# 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.

- Fluids. Density and pressure. Ideal fluids. Fluids at rest. Stevin's Law. Principle of Pascal. Archimedes' principle. Pipe flow. Bernoulli's principle and applications. Introduction to real fluids and viscosity.

Module 2

Problem solutions on the arguments of Module 1.

• 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

## Teaching methods

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

## Assessment methods

The exam is unique for both modules and consists of a written test and an oral test (divided into two parts).

The written test concerns the solution of two problems of mechanics, articulated in one or more questions, within 2 hours. The score is reported in 30 (max vote: 30/30).

The writing is preliminary to the oral test, of about 30 min. The threshold score of the writing to access the oral is 18/30. The result of the writing is valid for the oral appeals of the session.

The final grade is the average of the marks of the written test and of the two parts of the oral tests.

## Office hours

See the website of Maurizio Spurio

See the website of Luigi Guiducci