# FISICA GENERALE I 2Module ESERCITAZIONI

**Academic Year 2023/2024**- Teacher:

**LUCIA OLIVA**

## Expected Learning Outcomes

**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 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.

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 related to classical mechanics and thermodynamics and understanding of their physical implications and their mathematical description, in order to develop an ability to reflect on scientific issues in a way that presents traits of originality.

**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.

** **

**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 (axiom, conservation principle, universal law, theorem, law in global/integral or local/differential form and its generality, properties of materials, etc.). Ability to be able to evaluate the Physical Model and the corresponding Mathematical Model that best apply to the description of a physical process and therefore to the solution of quantitative problems.

** **

**Communications kills**

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

** **

**Learning skills**

Ability to learn the scientific concepts of Physics, necessary to undertake subsequent studies with a high degree of autonomy.

## Course Structure

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.

(*) If specialist tutors are
available for the course during the academic year.

## Required Prerequisites

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.

## Attendance of Lessons

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

## Detailed Course Content

**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. Impulse 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 differential
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 wave: amplitude, period, frequency, wavelength, 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 agas. 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. Enthalpy.

**Exercises on real gases. **Transformations of areal gas described by the Van der Waals
equation of state: work calculations, internal energy variation, heat, entropy
variation for some transformations.

## Textbook Information

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/

## Course Planning

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 ofthermodynamics (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 |

## Learning Assessment

### Learning Assessment Procedures

## 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, the correctness 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 oral exam 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.

** **

**Or for attending students:**

the exam can be divided into two partial tests: one relating to mechanics and gravitation (first partial 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 3 problems 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 third session,
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 even in 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 agreement between the
teacher and students.

**Information for 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 webpages

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) 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.

### Examples of frequently asked questions and / or exercises

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 x_{p}=0 m and y_{p}=120 m. A body of mass m=2 kg is launched with
initial horizontal velocity v_{0}=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 V_{1}=8
dm^{3} and temperature T_{1}=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 T_{4}=80
K;

b) Calculate the cycle yield.

3) A body with mass m_{1}=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 m_{2}=2.00
kg, is launched from the ground with speed v_{20},
along the same vertical. Given the conditions, the two bodies will collide; we
indicate with t_{c} and y_{c} 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 m_{1}+m_{2}) reaches a maximum height equal to h, determine:

a) the speed v_{20} with which the body 2 was launched;

b) the instant t_{c} and the altitude y_{c} 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 m_{1}=1000 kg, is launched in a radial direction from
the earth's surface with an initial velocity v_{0} equal to 3/5 of its escape velocity, v_{fuga}.

a) Determine the maximum
distance r_{max} from the center of the
Earth that reaches the body.

At the exact moment in which
the body is at the distance r_{max} (the one
calculated in the previous point), itis hit by a meteorite of mass m_{2}=2m_{1}.
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 r_{max}, determine:

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

c) the velocity v_{2} 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
groundmass and radius use the following values: M=5.98 · 10^{24} kg,
R=6.37 · 10^{6} m]

5) A parallelepiped-shaped body
floats in a container partially filled with mercury (density 13.6 g/cm^{3}), 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 V_{A}=40.00 dm^{3} at the pressure p_{A}=1.00
· 10^{5} Pa and at the temperature T_{A}=300.0
K. With a reversible isothermal compression the gas reaches state B with volume
V_{B}=(1/3)V_{A};
during this transformation the gas performs a job L_{AB}=−4.394
• 103 J. Then, through a reversible isochore it reaches state C at temperature T_{C}=600 K. Subsequently, in an irreversible
adiabatic way, the gas is brought to state D with volume V_{D}=V_{A} and
temperature T_{D}>T_{A}:
in this expansion the gas does the job L_{CD}=5.
894 · 10^{3} 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 Q_{AB}, Q_{BC}
and Q_{DA};

b) if the gas is monoatomic or
diatomic;

c) the value of T_{D};

d) the change in entropy ΔS_{CD}.

7) To a solid state body of
mass m=2 kg and at the initial temperature T_{0}=282.2
K, an amount of heat Q_{1}=15.5 kcal is
transferred and, correspondingly, its temperature rises to the value T_{1}=317.2 K. Now that the body is at temperature T_{1}, the quantity of heat Q_{2}=7.9
kcal is subtracted from it and, correspondingly, its temperature drops
to the value T_{2}=302.2 K. If, on the
other hand, a quantity of heat Q is transferred to the body at temperature T_{1}<Q_{2} it
is observed that its temperature remains constant at the T_{1} 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 ΔT=160 K. If the enthalpy change is ΔH=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 v_{cm} of its
center of mass;

b) the ratio between v_{cm} 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.