## Learning objectives

The objectives of the Course are:

• to provide a conceptual understanding of the fundamental laws of classical Mechanics, including systems dynamics, and of Thermodynamics, with particular focus on kinematics, Newton’s laws and conservation principles;

• to develop some understanding of main aspects of the dynamics of rigid bodies and of gravitation;

• to treat the mechanics of continuum systems (fluids and elastic properties of solids), the thermology and the thermodynamics from a phenomenological viewpoint;

• to initiate the description of oscillatory and wave phenomena.

The experimental method is the basis for understanding the behaviour of systems and phenomena described above. The most important physical theories will be learned in terms of logical and mathematical structure and experimental evidence. At the end of the course, the student will be able to assess similarities and differences between physical systems, methodologies to be applied, approximations and mathematical methods to be used and must have acquired the ability to apply knowledge and understanding by solving exercises and problems.

The aim of the course is, from one hand, to give the analytical instruments that allow describing the dynamics of the simplest mechanical and thermodynamical systems and examining their qualitative behaviour, even by 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 preparatory to the formalizations described in more advanced courses.

## Prerequisites

- Working knowledge of high school level algebra and trigonometry;

- Differential and integral calculus

- Principles of analytical geometry and of elementary vector analysis

## Course unit content

Part I

1. Mechanics: introduction and vector calculus

2. Kinematics of material Point: one-dimension

3. Dynamics of material point: Force and Newton’s laws

4. Kinematics of material Point: two- and three-dimension

5. Applications of Newton’s laws

6. Relative motion

7. Work and mechanical Energy

Part II

8. Dynamics of the systems of material points I

9. Dynamics of the rigid body I: moment of inertia and Newton’s 2nd law

10. Dynamics of the rigid body II: statics, rolling motion

11. Dynamics of the systems of material points II: angular momentum

12. Energy conservation

13. Collisions

Part III

14. Gravitation: phenomenology and Newton’s law

15. Elastic properties of solids

16. Fluid statics and dynamics

17. Oscillatory phenomena

18. Wave phenomena: elastic waves

19. Thermology - Ideal and real gases

20. Heat and first law of thermodynamics

21. Second law of thermodynamics and Entropy

## Full programme

Part I

1. Mechanics: introduction

Classical Mechanics and Thermodynamics; Physics and measurements; physical quantities and units. Basic vector operations: general properties of vectors; unit vectors; vector components; dot product and cross product; rectangular coordinates in 2-D and 3-D; vector derivatives.

2. Kinematics of Material Point: one-dimension

Material Point scheme. Position, velocity, acceleration vectors: constant-velocity and constant-acceleration motion. Free body fall. 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. Kinematics of Material Point: two- and three-dimension

Cartesian representation and intrinsic representation of position, velocity and acceleration. 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 motion

Inertial frames of reference: galileian relativity. Non-inertial frames of reference, fictitious forces. Rotating frames of reference: Coriolis’ force. The earth frame of reference. Roto-translational motion.

7. Work and mechanical Energy

Work of a constant and of a variable force; work-energy theorem for a particle. Power. Conservative and non-conservative forces; potential energy: elastic, gravitational; mechanical energy and its conservation in isolated conservative systems.

Part II

8. Dynamics of the systems of material points I

Motion of a system of particles; centre of mass and its motion; Newton’s 2nd law for a system of particles; conservation of linear momentum; centre of mass reference system; work-energy theorem. Koenig theorem for kinetic energy; kinetic energy and reference frames. Short account on variable mass systems.

9. Dynamics of the rigid body I: moment of inertia and Newton’s 2nd law

Rigid body scheme, density, centre 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;

10. Dynamics of the rigid body II: statics, rolling motion

Centre of gravity. Static equilibrium of a rigid body. Rolling motion of rigid bodies. Work and kinetic energy in the rotational and roto-translational motions.

11. Dynamics of the systems of material points II: angular momentum

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. Short account on precessional motion.

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 centre of mass.

13. Collisions

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.

Part III

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. Energy and orbits. Short account on gravitational field and potential.

15. Elastic properties of solids

Compression and tension, generalized Hooke’s law; Poisson law, volume deformation; shear deformation; torsion and torsion balance; uniform compression, pressure; plastic deformation.

16. Fluid statics and dynamics

Static equilibrium of a fluid; Stevin and Pascal laws; atmospheric pressure: barometric equation; Archimedean principle and buoyancy. Short account on surface phenomena: surface tension; Laplace formula; capillary phenomena; Jurin’s law. Motion of an ideal fluid, lines of flow and tubes of flow; continuity equation; Bernoulli theorem. Short account on real fluids: laminar flow; viscosity; Hagen-Poiseuille law; turbulent flow, Reynolds number; motion of a body immersed in a fluid; mean resistance.

17. Oscillatory phenomena

One-dimensional oscillating systems; simple harmonic motion; energy in the simple harmonic motion; connection with the uniform circular motion; applications: simple, physical and torsion pendulums; damped free oscillations; forced oscillations and resonance.

18. Wave phenomena: elastic waves

Wave and wave function; phase and phase velocity; harmonic waves, plane waves; D’Alembert equation and its solutions; superposition principle; interference of harmonic waves; standing waves; beats. Propagation of a transverse wave on a string; standing waves on a string, harmonic series. Propagation of a pressure longitudinal wave in a gas; sound speed; sound wave intensity; standing longitudinal waves.

19. Thermology and gases

Thermodynamic system and coordinates; equations of state; thermodynamic processes. Zero-th law of thermodynamics, thermal equilibrium. Temperature: scales and methods of measurements. Thermal expansion of solids. Macroscopic properties of gases. Kelvin temperature scale. Equation of state of an ideal gas. Constant-volume gas thermometer. Kinetic theory of gases: pressure and temperature of ideal gases. Mean free path of molecules. Real gases: pV diagrams, phase transitions and critical parameters.

20. Heat and first law of thermodynamics

Joule experiments; mechanical equivalent of heat. Reversible and irreversible processes. Heat; specific, molar and latent heat. Phase transitions. Calorimetry. Heat propagation. Work in thermodynamic processes. First law of thermodynamics. Examples: thermodynamic processes and cycles. Internal energy of an ideal gas. Molar heat of ideal gases. Molecular degrees of freedom and theorem of energy equipartition. Mayer relation. Isothermal, isobaric, isochoric and adiabatic process of an ideal gas.

21. Second law of thermodynamics and Entropy

Heat engines and heat pumps. Thermal efficiency. Kelvin-Planck and Clausius enunciations of second law. Reversible Carnot cycle. Thermal efficiency of the Carnot cycle. Carnot’s theorem. Absolute temperature scale. Clausius’ theorem. Entropy and second law: the entropy-increase principle. Examples of determination of entropy variation for reversible and irreversible processes.

## Bibliography

Elementi di Fisica – Meccanica - Termodinamica

P. Mazzoldi, M. Nigro e C. Voci

II edizione

Edizioni Scientifiche ed Universitarie (EdiSES), Napoli, 2008

ISBN: 9788879594189

FISICA 1

Meccanica - Acustica - Termodinamica

R. Resnick, D. Halliday, K. S. Krane

V edizione

Casa Editrice Ambrosiana (CEA), Milano, 2003

ISBN 8840812547

Fisica Generale: Meccanica e Termodinamica

S. Focardi, I. Massa e A. Uguzzoni

I edizione

Casa Editrice Ambrosiana (CEA), Milano, 1999

ISBN 8840812725

## Teaching methods

Frontal lesson with help of audio-visual multimedial instruments

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.

## Assessment methods and criteria

Evaluation methods:

Mid-term exams (in itinere evaluations) in written form and a final exam in oral and (eventual) written 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 (equal to or higher than 18/30). In order to sustain the oral exam, which will aim to assign the final grade, students must enrol for exams, despite being exempted from the written test examination.

The final exam, in written and oral form, is mandatory for the students having an insufficient grade of mid-term exams or do not giving the intermediate exams. In such a case will be considered eligible for the oral exam students who reach the written test examination an assessment equal to or greater than 18/30.

The written mid-term exams will require the solution of some exercises and problems relating to specific course topics and the answer to some questions on the theoretical aspects of these topics. The written final exam will have a similar structure but problems and questions will cover all the topics of the course program. The oral final exam will consist of the discussion of the carried out written exams (final or mid-term exams) and deepening of theoretical arguments chosen in the whole program.

## Other information

Office hours: Wednesday, 10.30-11.30 or upon appointment