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Module ESERCITAZIONI

Teaching staffFRANCESCO RUFFINOEmail: francesco.ruffino@ct.infn.itAddress: Dipartimento di Fisica ed Astronomia- Via S. Sofia 64- Buiding 6- Office 244(second floor)Phone: 0953785461Office hours: Monday 15:00-17:00, Wednesday 15:00-17:00.The teacher is also available for meetings in electronic mode, by appointment. Any notices of unavailability will be branched through Microsoft Teams and/or Studium.

** **

**LEARNING
OBJECTIVES**

The
course aims to provide students with basic quantitative knowledge on the topics
of classical mechanics and thermodynamics and to instruct them, specifically,
to face and solve real problems of increasing complexity. The course
favors the use of algebraic and analytical techniques in solving the proposed
problems, providing the ability to apply the Scientific Method to the
resolution of real and concrete problems. Furthermore, through the display of
numerous examples and the guided and autonomous resolution of the proposed
problems, the course aims to provide useful tools for the schematization and
analysis of complex real physical situations and to show how it is possible to
exploit simple models of real situations. complex to make predictions on the
evolution of physical systems. The course also aims to stimulate communication
between students around physics topics through joint efforts, using group work,
for the resolution of proposed and assigned problems.

In particular, and with reference to the so-called Dublin Descriptors, the course aims to provide the following knowledge and skills.

** **

**Knowledge
and understanding abilities**

Knowledge
of the main phenomenological aspects relating to classical mechanics and
thermodynamics and understanding of their physical implications and their
mathematical description with particular reference to the resolution of real
problems, justifying the steps necessary for solving these problems. Ability
to understand the sense of the scientific method for analyzing the
physical phenomena involved in a problem, also acquiring the ability to analyze
and synthesize the situations under investigation.

** **

**Applying
knowledge and understanding ability**

Ability to recognize the main physical laws that govern a phenomenon in mechanics and thermodynamics, and to apply them to solve problems and exercises in different fields and at different levels of complexity, and therefore of approximation, with the use of appropriate mathematical tools. Understanding of the utility and limitations associated with the use of schematizations and models in quantitative problems.

**Ability
of making judgements**

Ability
to estimate and calculate the order of magnitude of the variables that describe
a physical phenomenon (in mechanics and in thermodynamics). Ability to discern
the level of importance of a physical law in solving problems. Ability to be
able to evaluate the Physical Model and the corresponding Mathematical Model
that are best applied to the description of a physical process and therefore to
the solution of quantitative problems.

** **

**Communication
skills**

Ability to present scientific concepts of physics but also, and more generally, information, ideas, problems and solutions with properties and inambiguity of language, at different levels and to different interlocutors, both specialists and non-specialists.

**Learning
skills**

Ability
to learn the scientific concepts of Physics by applying them to problem solving
so as to provide a high degree of autonomy in subsequent studies.

The teaching activity consists of exercises (for a total of 2 ECTS, corresponding to 30 hours), accompanied by tutoring activities(*). The exercises provide for the resolution, both guided and autonomous, of tasks and exercises. Where possible, innovative teaching and learning strategies are

used. During each lesson, moreover, space is left to students for questions, curiosities and comments, in order to maximize teacher-student interaction.

** **

It is fundamental for the student to have
mastery of the subjects of elementary mathematics (algebra, geometry,
trigonometry, analytical geometry) and knowledge of those of mathematical analysis
(differential and integral calculus). In fact, for the presentation of the
physical concepts included in the course content, the following mathematical
tools are used: equations and systems of 1^{st} and 2^{nd}
degree equations, trigonometric functions and their properties, exponential
functions and their properties, logarithmic functions and their properties, equations
of loci in the plane and in space, derivatives and integrals of functions of
one variable, constant coefficient linear differential equations. For the
self-paced learning, and/or consolidation, of the required preliminary
knowledge, the mathematics and basic calculus courses available on e-learning
platforms such as, for example, Federica Web Learning and Coursera for Campus,
to which students of the University have access, may be useful.

Normally mandatory (as stated in the Didactic
Regulation of the course)

**MECHANICS**

**Exercises
on kinematics. **Speed, velocity, acceleration and time
dependence of motion. Straight and uniformly accelerated rectilinear motion.
Vertical motion. Simple harmonic motion. Rectilinear motion exponentially damped.
Motion in a plane: velocity and acceleration. Circular motion. Parabolic
motion.

**Exercises
on dynamics of the material point. **Principle of inertia
and the concept of force. Second and third Newton's law. mpulse and momentum.
Resulting force: binding reactions and equilibrium. Examples of forces: weight
force, sliding friction force, viscous friction force, centripetal force,
elastic force. Inclined plane. Simple pendulum. Wire tension. Reference
systems. Relative speed and acceleration. Inertial reference systems.

**Exercises
on work and energy. **Work, power and kinetic energy. The
theorem of the kinetic energy. Examples of works done by forces. Conservative
forces and potential energy. Non-conservative forces. Principle of conservation
of mechanical energy. Relationship between force and potential energy. Angular momentum.
Torque. Central forces.

**Exercises
on dynamics of systems of material points. **Systems of
points. Internal and external forces. Center of mass and its properties.
Principle of conservation of the momentum. Principle of conservation of the angular
momentum. The König theorems. Theorem of the kinetic energy. Collisions.

**Exercises
on dynamics of the rigid body. **Motion of a rigid body
and basic motion equations. Continuous bodies, density and the position of the
center of mass. Rigid rotations around an axis in an inertial reference system.
Rotational energy and work. Moment of inertia. Huygens-Steiner's theorem. Compound
pendulum. Pure rolling motion. Energy conservation in the motion of a rigid
body.

**Exercises
on dynamics of fluids. **Fluids
properties and pressure. Pascal’s Principle. Stevin’s law. Archimede’s law.
Motion of a fluid and Bernoulli’s theorem.

**Exercises
on oscillations and waves. **Properties of the differntial
equation of the harmonic oscillator. Simple harmonic oscillator: motion
equation and its solution. Motion of a mass connected to a spring. Energy of a simple
harmonic oscillator. Damped and forced harmonic oscillators. Characteristics of
a wave: amplitude, period, frequency, wavelenght, intensity. Wavefunction. Waves
on a string. Waves on gases.

**Exercises
on gravitation. **Kepler's laws. The Universal Gravitation
Law. Gravitational field and gravitational potential energy. Escape velocity,

** **

**THERMODYNAMICS**

**Exercises
on the first principle of thermodynamics. **Thermodynamic
systems and states. Thermodynamic equilibrium and the Principle of Thermal
Equilibrium. Temperature and thermometers. Equivalence of work and heat. First
Principle of Thermodynamics. Internal energy. Thermodynamic transformations. Work
and heat. Calorimetry. Phase transitions. Heat transmission.

**Exercises
on ideal gases. **Laws of the ideal gas. Equation of state
of the ideal gas. Transformations of a gas. Work. Specific heat and internal
energy of the ideal gas. Analytical study of some transformations. Ciclic transformations.
The Carnot cycle. Kinetic theory of gases. Equipartition of energy.

**Exercises
on the second principle of thermodynamics.** Reversibility
and irreversibility. Carnot's theorem. Clausius theorem. Entropy state
function. The principle of increasing entropy of the universe. Entropy
variations' calculations. Entropy of the ideal gas.

**Exercises on
thermodynamics potentials. **Gibbs free energy. Helmholtz free
energy. Henthalpy.

1) P. Mazzoldi, A. Saggion, C. Voci, Problemi di Fisica Generale-Meccanica, Termodinamica (Edizioni Libreria Cortina Padova 1996): Esercizi

2) M. Fazio, Problemi di Fisica (Springer, 2008)

**FURTHER DIDACTIC
MATERIAL**

Collections of exercises carried out and organized by levels of increasing difficulty, up to the level required to pass the preliminary exam (s), and presentations (if used by the teacher during the lessons) are published in PDF format in the "Documenti" section of the course page on the Studium portal and, also, on the website https://nanostar.jimdofree.com/didattica-fisica-1/

Subjects | Text References | |
---|---|---|

1 | Exercises on kinematics (2 hours) | Textbook 1-Chapter 1; Textbook 2-Chapter 2 |

2 | Exercises on dynamics of the material point (3 hours) | Textbook 1-Chapter 2; Textbook 1-Chapter 3; Textbook 2-Chapter 3; Textbook 2-Chapter 6 |

3 | Exercises on work and energy (4 hours) | Textbook 1-Chapter 2; Textbook 2-Chapter 4 |

4 | Exercises on dynamics of systems of material points (2 hours) | Textbook 1-Chapter 5; Textbook 2-Chapter 5 |

5 | Exercises on gravitation (2 hours) | Textbook 1-Chapter 7; Textbook 2- Chapter 9 |

6 | Exercises on dynamics of the rigid body (4 hours) | Textbook 1-Chapter 6; Textbook 2-Chapter 7 |

7 | Exercises on dynamics of fluids (2 hours) | Textbook 1-Chapter 8; Textbook 2-Chapter 8 |

8 | Exercises on oscillations and waves (2 hours) | Textbook 1-Chapter 4; Textbook 2-Chapter 15 |

9 | Exercises on the first principle of thermodynamics (2 hours) | Textbook 1-Chapter 9; Textbook 1-Chapter 10; Textbook 2-Chapter 10; Textbook 2-Chapter 12 |

10 | Exercises on ideal gases (2 hours) | Textbook 1-Chapter 9; Textbook 1-Chapter 10; Textbook 2-Chapter-11 |

11 | Exercises on the second principle of thermodynamics (3 hours) | Textbook 1-Chapter 10; Textbook 2-Chapter 13 |

12 | Exercises on thermodynamics potentials (1 hour) | Textbook 2-Chapter 14 |

13 | Excercises on real gases (1 hour) | Textbook 2-Chapter 14 |

The exam consists of a written test and an oral interview. The written test consists of 3 (or 4)

problems to be solved in a maximum time of 2 hours. To know the type of problems proposed,

consult the website http://nanostar.jimdo.com/.

The evaluation of the written test will take into account the problem solving approach, thecorrectness of the numerical calculations and significant values, the arguments supporting the

procedure followed. The minimum mark for admission to the oral exam is 18/30.

The evaluation of the oral interview will take into account the student's ability to use orders of

magnitude in the analysis of a phenomenon, the ability to critically evaluate similarities and

differences between physical systems, the level of depth of the contents exposed and its

properties of language and of exposure.

The written test has limited validity, it is necessary to complete the exam by passing the oralexam in the same calendar year as the written test. If the student does not complete the exam

within the calendar year, he must repeat the written test.

In addition for the attending students:

the exam can be divided into two partial tests: one relating to mechanics and gravitation (firstpartial test) and the second relating to thermodynamics, and fluid mechanics (second partial test).

Passing both partial tests will determine the achievement of the exam. These partial tests are to

be considered additional opportunities with respect to the exams and do not preclude

participation in the ordinary exam sessions.

Both partial tests consist of a written test and an oral interview. The written test consists of 3problems to be solved in a maximum time of 2 hours (http://nanostar.jimdo.com/).

The minimum mark for admission to the respective oral interviews is 15/30.

The first test takes place at the end of the first teaching period, in the February exam session.Students who pass the written test will have access to the oral interview which will determine

admission to the second partial test.

The second partial test can be held in each of the ordinary sessions of the second and thirdsession, according to the official calendar. The student who has passed the second written test

will have access to the oral interview which will determine the final result of the exam.

The student who passes the second written test is allowed to take the second oral interview evenin a subsequent session, as long as it is within the calendar year of the written test.

The two partial written tests can be replaced by ongoing tests to be scheduled in agreementbetween the teacher and students.

Information fo students with disabilities and/or SLD

In order to guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and/or dispensatory

measures, according to the educational objectives and specific needs. It is also possible to contact the CInAP (Centro l'Integrazione Attiva e Partecipata - Servizi per le Disabilità e/o DSA)

contact-person of the Department, Prof. Catia Petta.

**Dates of the exams**

Check the following web pages

http://portalestudente.unict.it

https://www.dfa.unict.it/corsi/L-30/esami

and news on the course page on the Studium portal (http://studium.unict.it) or teacher’s website (https://nanostar.jimdo.com/) for details on the time and place of the exams and any changes.

Exam booking through the Smart_Edu platform is mandatory. Non-booked students will not be able to do exams.The problems listed
below are not an exhaustive list but are just a few examples:

1) Taking a coordinate system with horizontal x-axis and ascending vertical y-axis, let P be the point of coordinates xp = 0 m and yp = 120 m. A body of mass m = 2 kg is launched with initial horizontal velocity v0 = 15 m / s from point P. Determine:

a) the time of flight and the point of fall of the body on the ground;

b) the expression of the forces acting on the body, tangential and normal to the trajectory as a function of time and their value at instant t* corresponding to half of the flight time;

c) the expression, as a function of time, of the angular momentum and the moment of force with respect to the launch point P and their value at instant t*.

2) A mole of ideal monoatomic gas with initial volume V1 = 8 dm3 and temperature T1 = 350 K completes a reversible cycle consisting, in sequence, of: 1 → 2 isothermal expansion, 2 → 3 isochoric with pressure decrease, 3 → 4 isobaric compression, 4 → 1 adiabatic which brings the gas back to the initial conditions.

a) Determine the thermodynamic coordinates (pressure, volume, temperature) of states 2, 3, 4 in such a way that the entropy change of the gas from state 1 to state 2 is equal to 19 J / K and that the temperature of state 4 let T4 = 80 K;

b) Calculate the cycle yield.

3) A body with mass m1 = 1.00 kg, initially at rest, is dropped from a height h, with h = 12.0 m. At the same instant in which the body 1 starts its motion, a second body, with mass m2 = 2.00 kg, is launched from the ground with speed v20, along the same vertical. Given the conditions, the two bodies will collide; we indicate with tc and yc the instant and the altitude at which the collision occurs. Knowing that the collision between the two bodies is completely inelastic and that after the collision the body resulting from the union of the two (of mass m1 + m2) reaches a maximum height equal to h, determine:

a) the speed v20 with which the body 2 was launched;

b) the instant tc and the altitude yc at which the impact occurs;

c) the energy lost in the collision.

[Treat the bodies as point-like, neglect any friction and assume an instant impact]

4) A body, of mass m1 = 1000 kg, is launched in a radial direction from the earth's surface with an initial velocity v0 equal to 3/5 of its escape velocity, vfuga.

a) Determine the maximum distance rmax from the center of the Earth that reaches the body.

At the exact moment in which the body is at the distance rmax (the one calculated in the previous point), it is hit by a meteorite of mass m2 = 2m1. Knowing that the collision with the meteorite is completely inelastic and that the resulting body begins to rotate around the Earth on a circular orbit of radius rmax, determine:

b) the time it takes the body to make a complete circle around the Earth;

c) the velocity v2 that the meteorite had before the collision, specifying its direction;

d) the energy lost in the impact.

[In the calculations, neglect both the resistance of the atmosphere and the earth's rotation. For the ground mass and radius use the following values: M = 5.98 • 1024 kg, R = 6.37 • 106 m]

5) A parallelepiped-shaped body floats in a container partially filled with mercury (density 13.6 g / cm3), remaining immersed only for two thirds of its height. Subsequently, water is added (immiscible with mercury) in order to abundantly cover the emerging part of the parallelepiped. Calculate the x height of the part immersed in mercury in the new conditions, knowing that the total height of the parallelepiped is h = 20 cm.

6) An ideal gas is contained in the volume VA = 40.00 dm3 at the pressure pA = 1.00 • 105 Pa and at the temperature TA = 300.0 K. With a reversible isothermal compression the gas reaches state B with volume VB = (1/3) VA ; during this transformation the gas performs a job LAB = −4.394 • 103 J. Then, through a reversible isochore it reaches state C at temperature TC = 600 K. Subsequently, in an irreversible adiabatic way, the gas is brought to state D with volume VD = VA and temperature TD> TA: in this expansion the gas does the job LCD = 5.894 • 103 J. Finally, with a reversible isochore the gas returns to the initial state A. Knowing that the cycle efficiency is η = 0.150, to determine:

a) the heat QAB, QBC and QDA;

b) if the gas is monoatomic or diatomic;

c) the value of TD;

d) the change in entropy DSCD.

7) To a solid state body of mass m = 2 kg and at the initial temperature T0 = 282.2 K, an amount

of heat Q1 = 15.5 kcal is transferred and, correspondingly, its temperature rises to the value

T1 = 317.2 K. Now that the body is at temperature T1, the quantity of heat Q2 = 7.9 kcal is

subtracted from it and, correspondingly, its temperature drops to the value T2 = 302.2 K. If, on

the other hand, a quantity of heat Q is transferred to the body at temperature T1 <Q2 it is

observed that its temperature remains constant at the T1 value. Assuming that the specific heat c of the body is independent of the temperature, calculate c

and the latent heat l in the situation described by the text.

8) A mass m = 50 g of an ideal monoatomic gas is subjected to a reversible isochoric transformation in which the temperature increases by DT = 160 K. If the enthalpy change is

DH = 8310 J, say which gas it is .

9) A homogeneous thin rod with length l = 4.00 m and mass m = 4.00 kg is hinged at one end with a frictionless hinge. The rod is initially in a vertical position as shown in the figure.

A very small perturbation causes the rod to rotate and fall. When the shaft has covered a quarter

of a turn and is in a horizontal position, calculate:

(a) the angular velocity ω of the rod and the velocity vcm of its center of mass;

(b) the ratio between vcm and the speed v that the rod would have if, instead of rotating, it were in free fall between the same altitudes of the center of mass;

(c) the angular acceleration α of the rod.

10) Two homogeneous spheres with radius R = 1.00 cm, having the same mass m = 100 g, descend along an inclined plane, with an inclination θ = 1.72 °. The first sphere slides without

rolling in the absence of any form of friction; the second sphere rolls down without sliding, in

the absence of rolling friction.

Determine the accelerations with which the 2 spheres descend.