# Matematica | RATIONAL MECHANICS

## Matematica RATIONAL MECHANICS

 0512300015 DIPARTIMENTO DI MATEMATICA EQF6 MATHEMATICS 2022/2023

 OBBLIGATORIO YEAR OF COURSE 3 YEAR OF DIDACTIC SYSTEM 2018 SPRING SEMESTER
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/07 9 72 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
 FRANCESCA PASSARELLA T
ExamDate
APPELLO MECCANICA RAZIONALE08/02/2023 - 09:00
Objectives
THE COURSE AIMS TO PROVIDE AND DEVELOP USEFUL TOOLS FOR A MATHEMATICAL TREATMENT OF THE PROBLEMS AND PHYSICAL PHENOMENA IN THE CONTEXT OF CLASSICAL MECHANICS.
THE COURSE AIMS AT THE FOLLOWING EDUCATIONAL OBJECTIVES:

1) KNOWLEDGE AND UNDERSTANDING
THE STUDENT HAS TO LEARN THE FUNDAMENTAL FACTS OF CINEMATICS AND DYNAMICS OF MATERIAL POINT AND MATERIAL SYSTEM, FREE AND WITH CONSTRAINT.

2) APPLYING KNOWLEDGE AND UNDERSTANDING – ENGINEERING ANALYSIS:
THE AIM OF THE COURSE IS THE ACQUISITION OF GOOD CAPACITY OF FORMULATION AND SOLUTION OF DIFFERENTIAL EQUATIONS DESCRIBING THE DYNAMICS OF MATERIAL SYSTEMS (MATERIAL SYSTEMS PROPERLY MODELED AS: THE MATERIAL POINT, THE RIGID BODY WITH A FIXED AXIS).
3) COMMUNICATION SKILLS
ABILITY TO APPLY KNOWLEDGE IN DIFFERENT SITUATIONS THAN THOSE PRESENTED IN THE COURSE AND ABILITY TO REFINE OWN KNOWLEDGE.

4) COMMUNICATION SKILLS – TRANSVERSAL SKILLS:
ABILITY TO EXPOSE ORALLY, WITH APPROPRIATE TERMINOLOGY, THE TOPICS OF THE COURSE.

Prerequisites
FOR THE SUCCESSFUL ACHIEVEMENT OF OBJECTIVES, STUDENTS ARE REQUIRED BASIC MATHEMATICAL KNOWLEDGE, WITH PARTICULAR REFERENCE TO THE CONCEPTS AND TECHNIQUES FOR SOLUTIONS RELATED TO THE THEORY OF INTEGRATION AND RESOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS. KNOWLEDGE OF ALGEBRA AND VECTOR MATRIX THEORY IS ALSO REQUIRED.
Contents
VECTOR CALCULUS:
CARTESIAN REPRESENTATION OF VECTORS AND OPERATIONS. VECTOR-VALUED FUNCTIONS. APPLICATIONS TO DIFFERENTIAL-GEOMETRIC CURVES. FERNET FORMULAS.
RESULTANT AND RESULTANT MOMENT OF A SYSTEM OF APPLIED VECTORS. CENTRAL AXIS. SYSTEM OF APPLIED VECTORS EQUIVALENT. VECTOR SYSTEM PLANS AND PARALLEL.
(H 10; H4 )

KINEMATICS:
SPEED. ACCELERATION. MOTION IN A PLANE. PLANE MOTIONS. HARMONIC MOTION.
HOLONOMIC SYSTEMS. DEGREES OF FREEDOM AND LAGRANGIAN COORDINATES. KINEMATICS OF RIGID BODIES. POISSON'S FORMULAS. MOZZI'S THEOREM. INSTANTANEOUS AXIS OF ROTATION AND TRANSLATION. RIGID MOTIONS: TRANSLATIONAL MOTION, ROTATIONAL MOTION AND ROTARY AND TRANSLATORY MOTION. RIGID MOTION PLANS.KINEMATICS OF RELATIVE MOTION.
(H 10; H 2)

DYNAMICS AND STATICS OF A FREE MATERIAL POINT.
WORK OF A FORCE. CONSERVATIVE FORCES. ENERGY THEOREM FOR A FREE MATERIAL SYSTEM AND ENERGY CONSERVATION MECHANICS. DIFFERENTIAL EQUATIONS OF MOTION OF A FREE MATERIAL POINT. DIFFERENTIAL EQUATIONS OF MOTION OF A POINT WITH RESPECT TO TWO NON-INERTIAL REFERENCES (APPARENT FORCES, GRAVITATIONAL FORCE). STATIC FREE MATERIAL POINT. HARMONIC OSCILLATOR.
STATICS AND DYNAMICS OF A CONSTRAINED POINT:
EQUATIONS OF MOTION OF A POINT CONSTRAINED. STATICS AND DYNAMICS OF POINT CONSTRAINED TO A CURVE. SIMPLE PENDULUM.
(H 8; H 4)

GEOMETRY OF THE MASSES:
CENTER OF GRAVITY AND PROPERTIES. CENTERS OF GRAVITY OF PLANE SYSTEMS. RADIUS OF INERTIA. MOMENTUM AND MOMENT OF MOMENTUM. THEOREM OF KONIG. KINETIC ENERGY AND MOMENT OF INERTIA. WAY TO VARY THE MOMENT OF INERTIA TO CHANGE THE AXIS AND APPLICATIONS.
(H 8; H 7)

DYNAMICS AND STATICS OF MATERIAL SYSTEMS: CARDINAL EQUATIONS OF DYNAMICS AND STATICS . THEOREM OF MOTION OF THE CENTER OF GRAVITY. WORK OF THE INTERNAL FORCES FOR A RIGID SYSTEM. ENERGY THEOREM AND CONSERVATION OF MECHANICAL ENERGY FOR A CONSTRAINED MATERIAL.
CONDITIONS OF EQUILIBRIUM OF A RIGID BODY: A FREE RIGID BODY, BODY WITH A FIXED POINT AND BODY WITH A FIXED AXIS.
MOTION OF A RIGID BODY WITH A FIXED AXIS FRICTIONLESS. MOTION OF A RIGID BODY WITH A FIXED POINT AND OF A RIGID BODY FREE.
(H 12; H 7)

Teaching Methods
THE COURSE COVERS THEORETICAL LESSONS, DURING WHICH WILL BE PRESENTED DURING THE COURSE TOPICS THROUGH LECTURES AND CLASSROOM EXERCISES, DURING WHICH PROVIDE THE MAIN TOOLS NEEDED FOR SOLVING EXERCISES RELATED TO THE CONTENT OF THE THEORETICAL ASPECTS.
Verification of learning
THE EXAM IS AIMED AT EVALUATING THE KNOWLEDGE AND THE ABILITY TO UNDERSTAND THE CONCEPTS EXPOSED DURING LESSONS AND THE ABILITY TO APPLY SUCH KNOWLEDGE AND FORMULATE THE DIFFERENTIAL EQUATIONS DESCRIBING THE DYNAMIC OF MATERIAL SYSTEMS.
THE EXAMINATION IS DIVIDED INTO A SELECTIVE WRITTEN TEST AND IN AN ORAL INTERVIEW.
WRITTEN PROOF: THIS LASTS 2 HOURS AND CONSISTS IN SOLVING TYPICAL PROBLEMS PRESENTED IN THE COURSE. IN THE CASE THAT THIS TEST IS SUFFICIENT, IT WILL BE EVALUATED BY THREE SCALES.
ORAL TEST: THIS TEST HAS A DURATION OF APPROXIMATELY 20 MINUTES AND EVALUATES THE ACQUIRED KNOWLEDGE.
IN THE FINAL EVALUATION, EXPRESSED IN THIRTIETHS, THE ASSESSMENT OF WRITTEN TEST WEIGHS FOR 40%, WHILE THE ORAL INTERVIEW WEIGHS FOR THE REMAINING 60%.
LAUDE FOLLOWS FROM BRILLIANT WRITTEN PROOF AND ORAL INTERVIEW.
Texts
M. FABRIZIO, ELEMENTI DI MECCANICA CLASSICA, ED. ZANICHELLI.
COURSE NOTES