CODE  98269 

ACADEMIC YEAR  2024/2025 
CREDITS 

SCIENTIFIC DISCIPLINARY SECTOR  MAT/07 
LANGUAGE  Italian 
TEACHING LOCATION 

SEMESTER  1° Semester 
MODULES  Questo insegnamento è un modulo di: 
TEACHING MATERIALS  AULAWEB 
AIMS AND CONTENT
LEARNING OUTCOMES
The course gives the competences (in physics and mathematics) necessary for the resolution of problems concerning the dynamics of material systems, with special focus on the constrained rigid body.
AIMS AND LEARNING OUTCOMES
Course attendance and of the class for will allow the students to:
 understand the mathematical foundation of Newtonian mechanics, in particular the kinematics and dynamics of rigid bodies
 be able to calculate the center of mass and the elements of the inertia matrix for systems with the rigidity constraint
 being able to deduce and use the most suitable reference and pole for the setting of the forcetorque equations for rigid bodies.
 being able to identify and apply the most suitable useful form of the forcetorque equations needed to describe an idealized rigid mechanical system
being able to discern the cases in which an aspect of the motion of a rigid mechanical system can be deduced using the work energy theorem
TEACHING METHODS
Classroom teaching, exercises in class. Attendance (and active participation) to the course is recommended.
SYLLABUS/CONTENT
Elements of vector algebra and of the theory of geometric curves:
Free and applied vectors. Vector quantities. Geometrical representation of free and applied vectors. Orthogonal projections. Scalar product. Orthonormal bases. Vector product. Mixed product, double vector product. Parallel and normal component of a vector with respect to another. Matrix algebra. Polar, spherical an cylindrical coordinates.
Absolute kinematics:
The concept of observer. Absolute space and absolute time axioms. Rectification formula. Arc length parameter. Frenet frame. Curvature and Torsion.Velocity, acceleration and their Cartesian representation. Examples of elementary motions (linear, uniform, harmonic, circular, in cartesian poral and spherical coordinates) and their relation with the arc length parameter.
Relative Kinematics:
Relative motion between frames. Transformation of vectors. Orthogonal matrices and change of orthonormal bases. Euler Angles. Angular velocity. Poisson formulas. Composition of angular velocities. Frame dragging motions. Theorem of composition of velocities and accelerations.
Dynamics:
First Principle of dynamics. Inertial mass. Momentum. Conservation of momentum for isolated systems. Second and third principle of dynamics. Work and Power of a force (including ). Conservative forces. Potential of conservative forces. Kinetic energy. Workenergy theorem. Conservation of energy. Rheonomic constraints on a point mass. Friction. Extension of the WorkEnergy theorem in the case of nonconservative forces.
Mechanics of systems of particles:
Systems of applied vectors. Resultant and total angular momentum of systems of vectors. Scalar invariant. Central axis. Reducible and irreducible systems of vectors. Center of a system of parallel applied vectors. Barycenter. Key mechanical quantities of systems of particles. Koenig theorem. Forcetorque equations. WorkEnergy theorem for systems of particles. Conservation laws of systems of particles.
Mechanics of the rigid body:
Reference frame comoving with a rigid body. Act of motion of a rigid body. Velocity and acceleration of the points of a rigid body. Examples of rigid motion. Composition of rigid motions. Key mechanical quantities of the rigid motion. Linear operators and their representation in terms of matrices. Symmetric linear operators and their matrix representation. Symmetric and antisymmetric linear operators. Eigenvalues and eigenvectors. Inertia tensor and its properties. Inertia matrices. Huygens’ theorem and parallelaxis theorem. ForceTorque equations for rigid bodies. Power of a system of forces acting on a rigid body. WorkEnergy theorem for rigid bodies. Poinsot motion and permanent rotations. Stability of permanent rotations. Poinsot Gyroscope. Motion of a free rigid body. Ideal constraints applied to a rigid body. Rigid body with fixed point. Rigid body with fixed axis. Compound pendulum. Pure rolling. Static and dynamic balancing.
RECOMMENDED READING/BIBLIOGRAPHY
The main topics of the course can be found in Biscari P. et al. “Meccanica razionale”, Monduzzi editore (2007)third edition. Lectures also contain elements of Massa E., “Appunti di meccanica razionale” (dipense); Grioli G. “Lezioni di meccanica razionale” Edizioni Libreria Cortina." Padua, Italy (1988); Demeio L. “Elementi di meccanica classica per l’ingegneria”, Città Studi edizioni (2016); Bampi F. , Zordan, C., “Lezioni di meccanica razionale” ECIG 1998; C. Cercignani, “Spazio, Tempo, Movimento”, Zanichelli; M.D. Vivarelli, “Appunti di Meccanica Razionale”, Zanichelli.
Reference for exercises: G. Frosali, F. Ricci “Esercizi di Meccanica Razionale“, Società editrice Esculapio (2013); Muracchini A. et al. ”Esercizi e temi d’esame di Meccanica Razionale” (2013); Bampi F. et al “Problemi di meccanica razionale” ECIG, (1984); G. Belli, C. Morosi, E. Alberti “Meccanica razionale. Esercizi”, edizioni Maggioli Editore (2008); S. Bressan, A. Grioli, “Esercizi di meccanica razionale”, edizioni Libreria Cortina (1981); B. Finzi, P. Udeschini “Esercizi di meccanica razionale”, editore: Massone (1986); V. Franceschini, C. Vernia “Meccanica razionale per l’ingegneria”, edizionei Pitagora (2011).
TEACHERS AND EXAM BOARD
Ricevimento: By appointment. Please send an email to sante.carloni@unige.it
LESSONS
Class schedule
The timetable for this course is available here: Portale EasyAcademy
EXAMS
EXAM DESCRIPTION
The final exam consists of a written and an oral part integrated into a group project. Students will have to solve a rigid body mechanics problem and deliver a paper containing its solution. Subsequently, this resolution will be exposed through an oral presentation which will be integrated, for each student, by some theoretical questions. The final grade is made up of the grade on the written paper (50%), on the presentation (10%), and on the oral questions (40%).
ASSESSMENT METHODS
The written exam consists in the resolution of a problem concerning the dynamics of a rigid body. The exam will allow to evaluate the capability of the students to determine the best resolution strategy of the problem, their ability to perform the calculations necessary, and their knowledge of the theory required to solve the problem.
The oral exam will consist in the presentation and demonstration of some aspects/theorems of Newtonian mechanics for point particles and material systems. The exam will allow to evaluate the capability of organization and presentation of the course material and to verify, via suitable questions, the degree of integration of the course material in the cultural background fo the student.