#
Mechanics

A.Y. 2021/2022

Learning objectives

Students will learn the Newtonian mechanics of a point mass, of extended systems (fluids and rigid bodies) and will be introduced to special relativity.

Expected learning outcomes

At the end of the course students are expected to:

· Know how to describe the kinematics of a point mass;

· Identify the system of forces acting on a point mass and deduce its motion;

· Know the main dynamics variables (momentum, energy, angular momentum) and their conservation laws;

· Address the dynamics of extended systems (systems of point masses, fluids, rigid bodies);

· Know the properties of the motion in a gravitational field;

· Know the basics of special relativity (space-time transforms, four-vectors, relativistic energy and momentum, massenergy transformations)

The knowledge must be both theoretical (ability to explain topics in details and to answer upon requests of clarification) and practical (ability in solving quantitatively specific problems)

· Know how to describe the kinematics of a point mass;

· Identify the system of forces acting on a point mass and deduce its motion;

· Know the main dynamics variables (momentum, energy, angular momentum) and their conservation laws;

· Address the dynamics of extended systems (systems of point masses, fluids, rigid bodies);

· Know the properties of the motion in a gravitational field;

· Know the basics of special relativity (space-time transforms, four-vectors, relativistic energy and momentum, massenergy transformations)

The knowledge must be both theoretical (ability to explain topics in details and to answer upon requests of clarification) and practical (ability in solving quantitatively specific problems)

**Lesson period:**
First semester

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### CORSO A

Responsible

Lesson period

First semester

Lectures will be held in synchronous mode, according to the timetable, and will be followed remotely, using the ZOOM platform. We don't foresee changes or reduction of the contents. The structure of the exams will be similar to the ordinary one: written test and oral exam, which, if necessary, will be held remotely on

ZOOM. All communication with students, as well as the distribution of materials of study, will take place through the ARIEL platform.

ZOOM. All communication with students, as well as the distribution of materials of study, will take place through the ARIEL platform.

**Course syllabus**

1) Physical quantities, systems of units and dimensional analysis

2) Vector calculus. Sum and difference of vectors. Multiplication of a vector by a scalar. Vector decomposition. Inner (dot) product and vector (cross) product of vectors. Vector as a function of a parameter: vector derivative and integration. Flux of a vector through a surface.

3) Kinematics of a point mass. Velocity. Acceleration. Linear motion: uniform and uniformly accelerated. Motion in the plane: velocity and acceleration in polar coordinates, tangent and normal acceleration. Circular motion. Angular velocity and angular acceleration (including vector notation). Projectile motion. Harmonic motion.

4) Dynamics of a point mass: the three laws of dynamics (Newton's laws of motion). Constant forces, force as a function of speed, position, time. Normal forces. Friction. Momentum. Impulse. Work, power, kinetic energy. Work-kinetic energy theorem. Conservative forces. Potential energy. Conservation of mechanical energy. Calculation of work and potential energy for weight and elastic force. Angular momentum and torques. Central forces and central force motion.

5) Systems of point masses. Centre of mass: properties and motion description. Euler's laws of motion for systems of point masses. Work-kinetic energy theorem. König theorems. Collisions. Ballistic pendulum.

6) Rigid body. Kinematics of rigid bodies. Momentum of inertia. Huygens-Steiner theorem. Dynamics of free and constrained rigid bodies. Rigid bodies rotating around an axis. Physical (compound) pendulum. Roto-translation of a rigid body. Rolling without slipping of spheres and cylinders on a plane. Kinetic energy of translating, rotating, and roto-translating rigid bodies. Work-Kinetic energy theorem and energy conservation for a rigid body.

7) Relative motions. Absolute and relative reference frames. Composition of velocities. Composition of accelerations: Coriolis' theorem. The Earth as a relative frame: variation of gravity acceleration with latitude, east deviation of falling objects, Foucault pendulum. Relative dynamics. Apparent forces.

8) Basics of special relativity theory. Crisis of Galilean relativity principle. Lorentz transformations and invariance of physical laws in inertial systems. Transformation of velocities. Lengths contraction and time dilation. Relativistic momentum. Force. Kinetic energy. Rest energy.

9) Universal gravitation. Gravitational force, gravitational field, work of the gravitational force and gravitational potential energy, gravitational potential. Discussion on potential energy curves. Kepler's laws. Gauss theorem and its applications.

10) Basics of fluid dynamics. Pressure, pressure forces, work of pressure forces. Hydrostatic equilibrium and Stevin's law. Archimedes' principle. Fluid motion: stationary regime and flow rate. Ideal fluid. Bernoulli's theorem.

2) Vector calculus. Sum and difference of vectors. Multiplication of a vector by a scalar. Vector decomposition. Inner (dot) product and vector (cross) product of vectors. Vector as a function of a parameter: vector derivative and integration. Flux of a vector through a surface.

3) Kinematics of a point mass. Velocity. Acceleration. Linear motion: uniform and uniformly accelerated. Motion in the plane: velocity and acceleration in polar coordinates, tangent and normal acceleration. Circular motion. Angular velocity and angular acceleration (including vector notation). Projectile motion. Harmonic motion.

4) Dynamics of a point mass: the three laws of dynamics (Newton's laws of motion). Constant forces, force as a function of speed, position, time. Normal forces. Friction. Momentum. Impulse. Work, power, kinetic energy. Work-kinetic energy theorem. Conservative forces. Potential energy. Conservation of mechanical energy. Calculation of work and potential energy for weight and elastic force. Angular momentum and torques. Central forces and central force motion.

5) Systems of point masses. Centre of mass: properties and motion description. Euler's laws of motion for systems of point masses. Work-kinetic energy theorem. König theorems. Collisions. Ballistic pendulum.

6) Rigid body. Kinematics of rigid bodies. Momentum of inertia. Huygens-Steiner theorem. Dynamics of free and constrained rigid bodies. Rigid bodies rotating around an axis. Physical (compound) pendulum. Roto-translation of a rigid body. Rolling without slipping of spheres and cylinders on a plane. Kinetic energy of translating, rotating, and roto-translating rigid bodies. Work-Kinetic energy theorem and energy conservation for a rigid body.

7) Relative motions. Absolute and relative reference frames. Composition of velocities. Composition of accelerations: Coriolis' theorem. The Earth as a relative frame: variation of gravity acceleration with latitude, east deviation of falling objects, Foucault pendulum. Relative dynamics. Apparent forces.

8) Basics of special relativity theory. Crisis of Galilean relativity principle. Lorentz transformations and invariance of physical laws in inertial systems. Transformation of velocities. Lengths contraction and time dilation. Relativistic momentum. Force. Kinetic energy. Rest energy.

9) Universal gravitation. Gravitational force, gravitational field, work of the gravitational force and gravitational potential energy, gravitational potential. Discussion on potential energy curves. Kepler's laws. Gauss theorem and its applications.

10) Basics of fluid dynamics. Pressure, pressure forces, work of pressure forces. Hydrostatic equilibrium and Stevin's law. Archimedes' principle. Fluid motion: stationary regime and flow rate. Ideal fluid. Bernoulli's theorem.

**Prerequisites for admission**

Good knowledge of basic mathematics, trigonometry, exponential and logarithmic functions, differential and integral calculus

**Teaching methods**

Front lectures on theory and exercises. Attending is strongly encouraged

**Teaching Resources**

- Reference textbook: Mazzoldi, Nigro, Voci "Fisica Volume I" edizione EdiSES

- Didactic material by the teachers, regularly made available on the Ariel website

- Exercises and examples, made available online.

- Didactic material by the teachers, regularly made available on the Ariel website

- Exercises and examples, made available online.

**Assessment methods and Criteria**

Written and oral exam. It is possible to take partial written tests, which, in case of positive outcome, are equivalent to the full written exam. The final mark will account for both the knowledge of the topics treated in the lectures, and the ability to solve problems.

FIS/01 - EXPERIMENTAL PHYSICS - University credits: 8

Practicals: 60 hours

Lessons: 24 hours

Lessons: 24 hours

Professors:
Basilico Davide, Bersanelli Marco Rinaldo Fedele

### CORSO B

Responsible

Lesson period

First semester

The structure of the exams will be similar to the ordinary one: written test and oral exam, which, if necessary, will be held remotely on ZOOM.

All communication with students, as well as the distribution of materials of study, will take place through the ARIEL platform.

**Course syllabus**

1) Physical quantities, systems of units and dimensional analysis

2) Vector calculus. Sum and difference of vectors. Multiplication of a vector by a scalar. Vector decomposition. Inner (dot) product and vector (cross) product of vectors. Vector as a function of a parameter: vector derivative and integration. Flux of a vector through a surface.

3) Kinematics of a point mass. Velocity. Acceleration. Linear motion: uniform and uniformly accelerated. Motion in the plane: velocity and acceleration in polar coordinates, tangent and normal acceleration. Circular motion. Angular velocity and angular acceleration (including vector notation). Projectile motion. Harmonic motion.

4) Dynamics of a point mass: the three laws of dynamics (Newton's laws of motion). Constant forces, force as a function of speed, position, time. Normal forces. Friction. Momentum. Impulse. Work, power, kinetic energy. Work-kinetic energy theorem. Conservative forces. Potential energy. Conservation of mechanical energy. Calculation of work and potential energy for weight and elastic force. Angular momentum and torques. Central forces and central force motion.

5) Systems of point masses. Centre of mass: properties and motion description. Euler's laws of motion for systems of point masses. Work-kinetic energy theorem. König theorems. Collisions. Ballistic pendulum.

6) Rigid body. Kinematics of rigid bodies. Momentum of inertia. Huygens-Steiner theorem. Dynamics of free and constrained rigid bodies. Rigid bodies rotating around an axis. Physical (compound) pendulum. Roto-translation of a rigid body. Rolling without slipping of spheres and cylinders on a plane. Kinetic energy of translating, rotating, and roto-translating rigid bodies. Work-Kinetic energy theorem and energy conservation for a rigid body.

7) Relative motions. Absolute and relative reference frames. Composition of velocities. Composition of accelerations: Coriolis' theorem. Earth as a relative frame: variation of gravity acceleration with latitude, east deviation of falling objects, Foucault pendulum. Relative dynamics. Apparent forces.

8) Basics of special relativity theory. Crisis of Galilean relativity principle. Lorentz transformations and invariance of physical laws in inertial systems. Transformation of velocities. Lengths contraction and time dilation. Relativistic momentum. Force. Kinetic energy. Rest energy.

9) Universal gravitation. Gravitational force, gravitational field, work of the gravitational force and gravitational potential energy, gravitational potential. Discussion on potential energy curves. Kepler's laws. Gauss theorem and its applications.

10) Basics of fluid dynamics. Pressure, pressure forces, work of pressure forces. Hydrostatic equilibrium and Stevin's law. Archimedes' principle. Fluid motion: stationary regime and flow rate. Ideal fluid. Bernoulli's theorem.

2) Vector calculus. Sum and difference of vectors. Multiplication of a vector by a scalar. Vector decomposition. Inner (dot) product and vector (cross) product of vectors. Vector as a function of a parameter: vector derivative and integration. Flux of a vector through a surface.

3) Kinematics of a point mass. Velocity. Acceleration. Linear motion: uniform and uniformly accelerated. Motion in the plane: velocity and acceleration in polar coordinates, tangent and normal acceleration. Circular motion. Angular velocity and angular acceleration (including vector notation). Projectile motion. Harmonic motion.

4) Dynamics of a point mass: the three laws of dynamics (Newton's laws of motion). Constant forces, force as a function of speed, position, time. Normal forces. Friction. Momentum. Impulse. Work, power, kinetic energy. Work-kinetic energy theorem. Conservative forces. Potential energy. Conservation of mechanical energy. Calculation of work and potential energy for weight and elastic force. Angular momentum and torques. Central forces and central force motion.

5) Systems of point masses. Centre of mass: properties and motion description. Euler's laws of motion for systems of point masses. Work-kinetic energy theorem. König theorems. Collisions. Ballistic pendulum.

6) Rigid body. Kinematics of rigid bodies. Momentum of inertia. Huygens-Steiner theorem. Dynamics of free and constrained rigid bodies. Rigid bodies rotating around an axis. Physical (compound) pendulum. Roto-translation of a rigid body. Rolling without slipping of spheres and cylinders on a plane. Kinetic energy of translating, rotating, and roto-translating rigid bodies. Work-Kinetic energy theorem and energy conservation for a rigid body.

7) Relative motions. Absolute and relative reference frames. Composition of velocities. Composition of accelerations: Coriolis' theorem. Earth as a relative frame: variation of gravity acceleration with latitude, east deviation of falling objects, Foucault pendulum. Relative dynamics. Apparent forces.

8) Basics of special relativity theory. Crisis of Galilean relativity principle. Lorentz transformations and invariance of physical laws in inertial systems. Transformation of velocities. Lengths contraction and time dilation. Relativistic momentum. Force. Kinetic energy. Rest energy.

9) Universal gravitation. Gravitational force, gravitational field, work of the gravitational force and gravitational potential energy, gravitational potential. Discussion on potential energy curves. Kepler's laws. Gauss theorem and its applications.

10) Basics of fluid dynamics. Pressure, pressure forces, work of pressure forces. Hydrostatic equilibrium and Stevin's law. Archimedes' principle. Fluid motion: stationary regime and flow rate. Ideal fluid. Bernoulli's theorem.

**Prerequisites for admission**

Good knowledge of basic mathematics, trigonometry, exponential and logarithmic functions, differential and integral calculus

**Teaching methods**

Front lectures on theory and exercises. Attending is strongly encouraged

**Teaching Resources**

M.Fanti, "Meccanica" -- freely downloadable PDF : https://libri.unimi.it/index.php/milanoup/catalog/book/19

Exercises provided by the lecturers.

Exercises provided by the lecturers.

**Assessment methods and Criteria**

Written and oral exam. It is possible to take partial written tests, which, in case of positive outcome, are equivalent to the full written exam. The final mark will account for both the knowledge of the topics treated in the lectures, and the ability to solve problems.

FIS/01 - EXPERIMENTAL PHYSICS - University credits: 8

Practicals: 60 hours

Lessons: 24 hours

Lessons: 24 hours

Professors:
Fanti Marcello, Ferraro Federico

Educational website(s)

Professor(s)

Reception:

Monday, 10-12 a.m. (by appointment)

LITA, 3rd floor, A/3/C11