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Mathematical Physics
REGGIO DI CALABRIA
Overview
Date/time interval
Syllabus
Course Objectives
Learning objectives
The discipline lies in between applied mathematics and experimental sciences, being the union of both mathematical and physical point of views, letting a physical problem to be transformed into a mathematical one, and allowing a physical interpretation of the problem solution. At the end of the course the student will be able to tackle and solve several problems related to the motion and the equilibrium of both material point systems and rigid body, within inertial and non inertial reference frames.
Course Prerequisites
Compulsory preliminary teaching:
Mathematical Analysis, Geometry, Physics
Teaching Methods
The course is carried out entirely in the classroom including exercises which will be interactive.
Assessment Methods
Examination modality: Written and oral exams
Assessment and evaluation methods:
The exam consists of a written and an oral test. The written test lasts 150 minutes, with the binding outcome for the next oral test: it is based on 4 closed-ended questions and 4 open-ended questions, and focuses on solving one or more practical problems related to motion and equilibrium of material point systems and rigid material bodies in inertial and non-inertial reference systems. The oral test focuses instead on a discussion of the theoretical foundations necessary to solve the same problems.
The outcome of the exam will result from the sum of partial scores reported in the written and oral test.
The written test aims to ascertain the student's ability to apply the knowledge acquired during the course; the oral exam aims to verify the level of knowledge and understanding of the course contents and to evaluate the autonomy of judgment, learning ability and communication ability.
The final grade will be awarded according to the following evaluation criteria:
30 - 30 cum laude: complete, in-depth and critical knowledge of the topics, excellent language skills, complete and original interpretative skills, full ability to independently apply knowledge to solve the proposed problems;
26 - 29: complete and in-depth knowledge of the topics, excellent language properties, complete and effective interpretative skills, able to independently apply knowledge to solve the proposed problems;
24 - 25: knowledge of the topics with a good degree of command, good language skills, correct and sure interpretative skills, good ability to correctly apply most of the knowledge to solve the proposed problems;
21 - 23: adequate knowledge of the topics but limited mastery of them, satisfactory language skills, correct interpretative ability, more than sufficient ability to independently apply the knowledge to solve the proposed problems;
18 - 20: basic knowledge of the main topics, basic knowledge of technical language, sufficient interpretative ability, sufficient ability to apply the basic knowledge acquired;
Insufficient: does not have an acceptable knowledge of the topics covered during the course.
Texts
Resources and main references
1. P. Giovine & A. Francomano: Notes on Rational Mechanics for the Undergraduate Courses, EquiLibri S.a.s., Reggio Calabria, 2nd edition October 2010;
2. P. Giovine & A. Francomano: Exams in Rational Mechanics for the Undergraduate Courses, EquiLibri S.a.s., Reggio Calabria, June 2009.
(texts in Italian; handouts in English will be provided)
Other references
3. M. Fabrizio: Introduction to Rational Mechanics, Zanichelli (BO) 1994;
4. T. Manacorda: Notes on Rational Mechanics, Pellegrini (PI) 1996;
5. S. Bressan & A. Grioli: Exercises in Rational Mechanics, Cortina (PD) 1990;
6. P.Giovine et al: Completed Exam Papers in Rational Mechanics, (RC) 2002.
Contents
Detailed course program
1. Basic vectorial calculus (1 UFC)
Frames of reference and free vectors in space - Dyads – Vector operations – Scalar and vector product – Mixed product, double vector product, vector division – Applied vectors – Resultant force and resultant angular momentum – Axial momentum – Law of variation of the resultant angular momentum - Torque – Continuous systems – Characteristic vectors and scalar invariant – Central axis – Equivalent and balanced systems – Varignon’s theorem for systems of incident vectors – Equivalence theorems – Elementary operations - Mutual reduction of two systems of applied vectors – Systems of plane applied vectors and funicular polygon – Systems of parallel applied vectors – Centre of a system of parallel applied vectors – Graphic reduction of two parallel applied vectors.
2. Mass geometry (1 UFC)
Mass of a system of material points - One-, two- and three-dimensional continua – Mass density – Centre of gravity of a material system and its characteristics – Material symmetry plane – Moment of inertia of a material system – Law of variation of the moment of inertia with respect to parallel straight lines: Huygens-Steiner theorem - Law of variation of the moment of inertia with respect to converging straight lines – Skew moment and its law of variation with respect to parallel planes – Matrix of inertia – Principal (central) axes and moment of inertia – Body with gyroscopic structure and gyroscope – Law of variation of the central inertial matrix – A priori criteria for principal (or central) inertia axes.
3. Mass kinematics and constraints (1 UFC)
Single point kinematics – Unilateral, bilateral, scleronomic, rheonomic, holonomic, non holonomic constraints – Degrees of freedom of a holonomic material system – Lagrangian coordinates – Rigid motion and rigid body – Rigid body velocity and acceleration – Poisson’s formulae – Particular types of rigid motion: translational, rotational, with sliding axis (helicoidal), polar (with a fixed point) - Euler angles – Outline of relative kinematics - Galileo’s principle – Coriolis theorem - Mutual (and pure) rolling of two rigid surfaces - Mass kinematics: linear momentum, angular momentum and kinetic energy for a material system – Relative motion with respect the centre of gravity of a material system - König’s theorems – Angular momentum and kinetic energy of a rigid motion – Particular cases of rigid motion.
4. Mechanics of free and constrained systems (1,5 UFCs)
Newtonian point dynamics – Point dynamics in a non inertial reference frame and fictitious forces – Internal and external forces with respect to a material system – Zero reduction of the internal forces of a material system – Constraining reaction - Constraining reaction postulate- Coulomb-Morin laws on static and dynamic friction – Constant, positional and resistive forces – Distributed forces – Dynamics fundamental equations- Centre of gravity motion theorem - Axial angular momentum theorem – Dynamics fundamental equations sufficiency in the study of a rigid body system (without proof) – First integrals of motion.
5. Displacements, work, energy and outline of statics of the systems (1,5 UFCs)
Effective, elementary, virtual (reversible and irreversible) displacements – Elementary and virtual displacements expressed as a function of the Lagrangian coordinates – Power and work of a force system – Gyroscopic forces – Stress work acting on a rigid body – Work of internal forces – Perfect constraints and their characterization: point constrained over a fixed curve, a fixed surface, not crossing a fixed surface; rigid body having a fixed point, a fixed or a sliding axis; stiffness and pure rolling constraints (rolling friction) – Zero equality of the constraining reaction elementary work done by perfect and fixed constraint– Motion and equilibrium pure equations – Conservative forces and potential and their expression as a function of the Lagrangian coordinates – Work and kinetic energy theorem – Mechanical energy conservation theorem for bound systems – On the equilibrium of a material system – Statics fundamental equations - Statics fundamental equations sufficiency for the equilibrium of a rigid system – Holonomic system equilibrium – Potential stationariness principle (without proof) – Equilibrium stability – Dirichlet’s theorem (w.d.).
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