduction to special relativity The departure from an ether-wave picture of light; the postulates of the theory of special relativity; the experimental evidence; Simultaneity in an inertial frame Definition of simultaneity; synchronisation of clocks in an inertial frame; the relativity of simultaneity; Space-time diagrams Representation of coordinate system of different reference frames on the same space-time diagram; point events; Lorentz transformations for coordinates Derivation; the space-time invariant; Measurements in relativity Observers and measurements; length contraction; time dilation; rest frames; and proper and nonproper measurements; The symmetry of relativity The symmetry of length contraction; the symmetry of time dilation; Causality Space-time intervals; the subdivision of the space-time diagram into past, future and elsewhere; space like and time like intervals; Relativistic kinematics Transformation of velocities; transverse motion and stellar aberrations; the Doppler effect along a line and at an angle; transformation of acceleration; the twin paradox; Looking as against measuring Looking at the length of moving objects; looking at moving clocks; Momentum Definition of momentum; the law of conservation of momentum; Energy Definition of total energy; the rest energy; the law of conservation of energy; the kinetic energy of the system; Collisions Absorption and emission of photons, the Mössbauer effect, elastic scattering of identical particles in two dimensions, Compton’s effect; Particle productions Need and limitations of particle production; conditions for production; threshold energy; Books The recommended books are: Waves: French, A. P., Vibrations and Waves, Van Nostrand Reinhold Ingard, K. U., Fundamentals of waves and oscillations, Cambridge press Pain, H. J., The Physics of Vibrations & Waves, John Wiley & Sons Special relativity: French, A. P., Special Relativity, Van Nostrand Reinhold The use of other books with similar content would be equivalently good. |
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| ADDITIONAL NOTES | Prerequisites: PHY1140 Electricity and Magnetism PHY1160 Waves and Optics or courses with an equivalent content. |
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| STUDY-UNIT TYPE | Lecture | ||||||
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| LECTURER/S | Pierre Sandre Farrugia |
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The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints. Units not attracting a sufficient number of registrations may be withdrawn without notice. It should be noted that all the information in the description above applies to study-units available during the academic year 2025/6. It may be subject to change in subsequent years. |
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