# UNIT 2

## 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 systems dynamics, and of Thermodynamics, with particular focus on Newton’s laws and conservation principles;

- to develop some understanding of main aspects of the dynamics of rigid bodies;

- 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 and 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 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

Suggested prerequisites:

- Working knowledge of high school level algebra and trigonometry;

- Differential and integral calculus

- Principles of analytical geometry and of elementary vector analysis

## Course contents summary

Part II

8. 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; work-energy theorem; Koenig theorem for kinetic energy; kinetic energy and reference systems. Variable-mass systems.

9. 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. Work and kinetic energy in the rotational and roto-translational motions.

10. 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.

11. 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.

12. 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

13. Short account on special relativity

Difficulties of classical physics: time, length, speed, energy, light. The postulates of special relativity; consequences of the postulates: time dilation and length contraction; relativistic composition of velocities. Lorentz transformations; measurement of the space-time coordinates of an event; speed transformation; relativity of simultaneity. Relativistic linear momentum; relativistic energy and mass; conservation of energy.

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

Motion equation for a system of two bodies; orbits and Kepler’s laws; energy and orbits. Gravitational field and potential; Gauss theorem and its application to the problem of spherical mass distribution.

16. Elastic properties of solids

Atomic model of elasticity; compression and tension, generalized Hooke’s law; Poisson law, volume deformation; shear deformation; torsion and torsion balance; uniform compression, pressure; relation between elastic modules; plastic deformation.

17. Fluid statics

Static equilibrium of a fluid; Stevin and Pascal laws; atmospheric pressure: barometric equation; Archimedean principle and buoyancy. Surface phenomena: surface tension; non-flat free surfaces: Laplace formula. Capillary phenomena, Jurin’s law.

18. Fluid dynamics

Motion of an ideal fluid, lines of flow and tubes of flow; continuity equation; Bernoulli theorem. Real fluids: laminar flow; viscosity; Hagen-Poiseuille law; turbulent flow, Reynolds number; motion of a body immersed in a fluid; mean resistance, lift force.

19. 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 IV

20. Wave phenomena

Wave and wave function; phase and phase velocity; harmonic waves, plane waves; D’Alembert equation and its solutions; polarization; superposition principle and Fourier theorem; interference of harmonic waves; standing waves; beats.

21. Elastic waves

Propagation of a transverse wave on a string; energy, power, intensity; reflection, refraction, standing waves on a string, harmonic series. Propagation of a pressure longitudinal wave in a gas, displacement wave; sound speed, pressure and density wave; power, intensity; standing longitudinal waves.

22. 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.

23. 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. Molecular speed distribution. Real gases: pV diagrams, phase transitions and critical parameters; the virial equation of state; the Van der Waals equation of state.

24. 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. The black body. Work in thermodynamic processes. First law of thermodynamics. Examples: thermodynamic processes and cycles.

25. Applications of the first law of thermodynamics

Internal energy of an ideal gas. Molar heat of ideal gases. Mayer relation. Molecular degrees of freedom and equipartition of energy theorem. Isothermal, isobaric, isochoric and adiabatic process of an ideal gas. Specific heat of solids; elastic properties of ideal gases.

26. Entropy and 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. Entropy and second law: the entropy-increase principle. Examples of determination of entropy variation for reversible and irreversible processes. Entropy and statistics. Third law of thermodynamics.

## Recommended readings

Suggested textbooks

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

Teaching methodology:

Frontal lesson with help of audio-visual multimedial instruments

esercitazioni in aula (soluzione problemi ed esercizi proposti)

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.

Evaluation methods:

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.