Learning outcomes of the course unit
A good knowledge of the fundamental principles of the classical mechanics applied to many particles including: free and bound ( rigid bodies and fluids or gases) systems. A satisfying mastery of the fundamental elements of the calculus in order to solve the motion’s equation by determining it as a function of the particular hypothesis (boundary conditions) An efficient ability in distinguishing between a particular and a general approach
corse unit : Mechanics of a single particle ( Autumn term)
course unit: Calculus I ( Autumn term)
Course contents summary
1. Dynamic of many particles systems ( 4 lectures): Center of mass. Linear momentum of a many particles system. Torque and angular momentum of a many particles system. Fundamental equations of dynamics: conditions for the linear and angular momentum conservation. Kinetic energy of a many particles system (Konig ‘s theorem).
2) Dynamics of collisions ( 5 lectures): Impulsive forces and Impulse’s theorem. Analysis of collisions in the center of mass and in the laboratory frame. Elastic and non-elastic collision. Fully nonelastic collisions. Central and oblique collision.
3. Dynamic of rigid bodies (6 lectures): Linear momentum, torque, angular momentum. Tensor o inertia and momentum of inertia. Calculation of the momentum of inertia for regular solids (rod, cylinder, sphere). Rotational Kinetic energy. Theorem of Huygens-Steiner. Dynamic of a rigid body rotating around a fixed and variable axis. Outlines on the precession of the angular momentum. Examples of rigid body dynamics: (i) torque pendulum, (ii) physical pendulum, (iii) ballistic pendulum, (iv) rotation of a wheel (friction effect), (v) rolling sphere.
4. Fields (4 lectures): Scalar and vector fields. Gradient of a scalar function and properties. Curl of a vector function and a sufficient condition for a conservative field. Flux of the field through a surface and divergence of the vector field. Harmonic field . The Laplacian and the Laplace’s equation.
5. Static fluids ( 2 lectures): Definitions of pressure, density and viscosity in a fluid. Stress in a fluid at rest. Fundamental law of the static fluid. Applications: (i) the Pascal’s principle. (ii) pressure in a fluid (Stevino’s law), (iii) pressure measurement by a U-tube and the Torricelli’s barometer, (iv) buoyancy and Archimedes’ principle.
6. Fluid Dynamics (3 lectures): Ideal fluid approximation. Strema line and tube of flow. The continuity equation and the Leonardo’s principle. Bernoill’s theorem and its applications: (i) the Pitot’s tube. (ii) the Venturi meter, (iii) the Torricelli’s theorem. Non ideal fluids: Pouiselle’s equation. Study of the emptying law of a container in the Torricelli’s (ideal fluid) and Poiseuille’s (non ideal fluid) regimes.
7. Oscillations. Mechanic waves (8 lectures): Restoring forces, periodic oscillations and potential energy function. The free harmonic oscillator. Some outlines on damped and forced harmonic oscillator. General considerations on the mechanical waves. Propagation of a mechanical wave and D’ Alanbert’s equation. Harmonic waves. Propagating waves in homogeneous media and phase velocity. Examples of propagating waves in: string, gas, solid rod. Standing waves and resonance modes Examples: (i) vibrating string, (ii) sound in a closed/open tube. Experimental determination of the sound sopeed through the study of the resonance modes in a closed/open tube ( Kundt’s tube). Sound waves propagation and Doppler effect. Supersonic speeds and Mach’s cone.
1) Lecture’s notes
2) Fisica Generale, S. Rosati, Casa Editrice Ambrosiana, Milano
3) Fisica Generale: Meccanica e Termodinamica, R.G.M. Caciuffo, S. Melone, Ed.Masson, Milano
4) Fisica I, D. Halliday, R. Resnick, K.S. Krane, Casa Editrice Ambrosiana, Milano
Planning of the course: The overall course is scheduled in three course units (modules) , 4 credits for each one, for a final assignment of 12 credits. The present second course unit or module is scheduled in the second period ( spring term) and it consists of 32 hours corresponding to 4 credits.
The present course unit includes: lectures, problem class and tests (continuous assessment). Extra hours are also available for tutorial or small group teaching .
Also the lectures are organised on the basis of an interactive approach to the class: collective discussion on particular topics, illustration of attractive examples and frequent references to practical applications and to experiments.
All the course unit activity is strongly correlated to the parallel Laboratory of Mechanics course units.
Exam: Continuos assessment,consisting in a number of written and oral tests tperformed during the teaching period and used to monitor the student progress and contribute to the overall assessment of the three course units (final exam).