GENERAL PHYSICS I 2

Module EXERCISES

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 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 with proper language, at different audiences, both specialists and non-specialists.

Learning skills

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

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.

It is fundamental for the student to have mastery of the subjects of elementary mathematics (algebra, geometry, trigonometry, analytical geometry).

The mathematical analysis knowledge necessary for understanding the topics covered in the course will be progressively introduced by the teachers, so that all students, regardless of their prior diploma, can tackle the subjects discussed in class.

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

 

Exercises on elastic properties of solids. Traction and compression. Scrolling. Compressibility. Torsion balance.

 

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

 

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.

1) “Meccanica e termodinamica. Guida alla Soluzione degli Esercizi da Mazzoldi, Nigro, Voci - Fisica Mazzoldi, Nigro, Voci - Elementi di Fisica” (EdiSES, 2023)

2) M. Zani, L. Duò, P. Taroni, “Esercizi di Fisica. Meccanica e Termodinamica” (EdiSES, 2021)

FURTHER DIDACTIC MATERIAL

Collections of exercises carried out and organized by levels of increasing difficulty, up to the level required to pass the written exam, are published in the "Documenti" section of the course page on the Studium portal.

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 https://nanostar.jimdofree.com/didattica-fisica-1.
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 15/30.
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.
 
In order to facilitate the process, the exam can be divided into two partial tests: one relating to mechanics, gravitation and elastic properties of solids  (first partial test) and the second relating to oscillations and waves, fluid mechanics and thermodynamics (second partial test). Both partial tests consist of a written test (3 problems to be solved in a maximum time of 2 hours) and an oral interview. 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. It is necessary to complete the exam by passing the oral exam in the same calendar year as the first partial written test. To know the type of problems proposed, consult the website https://nanostar.jimdofree.com/didattica-fisica-1.
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.
 
Moreover, for attending students, 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 per l’Inclusione Attiva e Partecipata - Servizi per le Disabilità e/o DSA) contact-person of the Department, Prof. Catia Petta.
 
DATES OF THE EXAMS
 
Check the webpages http://portalestudente.unict.it, https://www.dfa.unict.it/corsi/L-30/esami for the dates of the exams.
Check news on the webpage https://www.dfa.unict.it/corsi/l-30/avvisi and 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.

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 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 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 m₁+m₂) reaches a maximum height equal to h, determine:

a) the speed v₂₀ 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 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, 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 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 r_max, 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 groundmass 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 V_A=40.00 dm³ at the pressure p_A=1.00 · 10⁵ 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 · 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 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 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 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.