## 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 aim of the course is, from one hand, to give the analytical instruments that allows describing the dynamics of the most simple mechanical and thermodynamic 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 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

2. Kinematics of Material Point

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

4. Applications of Newton’s laws

5. Work and mechanical Energy

6. Dynamics of the systems of material points

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

Part II

8. Dynamics of the rigid body II: angular momentum, statics

9. Energy conservation

10. Collisions

11. Gravitation: phenomenology and Newton’s law

12. Elastic properties of solids

13. Fluid statics

14. Fluid dynamics

15. Oscillatory phenomena

Part III

16. Wave phenomena: elastic waves

17. Thermodynamic systems and Thermology

18. Ideal and real gases

19. Heat and first law of thermodynamics

20. Applications of the first law of thermodynamics

21. Second law of thermodynamics

22. Entropy

## Full programme

Part I

1. Mechanics: introduction

Classical Mechanics and Thermodynamics; Physics and measurements; physical quantities and units. Elements of vector algebra: 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

Material Point scheme. Position, velocity, acceleration vectors: constant-velocity and constant-acceleration motion. Free body fall. Planar motions: projectile motion; circular motion; centripetal acceleration; angular Kinematics. 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. 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. Inertial frames of reference: galileian relativity. Non-inertial frames of reference, fictitious forces. Rotating frames of reference: Coriolis’ force. The earth frame of reference.

5. 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; general treatment of one-dimensional and three-dimensional conservative systems.

6. Dynamics of the systems of material points

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.

7. 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; centre of gravity. Rolling motion of rigid bodies. Work and kinetic energy in the rotational and roto-translational motions.

Part II

8. Dynamics of the rigid body II: angular momentum, statics

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; angular momentum conservation. Static equilibrium of a rigid body.

9. 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.

10. 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.

11. 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.

12. 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.

13. Fluid statics

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.

14. Fluid dynamics

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, lift force.

15. 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.

Part III

16. 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.

17. Thermodynamic systems and Thermology

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.

18. Ideal and real gases

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; the Van der Waals equation of state.

19. 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.

20. Applications of the first law of thermodynamics

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

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.

22. Entropy

Entropy and second law: the entropy-increase principle. Examples of determination of entropy variation for reversible and irreversible processes. Third law of thermodynamics (short account).

## 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

Mid-term exams (in itinere evaluations) 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.

## Other information

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## 2030 agenda goals for sustainable development

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