Unit RATIONAL MECHANICS I
- Course
- Mathematics
- Study-unit Code
- 55031206
- Curriculum
- In all curricula
- Teacher
- Diletta Burini
- Teachers
-
- Diletta Burini
- Hours
- 63 ore - Diletta Burini
- CFU
- 9
- Course Regulation
- Coorte 2024
- Offered
- 2026/27
- Learning activities
- Caratterizzante
- Area
- Formazione modellistico-applicativa
- Sector
- MAT/07
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- Kinematics and dynamics of material points and rigid bodies. Relative motions and constraints. Fundamental principles of classical mechanics. Mass geometry, cardinal equations of dynamics and first integrals. Statics of mechanical systems. Principle of virtual work and Lagrangian formulation of mechanics.
- Reference texts
- P. Biscari, T. Ruggeri, G. Saccomandi, M. Vianello. Meccanica razionale (Vol. 81). Springer. 2014
- Educational objectives
- The course aims to provide students with the fundamental tools for the mathematical description and analysis of mechanical systems. At the end of the course, students will be able to model simple mechanical systems, analyse their motion and equilibrium using the principles of classical mechanics, and apply the main methods of rational mechanics to problem solving. The course also aims to develop the ability to interpret mechanical phenomena through mathematical models and to use rigorous mathematical language in the formulation and analysis of problems.
- Prerequisites
- Students are expected to have a basic knowledge of mathematical analysis, linear algebra and geometry acquired during the Bachelor's degree programme. In particular, familiarity with differential and integral calculus in one and several variables, vectors and matrices, Euclidean geometry, and the fundamental elements of classical mechanics is required.
- Teaching methods
- The course consists of classroom lectures devoted to the presentation of the theoretical topics and the main methods of rational mechanics. Lectures are complemented by the discussion of relevant examples and applications concerning the kinematics, dynamics and statics of mechanical systems. All teaching material used during the course, including lecture notes, is made available to students through the Unistudium platform.
- Other information
- Class attendance is not mandatory, but it is strongly recommended. All teaching materials used during the course are available on the Unistudium platform.
- Learning verification modality
- Learning assessment consists of a written examination and an oral examination.
The written examination consists of solving problems related to the topics covered in the course and is aimed at assessing the student's ability to apply the methods of rational mechanics to problem solving. A passing grade in the written examination is required in order to take the oral examination.
The oral examination is intended to assess the student's knowledge and understanding of the theoretical contents of the course, the ability to establish connections among the different topics, and the ability to present them in a rigorous and appropriate manner. The final assessment takes into account both the problem-solving skills demonstrated in the written examination and the theoretical preparation shown during the oral examination.
Intermediate tests may be offered during the course according to arrangements communicated by the instructor.
For information on support services for students with disabilities and/or specific learning disorders (DSA), please visit http://www.unipg.it/disabilita-e-dsa. - Extended program
- Point-particle kinematics: description of motion, trajectory, velocity and acceleration. Intrinsic frame and motion in polar coordinates. Plane motions and central motions.
Rigid body kinematics. Body-fixed reference frames, rotation matrices and Euler angles. Angular velocity and Poisson formulas. Translational, rotational and roto-translational motions. Euler, Chasles and Mozzi theorems. Instantaneous centre of rotation. Constraints, degrees of freedom and rolling constraints.
Relative kinematics and dynamics. Composition of velocities and accelerations. Galileo's theorem and Coriolis theorem.
Fundamental principles of classical mechanics. Forces and constraint reactions. Static and dynamic friction. Equations of motion of a material point and equilibrium problems.
Mass geometry. Centre of mass, linear momentum, angular momentum and kinetic energy. König's theorem. Cardinal equations of dynamics and their applications.
Work, power and energy. Conservative forces, potential energy, kinetic energy theorem, conservation of mechanical energy and first integrals of motion.
Moments and products of inertia. Inertia matrix, principal axes and principal moments of inertia. Huygens–Steiner theorem. Dynamics and statics of rigid bodies.
Virtual displacements and virtual velocities. Ideal constraints. Principle of virtual work and its applications to the statics and dynamics of mechanical systems.
Holonomic systems. Potential stationarity theorem. Generalized coordinates and Lagrange equations. Conservative systems, first integrals and symmetries of the Lagrangian. - Obiettivi Agenda 2030 per lo sviluppo sostenibile
- Goal 4 - Quality Education
The course contributes to the achievement of Sustainable Development Goal 4 by providing skills in mathematical modelling and the analysis of mechanical systems, fostering quantitative reasoning, problem-solving abilities, and the scientific background required for further studies and lifelong learning.