Learning outcomes of the course unit
The objectives of the Course are:
- to provide a conceptual understanding of the fundamental laws of classical Mechanics including particle and systems dynamics, with particular focus on kinematics, Newton’s laws and conservation principles;
- to develop a basic understanding of main aspects of the dynamics of rigid bodies;
- to treat the mechanics of continuum systems (fluids and elastic properties of solids) from a phenomenological viewpoint;
- to initiate the description of oscillatory and wave phenomena and the treatment of gravitation.
The aim of the course is, from one hand, to give the analytical instruments that allows describing the dynamics of the most simple mechanical systems and examining their qualitative behaviour, even through the development of problem solving skill. On the other hand the course will provide the conceptual basis of the newtonian formulation of Mechanics, which is propedeutical to the formalizations described in more advanced courses.
- Working knowledge of high school level algebra and trigonometry;
- Differential and integral calculus
- Principles of analytical geometry and of elementary vector analysis
- Principles of general chemistry.
Course contents summary
1. Mechanics: introduction
Classical Mechanics; Physics and measurements; physical quantities and units.
2. Kinematics of Material Point: one-dimensional motion
Material Point scheme. Position, velocity, acceleration vectors: constant-velocity and constant-acceleration motion. Free body fall. Simple harmonic motion.
3. Dynamics of material point: Force and Newton’s laws
Interactions, the conception of force; Newton’s laws; inertial reference systems; mass and weight; linear momentum and its conservation, general form of the Newton’s 2nd law; impulse and impulse theorem.
4. Two- and three-dimensional motion
Intrinsic representation of the trajectory, velocity and acceleration; constant-velocity and constant-acceleration motion. Planar motions: projectile motion; circular motion; centripetal acceleration; angular Kinematics.
5. Applications of Newton’s laws
Contact forces: tension, normal force; forces of static and dynamic friction; elastic force and Hooke’s law. Dynamics of the uniform circular motion: centripetal force. Simple pendulum and conical pendulum.
6. Relative motions
Inertial frames of reference: galileian relativity. Non-inertial frames of reference, fictitious forces. Rotating frames of reference: Coriolis force. The earth frame of reference. Frames of reference both rotating and translating.
7. Dynamics of the systems of material points
Motion of a system of particles; center of mass and its motion; Newton’s 2nd law for a system of particles; conservation of linear momentum; center of mass reference system; variable-mass systems.
8. Dynamics of the rigid body I
Rigid body scheme, density, center of mass; translation, rotation and roto-translation; torque and moment of force; moment of inertia; Newton’s 2nd law for rotational motions; Huygens-Steiner theorem; center of gravity; static equilibrium of a rigid body. Rolling motion of rigid bodies.
9. Dynamics of the rigid body II
Angular momentum of a particle, of a system of particles and of a rigid body; theorem of angular momentum; symmetry of bodies; angular momentum and frames of reference; Koenig theorem for angular momentum; angular momentum conservation. Precessional motions: gyroscopes, spinning top.
10. Work and kinetic energy
Work of a constant and of a variable force; work-energy theorem for a particle. System of particles and rigid body: work-energy theorem; Koenig theorem for kinetic energy; kinetic energy and reference systems. Work and kinetic energy in the rotational motion. Power.
11. Potential energy and conservation of mechanical energy
Conservative and non-conservative forces; potential energy: elastic, gravitational; mechanical energy and its conservation in isolated conservative systems; general treatment of one-dimensional and three-dimensional conservative systems.
12. Energy conservation
Generalization of the principle of conservation of mechanical energy; work of external forces; internal energy for a system of particles; energy conservation for a system of particles; energy associated to the center of mass. Heat and the first principle of thermodynamics.
Definition of collision; impact forces, conservation principles; one-dimensional elastic collisions; inelastic collisions; angular impulse, moment of body impulse; collisions between particles and rigid bodies.
14. Gravitation: phenomenology and Newton’s law
Motion of planets and satellites: Kepler laws; Newton’s gravitation law; measurement of constant G; inertial and gravitational mass; gravity near the Earth surface. Spherical distribution of mass (shells theorems). Gravitational potential energy, escape velocity: motion of artificial satellites. Central forces.
15.Gravitation: outline of the formal treatment
Meccanica - Acustica - Termodinamica
D. Halliday, R. Resnick, K. S. Krane
Casa Editrice Ambrosiana (CEA), Milano, 2003
Elementi di Fisica – Meccanica - Termodinamica
P. Mazzoldi, M. Nigro e C. Voci
Edizioni Scientifiche ed Universitarie (EdiSES), Napoli, 2008
Fisica Generale: Meccanica
S. Focardi, I. Massa e A. Uguzzoni
Casa Editrice Ambrosiana (CEA), Milano, 2003
A part of the course will be devoted to the solution of problems and exercises, under the supervision of the teacher. A selection of exercises and problems for each topics will be posted on the course web page.
Mid-term exams in written form and a final exam in (eventual) written and oral form will be given. A provisional grade will be proposed to the students if the comprehensive grade of mid-term exams is above a specific threshold. In such a case the final grade is assigned after an oral exam. The final exam, in written and oral form, is mandatory for the students having an insufficient grade of mid-term exams or don’t giving the intermediate exams.