Russia 2020

Problem 1 Find the charge of the conductor. Let ${\epsilon }_1={\epsilon }_2=100\text{V}$, $\epsilon =60\text{V}$, $R_1=100\Omega$, $R_2=300\Omega$, and $C=1\text{mF}$. Neglect the internal resistance of the voltage sources. Figure 1: Problem 1

circuito con sorgenti e condensatore

Topic: Circuits, Electrostatics Metodi: Kirchhoff’s Laws, Electric Potential Method, Equivalent Circuit Reduction Competenze: Physical Reasoning, Mathematical Modeling Objects: Battery, Resistor, Capacitor Fonte: Testo (PDF) — p.1

Problema 1 Trova la carica del conduttore. Let ${\epsilon }_1={\epsilon }_2=100\text{V}$, $\epsilon =60\text{V}$, $R_1=100\Omega$, $R_2=300\Omega$, and $C=1\text{mF}$. - Non si tratta di resistenza interna delle fonti di tensione. Figura 1: Problema 1

*circuito con sorgenti e condensatore *

Topic: Circuits, Electrostatics Metodi: Kirchhoff’s Laws, Electric Potential Method, Equivalent Circuit Reduction Competenze: Physical Reasoning, Mathematical Modeling Objects: Battery, Resistor, Capacitor Fonte: Testo (PDF) — p.1

Problem 2 There is one mole of monoatomic ideal gas and a heat engine. The heat engine has a cycle of (1)1-2-3-1 and (2)1-3-4-1. The coordinates of 1, 2, 3, 4 are given on the diagram. The efficiency of 1st cycle is ${\eta }_1$ and the efficiency of 2nd cycle is ${\eta }_2$. Find $\frac{{\eta }_1}{{\eta }_2}$ . Figure 2: Problem 2

diagramma P-V ciclo termodinamico

Topic: Thermodynamics Metodi: Thermodynamic Cycle Analysis, First Law of Thermodynamics, Ideal Gas Law Competenze: Mathematical Modeling, Physical Reasoning Objects: Gas, Heat Engine Fonte: Testo (PDF) — p.1

Il problema 2 C’è un mole di gas ideale monoatomico e un motore termico. Il motore termico ha un ciclo di (1)1-2-3-1 e (2)1-3-4-1. Le coordinate di 1, 2, 3, 4 sono indicate sul diagramma. L’efficienza del primo ciclo è ${\eta }_1$ e l’efficienza del secondo ciclo è ${\eta }_2$. Trova $\frac{{\eta }_1}{{\eta }_2}$ . Figura 2: Problema 2

diagramma P-V ciclo termodinamico

Topic: Thermodynamics Metodi: Thermodynamic Cycle Analysis, First Law of Thermodynamics, Ideal Gas Law Competenze: Mathematical Modeling, Physical Reasoning Objects: Gas, Heat Engine Fonte: Testo (PDF) — p.1

Problem 3 There is a thin ring made of nonconducting material with radius $R$. This ring has charge density given by $\lambda ={\lambda }_0\cos \varphi$. Find the Electric field $\vec{E}$ at the center of the ring. Figure 3: Problem 3

anello carico densità lambda cos phi

Topic: Electrostatics Metodi: Coulomb’s Law, Calculus-Integration, Superposition Principle Competenze: Mathematical Modeling, Physical Reasoning Objects: — Fonte: Testo (PDF) — p.2

Il problema 3 Si tratta di un anello sottile in materiale non conduttore con raggio $R$. Questo anello ha una densità di carica data da $\lambda ={\lambda }_0\cos \varphi$. Trova il campo elettrico $\vec{E}$ al centro dell’anello. Figura 3: Problema 3

*anello carico densità lambda cos phi *

Topic: Electrostatics Metodi: Coulomb’s Law, Calculus-Integration, Superposition Principle Competenze: Mathematical Modeling, Physical Reasoning Objects: — Fonte: Testo (PDF) — p.2

Problem 4 A proton at rest combines with another proton whose kinetic energy is $2m_p{c}^2$ and a new particle appears. Find the rest mass of the new particle. (Note that $m_p$ is the rest mass of the proton, and $c$ is the speed of light).

Topic: Special Relativity, Nuclear & Particle Physics Metodi: Relativistic Energy-Momentum, Conservation Laws, Mass-Energy Equivalence Competenze: Physical Reasoning, Mathematical Modeling Objects: Nucleus Fonte: Testo (PDF) — p.2

Problema 4 Un protone in riposo si combina con un altro protone la cui energia cinetica è $2m_p{c}^2$ e appare una nuova particella. Trova il resto . massa della nuova particella. (Ricorda che $m_p$ è la massa di riposo del protone e $c$ è la velocità della luce).

Topic: Special Relativity, Nuclear & Particle Physics Metodi: Relativistic Energy-Momentum, Conservation Laws, Mass-Energy Equivalence Competenze: Physical Reasoning, Mathematical Modeling Objects: Nucleus Fonte: Testo (PDF) — p.2

Problem 5 There is a mine created from one point of the Earth to the other point of the Earth along its diameter.Find the period of oscillation of an object which falls into this mine. Let $R_{\text{Earth}}=6400\text{km}$, $g=10\text{m}{\text{s}}^{-2}$, and $\pi =3.14$.

Topic: Oscillations & Waves, Newtonian Mechanics, Gravitation Metodi: Simple Harmonic Motion Analysis, Free-Body Diagram, Kinematic Equations Competenze: Mathematical Modeling, Physical Reasoning Objects: Planet Fonte: Testo (PDF) — p.2

Problema 5 C’è una miniera creata da un punto della Terra all’altro punto della Terra lungo il suo diametro. di oscillazione di un oggetto che cade in questa miniera. Si possono utilizzare $R_{\text{Earth}}=6400\text{km}$, $g=10\text{m}{\text{s}}^{-2}$ e $\pi =3.14$.

Topic: Oscillations & Waves, Newtonian Mechanics, Gravitation Metodi: Simple Harmonic Motion Analysis, Free-Body Diagram, Kinematic Equations Competenze: Mathematical Modeling, Physical Reasoning Objects: Planet Fonte: Testo (PDF) — p.2