60/64/183 - MECHANICS 1
Academic Year 2021/2022
Free text for the University
FRANCESCO DEMONTIS (Tit.)
- Teaching style
- Lingua Insegnamento
|[60/64] MATHEMATICS||[64/00 - Ord. 2017] PERCORSO COMUNE||8||64|
1. Acquiring knowledge and understanding.
The course is devoted to students in the second year of the Bachelor's degree in Mathematics. It aims to provide a working knowledge of the main facts on the kinematics and dynamics of the rigid bodies (in particular, the concepts of equilibrium position and the analytic criteria necessary to find them). These topics are presented by providing a rigorous theoretical justification.
2. Applying knowledge and understanding.
Possible applications of the methods treated during the course will be discussed whether for the solution of mathematical problems or for the solution of problems arising in in other disciplines such as Physics or Engeeniring.
3. Making informed judgements and choices.
This course allows assiduous students to achieve knowledge and understanding sufficient for applying the studied techniques to the solution of problems which typically can be encountered in Mechanics.
4. Communicating knowledge and understanding.
The evaluation of the written test keeps into account the ability of the student to give a methodical and consistent exposition of the topics of the program necessary to find the solutions of the assigned exercises. His communicating knowledge is further analyzed during the oral interview.
5. Capacities to continue learning.
This course allows assiduous students to acquire a basic expertise which is sufficient to understand advanced mathematical texts for widening autonomously their knowledge in Analytical Mechanics.
The course requires a good knowledge of the basic concepts of analysis (calculus 1 and 2), linear algebra (Geometry 1)and analytic geometry (Geometry 2).
1. Vectorial spaces: Elements of vectorial calculus. Vectorial equations. Scalar Invariant, vectorial invariant and central axis. Definition and properties of the centre of a system of applied and parallel vectors with a non zero sum. An overview on curves and surfaces.
2.Kinematic: velocity, acceleration, motion of a (given) point under a central force, harmonic and helicoidal motion of a point.
3. Kinematic of the rigid bodies: Poisson's formula and angular velocity. Fundamental formula of the kinematics of the rigid bodies. Mozzi's theorem: instantaneously axis of motion and instantaneously axis of rotation. Some particular rigid motion (translational, rotational, rototranslational and helicoidal motion).
4. Motion of a point with respect two different reference frame moving one respect to the other: theorems of velocity-addition. Coriolis's theorem.
5. Complements of kinematic of rigid bodies: Euler's angles. Motion of a rigid bodies around a fixed point, Poinsot's cones and regular precessions. Earth's regular precession and the precession of the equinoxes.
6. Center of mass and inertia momentum. Inertia Ellipsoid. Inertia tensor and principal axes.
7. Holonomic and anaholonomic constraints.
8. Kinetic energy for a discrete or continuous system: Koenig's theorem. Kinetic energy for a rigid body. Kinetic energy for an holonomic system in lagrangia coordinates. Momentum for a discret or continuous system. Motion around to the center of mass. Angular momentum for a discret or continuous system. Momentum and angular momentum for a rigid body: rigid body with a fixed point. Axial momentum for a rigid body rotating around a fixed axis.
9. Power and work. Conservative stress: potential and potential energy. Elementary work of the internal forces. Expression of the power and of work in the case of a rigid body. Elementary work for an holonomic system: lagrangian components of the forces. Virtual work.
10. Newton equations of the dynamics. Internal and external forces. Fundamental theorems on:
balance of momentum, of the angular momentum and of the kinetic energy (Huygens's theorem),
First integrals and conservation laws.
11. Elements of Analytic Statics. Equilibrium positions and Statics. The principle of virtual work and general conditions of equilibrium. Statics for holonomic system: Expression of the conditions of equilibrium by means of the lagrangian's coordinates.
12. Analytical Mechanics: Symbolic relation and equation of the dynamics (D'Alembert's principle). Lagrange's equations obtained by the symbolic equation of the dynamics. Energy function and conservation of total energy.
The course consists of 64 lecture hours. In order to make the teaching as much efficient as possible, the theoretical topics are immediately supported by exercises and solutions of grading written tests. Furthermore, the teacher will lead 15-20 hours of tutorial activity to assist the students while they study for the final grading. The teacher offers constant assistance to students during the whole year both by personal interviews and by means of e-mail messages.
The main tools to support teaching are the teacher's personal web site
It provides information updated in real time, including: lectures diary reporting the topics treated in each lecture, information on teaching activities, additional documents to support learning, texts and solutions of grading tests.
Verification of learning
The course is assessed by means of a written exam and an oral exam. The written exam includes some exercises which requires operative skills and knowledge of the theory (on the entire contents of the course) to be solved. Only those students who have taken at least 18/30 in the written exam will be admitted to the oral exam. The oral exam includes, eventually, a discussion on the exercises of the written exam where some difficulties has been detected, and some other questions to verify the global understanding of the program.
a) P. Biscari, T Ruggeri, G. saccomandi, M. Vianello, Meccanica Razionale, Springer
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