# MECHANICS

## 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.

## 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

- 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.

3. Dynamics of material moint: 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. Relative motions: galileian relativity.

5. Applications of Newton’s laws

Contact forces: tension, normal force; forces of static and dynamic friction; elastic force and Hooke’s law. Relative motions: non-inertial reference systems, forces of inertia, Coriolis force.

6. 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.

7. 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 weight; static equilibrium of a rigid body. Rolling motion of rigid bodies.

8. 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 reference systems; Koenig theorem for angular momentum; angular momentum conservation. Precessional motions: gyroscopes, spinning top.

9. 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.

10. Potential energy and mechanical energy conservation

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.

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.

13. Gravitation

Newton’s gravitation law; motion equation for a system of two bodies; measurement of constant G; gravity at the Earth surface; mass spherical distribution (shells theorems); gravitational potential energy, escape velocity; multi-body systems; motion of planets and satellites: Kepler laws; energy and orbits; gravitational field; inertial and gravitational mass.

14. 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.

15. 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.

16. 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.

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

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; wave packet; propagation in dispersive means; standing waves; beats; Doppler effect; shock waves.

19. 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.

## Recommended readings

Suggested textbooks:

FISICA 1

Meccanica - Acustica - Termodinamica

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

V edizione

Casa Editrice Ambrosiana (CEA), Milano, 2003

ISBN 8840812547

Fisica - Vol. I

P. Mazzoldi, M. Nigro e C. Voci

II edizione

Edizioni Scientifiche ed Universitarie (EdiSES), Napoli, 2000

ISBN: 8879591371

## Teaching methods

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 an eventual final exam in written and oral form will be given. A final grade will be proposed to the students if the comprehensive grade of mid-term exams is above a specific threshold. An integrative exam (oral or written and oral) will be requested instead if the comprehensive grade of mid-term exams is sufficient but under the threshold. 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.