15749 - Mechanics

Learning outcomes

At the completing the course , the student has the basic knowledge of Newtonian mechanics of point particles and particle systems.

He/she has learned the use of the vector calculus and is able to apply the general concepts and the basic laws to solve simple problems of kinematic and dynamic

Course contents

• Module 1

  - Math complements. Vector calculus: vectors, vector operations (addition, subtraction, scalar and vector multiplications), properties of vectors, unit vectors. Simple derivatives and integrals of one-variable functions; concepts of partial derivatives of multivariable functions.

  - Physical quantities. The concept of physical quantities, vector and scalar physical quantities. Dimensional analysis. Standard unit of measurement and systems of units of measurement ( S.I.). Homogeneity, isotropy and specular symmetry of space. Polar and axial vectors-

  - Kinematics in one and more dimensions. Position, displacement, velocity and acceleration. Rotational variables (displacement, velocity and angular acceleration). Uniformly accelerated motion. Motion of projectiles. Uniform circular motion and centripetal acceleration. Circular motion in polar coordinates. Galilean transformations for inertial systems (position, speed, acceleration).

  - Dynamics of material point. The laws of Newton for the forces. Mechanical forces (contact and not). Spring force. Static and dynamic friction.

  Work and energy. Work of forces. Kinetic energy and theorem of live forces. Conservative and non-conservative forces. Definition of power.

  - Potential energy for conservative forces. Circulation of a vector field of conservative forces Conservation of mechanical energy for a material point. Equilibrium: stable, unstable and indifferent.

  - Non-inertial reference systems. Real and fictitious forces (drag, centrifugal and Coriolis). Centrifugal force.

  - Static. Static equilibrium. Torque. Introduction on forces of compression and tension. Elasticity.

  - Systems of particles. The center of mass. Newton's laws for particle systems. Momentum. Conservation of momentum. Internal and external forces to the system of material points and change in internal energy. Conservation of energy.

  - Collisions between material bodies. Elastic and inelastic collisions. Collisions seen in different reference frames.

  - Newton's Gravitation. Inertial and gravitational messes. Spherical body: total mass in its center. The acceleration of gravity at the Earth's surface. Gravitational potential energy. Movement of the planets, and Kepler's laws.

  - Rotational dynamics: Rotational kinetic energy. Moment of inertia. Newton's laws for the rotary motion. The rotational kinetic energy conservation. Centripetal force. Work and power in rotations.
  Roto-translational motion and rolling friction. Angular momentum and conservation of angular momentum: inertial axes of rotation; the spinner. Determination of G.

   

  Module 2

  Problem solutions on the arguments of Module 1.

Readings/Bibliography

• Focardi, Massa, Uguzzoni, Villa: Fisica Generale (meccanica e termodinamica), C.E.A. Milano
• R. Resnick, D. Halliday, K. Krane: Fisica 1, C.E.A. Milano
• Mazzoldi, Nigro, Voci: Fisica (Volume I), EdiSES Napoli
• Villa, Uguzzoni: Esercizi di Fisica: Meccanica, C.E.A. Milano
• The The Feynman Lectures https://www.feynmanlectures.caltech.edu/

Teaching methods

Lectures at the blackboard and solution of problems in the classroom.

Assessment methods

The exam is unique for both modules and it consists of a written proof and an oral assessment. Six written proofs are scheduled in each academic year: three in the first Session (January/February); two in the second (June/July) and one in the third (September). The oral assessments start after the corrections of the writing proofs.

The written proof concerns the solution of two distinct mechanics exercises. Each exercise is divided into points (from 3 to 5). As a rule, the difficulty is increasing (the first point is easy, the last is the most difficult). The total time available for the written proof is 2 hours. Sufficiency (18/30) is achieved by completing one of the two proposed exercises. The maximum mark (31/30) will be attributed to the exact resolution of both exercises.

A sufficient score in the written proof is a prerequisite for accessing the oral assessment, in one of the defined dates of the Session. The overall length of the oral assessment is about 30 minutes, divided between the teachers responsible for the two modules (the student is examined first by the teacher of one module and then by the teacher of the other). In the oral exam, the accuracy of the answers and definitions, the clarity and synthesis with which the topics are presented and the ability to connect the various topics will be assessed. Each teacher independently evaluate the test with a grade expressed with a maximum score of 31/30. A necessary condition for passing the oral assessment is that the student is evaluated with score >= 18 by both teachers. The verbalized final mark is the average of the three marks obtained (in the written test and in the two parts evaluated by the two oral teachers).

The student has a maximum of 3 days to decide to accept the final grade. After this period, failure to confirm acceptance is equivalent to refusal (which is recorded as such). According to the university teaching regulations, the refusal of the grade is possible at least once by the teacher. In case of refusal, the student will have to start from the written test.

Teaching tools

 

Office hours

See the website of Maurizio Spurio

See the website of Nicolò Masi