Vector algebra. Coordinates systems, transformations. Kinematics of a material point and of rigid bodies. Static and dynamic of the material point, of the systems, and of the rigid bodies. Conservative forces fields. Universal gravitation. Shocks.
“Fisica Generale -Meccanica e Termodinamica” -Focardi, Massa, Uguzzoni, Villa - Casa editrice Ambrosiana
Learning Objectives
The main educational objectives of the course are the following:
Knowledge and understanding: In-depth knowledge of the laws of motion and of the classical mechanics of the material point and of the mechanical systems, understanding the motion of simple physical systems under the effect of the forces.
Ability to apply knowledge and understanding: ability to model simple physical systems, identifying the most important causes responsible for their motion, using rigorously the language of mathematics. Ability to apply the knowledge acquired theoretically to describe physical systems different from those already studied.
Communication skills: ability to use a precise mathematical and physical language, necessary for the description of mechanical systems
Learning skills: ability to demonstrate the fundamental theorems and laws of kinematics and mechanics, starting from the experimental physical elementary principles.
Prerequisites
Basic elements of calculus.
Teaching Methods
CFU: 12
Total hours of the course: 300
Number of hours for classroom lectures: 108
Classroom activities are divided into Lectures (72 hours) and Exercises (36 Hours).
In addition there is a Tutor assigned for the course, which helps students with 'Dynamic exercises', based on a deep interaction with the students.
The examination is separated in a written part and an oral part.
The written test consists in the solution of mechanical exercises, involving both material points and rigid bodies.
The oral examination consists in a series of questions on all the possible topics listed in the extended program, and can also consist in the solution of a simple mechanical example, to verify that the student is able to apply to the study of the motion of a physical system the acquired theoretical knowledge.
The evaluation parameters of the exam are:
1) the ability to demonstrate a thorough knowledge of the main topics of the course, with the demonstration of the theorems that are derived from the experimental principles of mechanics
2) the ability to critically analyze a mechanical system and to derive and solve the system's equations of motion
3) the ability to model a simple physical system by identifying the most important characteristics that are responsible for the motion
4) the ability to use a precise mathematical and physical language, necessary for the description of mechanical systems
The written test and the oral test have the same weight in the overall assessment of the exam.
Course program
Vector algebra:
Vectors and operations with vectors.
Unit Vectors.
Decomposition of vectors along oriented directions.
Orthogonal Cartesian systems, unit vectors i, j, k.
Moment of an applied vector.
Position vector.
Flat and spherical polar coordinates.
Kinematics of the material point:
Reference systems and schematization of the material point.
Trajectory and curvilinear abscissa.
Average speed, instantaneous speed and its Cartesian and intrinsic representation.
Average acceleration, instant acceleration and its Cartesian and intrinsic representation.
Straight and uniformly accelerated rectilinear motion
Circular motion, uniform and uniformly accelerated, in Cartesian, polar and intrinsic representation.
Centripetal acceleration.
Periodic phenomena, period and frequency.
Harmonic motion.
Parabolic motion.
Change of the reference system: law of transformation of velocities and accelerations.
Coriolis acceleration.
Kinematics of the rigid body:
The rigid body.
Fundamental formula of the kinematic of rigid bodies.
Number of degrees of freedom of a rigid body.
Examples of motion of rigid bodies:
1) translational motion
2) rotational motion around a fixed axis
3) pure rolling motion
Dynamics of the material point:
Operational definition of force.
First principle of dynamics.
Inertial reference systems.
The second principle of dynamics and examples of its application.
Momentum and angular momentum.
Impulse of a force.
Impulse theorem.
Conservation of angular momentum for central forces.
Applications of dynamics principles: constant forces, elastic forces, static and dynamic friction forces, viscous friction forces.
Universal gravitation law:
Derivation of the law of universal gravitation from the laws of Kepler
Inertial mass and gravitational mass.
Dynamics in non-inertial reference systems:
Apparent forces.
Centrifugal force.
Coriolis force.
The earth as a non-inertial reference system: dependence of g from latitude, deflection towards east of masses in free fall, Foucault pendulum.
Work and energy:
Work of a force.
Conservative forces.
Potential energy of the weight force, elastic force, central forces with spherical symmetry, gravitational force, centrifugal force.
Conservation of mechanical energy.
Potential energy in one-dimensional systems.
Stationarity of potential energy and equilibrium.
The harmonic oscillator as an example of transformation of potential energy into kinetic energy and vice-versa.
Time solution of the equation of motion with energy conservation.
Dynamics of material point systems:
First cardinal equation of the dynamic of the systems.
Center of mass.
Motion of the center of mass.
Properties of the center of mass.
Calculation of the center of mass for sets of material points and for extended bodies.
Motion of two isolated masses connected to a spring.
Problem of the two bodies:
Solution for the relative motion and for the motion of one of the two bodies relative to the center of mass. Reduced mass.
Relative Sun-Earth motion.
Second cardinal equation of the dynamic of the systems.
Angular momentum for a pair of two point-like masses at a fixed distance.
Konig's theorem of the center of mass.
Potential energy.
Rockets.
Shock:
Conservation of momentum and angular momentum in isolated systems.
Shock in the center of mass system.
Elasticity coefficient.
Single-dimensional elastic and non-elastic shocks.
Static and dynamic of rigid bodies:
Conditions of equilibrium of rigid bodies.
Stability of the equilibrium.
Momentum for rigid bodies translating and rotating.
Angular momentum for rigid bodies rotating around a fixed axis - Component parallel and perpendicular to angular velocity.
Moment of inertia.
Rotational dynamics of a rigid body around a fixed axis.
Examples of dynamics of rigid bodies: Flechter and Atwood apparatus, Physical pendulum, rotating projectile with and without viscous friction, Magnus effect.
Kinetic energy of rigid bodies.
Konig's theorem for rigid bodies.
Work of external forces on rigid bodies.
Dynamics of rigid systems in the presence of conservative forces only.
Shocks involving rigid bodies with impulsive and non-impulsive external forces.