IPhO 2026

Problem 1 Free fall on an exoplanet (MC problem) (2nd round towards IPhO 2022, problem group of the PhysicsOlympiad - Thomas Hellerl) On the surface of an extrasolar planet - exoplanet for short - the fall time of a body from a small height $h$ , neglecting all friction effects, is exactly twice as large as on Earth. Which of the following statements is consistent with this, assuming a spherically symmetric structure of the exoplanet? The exoplanet has … A … half the Earth’s mass and twice the Earth’s radius. B … exactly the Earth’s mass and four times the Earth’s radius. C … twice the Earth’s mass and twice the Earth’s radius. D … four times the Earth’s mass and four times the Earth’s radius. Answer section Calculations and explanations Correct answer:

Topic: Gravitation, Newtonian Mechanics Metodi: Newton’s Law of Gravitation, Kinematic Equations, Free-Body Diagram Competenze: Physical Reasoning Objects: Planet Fonte: Testo (PDF) — p.10

Il problema 1 è il problema del free fall on an exoplanet (MC). (II round towards IPhO 2022, problem group of the PhysicsOlympiad - Thomas Hellerl) On the surface of an extrasolar planet - exoplanet for short - the fall time of a body from a small height $h$ , neglecting all friction effects, is exactly twice as large as sulla Terra. Which of the following statements is consistent with this, assuming a spherically symmetric La struttura dell’esoplaneta? Il pianeta è morto … A … metà della massa terrestre e il doppio del raggio terrestre. B … esattamente la massa della Terra e quattro volte il raggio della Terra. C … Due volte la massa terrestre e due volte il raggio terrestre. D … Quattro volte la massa terrestre e quattro volte il raggio terrestre. Answer section Calcoli e spiegazioni Corretta risposta:

The problem is that the system is not yet fully functional. (second round towards IPhO 2022, problem group of the PhysicsOlympiad - Thomas Hellerl) On the surface of an extrasolar planet - exoplanet for short - the fall time of a body from a small height $h$ , neglecting all friction effects, is exactly twice as large as on Earth. Which of the following statements is consistent with this, assuming a spherically symmetrical The structure of the exoplanet? The exoplanet has … A … half the mass of the Earth and twice the radius of the Earth. B … exactly the mass of the Earth and four times the radius of the Earth. C … twice the mass of the Earth and twice the radius of the Earth. D … Four times the mass of the Earth and four times the radius of the Earth. Answer section Calculations and explanations Correct answer:

Problem 2 Pendulum in an elevator (MC problem) (2nd round towards IPhO 2022) Two elevator cars of masses $m_{A}$ and $m_{B}$ with $m_{A} < m_{B}$ hang from the ends of a long rope that is guided over a fixed pulley. The mass of the pulley and the rope can be neglected. In the left car hangs a simple pendulum of length $ℓ$ . When the cars are at rest and for small displacements, the period of the pendulum is $T$ . When the cars are released, they move without friction under the influence of gravity. How must the length $ℓ^{'}$ of the simple pendulum in the left car be chosen so that, after the car is released, it oscillates with the period $T$ ? A $ℓ^{'} = \frac{m _{A}}{m _{B}} ℓ$ B $ℓ^{'} = \frac{2 m _{A}}{m _{A} + m _{B}} ℓ$ C $ℓ^{'} = \frac{2 m _{B}}{m _{A} + m _{B}} ℓ$ D $ℓ^{'} = \frac{m _{B}}{m _{A}} ℓ$ Answer section Calculations and explanations Correct answer:

Topic: Newtonian Mechanics, Oscillations & Waves Metodi: Free-Body Diagram, Simple Harmonic Motion Analysis, Physical Modeling Competenze: Physical Reasoning, Mathematical Modeling Objects: Pendulum, String, Pulley Fonte: Testo (PDF) — p.11

Problema 2 Pendolo in ascensore (problema MC) (II round towards IPhO 2022) Due ascensori di masse $m_{A}$ e $m_{B}$ con $m_{A} < m_{B}$ pendono dalle estremità di una lunga corda che è guidata su una polla fissa. La massa della polla e della corda può essere trascurata. In la macchina sinistra appeso un semplice pendolo di lunghezza $ℓ$ . Quando le auto sono a riposo e per piccoli spostamenti, il periodo del pendolo è $T$ . Quando le auto vengono rilasciate, si muovono senza attrito sotto l’influenza della gravità. How must the length $ℓ^{'}$ of the simple pendulum in the left car essere scelto in modo che, dopo il rilascio della macchina, oscilla con il Periodo $T$ ? A $ℓ^{'} = \frac{m _{A}}{m _{B}} ℓ$ B $ℓ^{'} = \frac{2 m _{A}}{m _{A} + m _{B}} ℓ$ C $ℓ^{'} = \frac{2 m _{B}}{m _{A} + m _{B}} ℓ$ D $ℓ^{'} = \frac{m _{B}}{m _{A}} ℓ$ Answer section Calcoli e spiegazioni Corretta risposta:

Problem 2 Pendulum in an elevator (MC problem) (second round towards IPhO 2022) Two elevator cars of masses $m_{A}$ and $m_{B}$ with $m_{A} < m_{B}$ hang from the ends of a long rope that is guided over a fixed pulley. The mass of the pulley and the rope can be neglected. In the left car hangs a simple pendulum of length $ℓ$ . When the cars are at rest and for small displacements, the period of the pendulum is $T$ . When the cars are released, they move without friction under the influence of gravity. How must the length $ℓ^{'}$ of the simple pendulum in the left car be chosen so that, after the car is released, it oscillates with the period $T$ ? A $ℓ^{'} = \frac{m _{A}}{m _{B}} ℓ$ B $ℓ^{'} = \frac{2 m _{A}}{m _{A} + m _{B}} ℓ$ C $ℓ^{'} = \frac{2 m _{B}}{m _{A} + m _{B}} ℓ$ D $ℓ^{'} = \frac{m _{B}}{m _{A}} ℓ$ Answer section Calculations and explanations Correct answer:

Problem 3 Double spring pendulum (MC problem) (5 pts.) (2nd round towards IPhO 2023, idea: problem group of the PhysicsOlympiad - Thomas Hellerl) In each of the two spring pendulums shown in the figure, a body of mass $m$ oscillates without friction. However, the spring constants $D_{1}$ and $D_{2}$ of the two Hookean springs are different. Therefore the bodies, after being displaced, oscillate with different frequencies $f_{1}$ and $f_{2}$ . $m$ $D_{1}$ $f_{1} = 1, 2$ Hz $m$ $D_{2}$ $f_{2} = 1, 6$ Hz What is the oscillation frequency (natural frequency) of the system shown below, in which the springs are coupled? $m$ $D_{1}$ $D_{2}$ $f = ?$ A 1.4 Hz B 2.0 Hz C 2.4 Hz D 2.8 Hz Answer section Calculations and explanations Correct answer:

Topic: Oscillations & Waves Metodi: Simple Harmonic Motion Analysis, Hooke’s Law, Superposition Principle Competenze: Mathematical Modeling Objects: Spring Fonte: Testo (PDF) — p.12

Problema 3 PENDULO doppio spruzzatore (problema MC) (cfr. (II round towards IPhO 2023, idea: problem group of the PhysicsOlympiad - Thomas Hellerl) In each of the two spring pendulums shown in the figure, a body of mass $m$ oscilla senza attrito. Tuttavia, le costanti di primavera $D_{1}$ e $D_{2}$ delle due Hookean springs sono diverse. Pertanto, i corpi, dopo essere stati spostati, oscillavano con diverse frequenze $f_{1}$ e $f_{2}$ . $m$ $D_{1}$ $f_{1} = 1, 2$ Hz $m$ $D_{2}$ $f_{2} = 1, 6$ Hz Qual è la frequenza di oscillazione (natural frequency) del sistema mostrato qui sotto, in cui il sistema oscillazione è le sorgenti sono accoppiate? $m$ $D_{1}$ $D_{2}$ $f = ?$ A 1.4 Hz B 2.0 Hz C 2.4 Hz D 2.8 Hz Answer section Calcoli e spiegazioni Corretta risposta:

The problem is that the two-dimensional pendulum is not a single pendulum. (five points) (second round towards IPhO 2023, idea: problem group of the PhysicsOlympic - Thomas Hellerl) In each of the two spring pendulums shown in the figure, a body of mass $m$ oscillates without friction. However, the spring constants $D_{1}$ and $D_{2}$ of the two Hookean springs are different. Therefore the bodies, after being displaced, oscillate with different frequencies $f_{1}$ and $f_{2}$ . $m$ $D_{1}$ $f_{1} = 1, 2$ Hz $m$ $D_{2}$ $f_{2} = 1, 6$ Hz What is the oscillation frequency (natural frequency) of the system shown below, in which the

springs are coupled? $m$ $D_{1}$ $D_{2}$ $f = ?$ A 1.4 Hz B 2.0 Hz C 2.4 Hz D 2.8 Hz Answer section Calculations and explanations Correct answer:

Problem 4 Resonant circuits (MC problem) (2nd round towards IPhO 2024, problem group of the PhysicsOlympiad - Thomas Hellerl & Rolf Faßbender) A circuit consisting of an ideal coil and an ideal capacitor is called a resonant circuit. The two electrical resonant circuits shown above, with the same inductance $L$ but different capacitances $C_{i}$ , oscillate completely without resistance at the indicated frequencies. $C_{1}$ $L$ $f_{1} = f$ $L$ $C_{2}$ $f_{2} = \frac{4}{3} f$ What is the oscillation frequency $f_{12}$ (natural frequency) of the following coupled system? $L$ $C_{2}$ $C_{1}$ $f_{12} = ???$ A $\frac{2}{3} f$ B $\frac{3}{4} f$ C $\frac{4}{5} f$ D $\frac{5}{4} f$ Answer section Calculations and explanations Correct answer:

Topic: Circuits, Oscillations & Waves Metodi: Simple Harmonic Motion Analysis, Kirchhoff’s Laws, Equivalent Circuit Reduction Competenze: Mathematical Modeling Objects: Inductor, Capacitor Fonte: Testo (PDF) — p.14

Problema 4 Circuiti di risonanza (problema MC) (II round towards IPhO 2024, problem group of the PhysicsOlympiad - Thomas Hellerl & Rolf Faßbender) Un circuito composto da una bobina ideale e da un condensatore ideale è chiamato circuito risonante. I due circuiti elettrici risonanti mostrati sopra, con la stessa inductanza $L$ ma diversi capacità $C_{i}$ , oscillazione completamente senza resistenza alle frequenze indicate. $C_{1}$ $L$ $f_{1} = f$ $L$ $C_{2}$ $f_{2} = \frac{4}{3} f$ Qual è la frequenza di oscillazione $f_{12}$ (frequenza naturale) del seguente sistema accoppiato? $L$ $C_{2}$ $C_{1}$ $f_{12} = ???$ A $\frac{2}{3} f$ B $\frac{3}{4} f$ C $\frac{4}{5} f$ D $\frac{5}{4} f$ Answer section Calcoli e spiegazioni Corretta risposta:

The following is the list of the problems: (Third round towards IPhO 2024, problem group of the PhysicsOlympiad - Thomas Hellerl & Rolf Faßbender) A circuit consisting of an ideal coil and an ideal capacitor is called a resonant circuit. The two electrical resonant circuits shown above, with the same inductance $L$ but different capacitances $C_{i}$ , oscillate completely without resistance at the indicated frequencies. $C_{1}$ $L$ $f_{1} = f$ $L$ $C_{2}$ $f_{2} = \frac{4}{3} f$ What is the oscillation frequency $f_{12}$ of the following coupled system? $L$ $C_{2}$ $C_{1}$ $f_{12} = ???$ A $\frac{2}{3} f$ B $\frac{3}{4} f$ C $\frac{4}{5} f$ D $\frac{5}{4} f$ Answer section Calculations and explanations Correct answer:

Problem 5 Coulomb force (MC problem) (1st round towards IPhO 2017) Two equally sized charged metal spheres are located at a very large distance from each other. The charge of one sphere is three times as large as that of the other. The force that the spheres exert on each other is $F$ . Now the spheres are brought into contact with each other and then positioned at a distance that is twice as large as the initial one. What is now approximately the force between them? A $0, 25 F$ B $0, 33 F$ C $0, 50 F$ D The force remains the same. Answer section Calculations and explanations Correct answer:

Topic: Electrostatics Metodi: Coulomb’s Law, Conservation Laws Competenze: Physical Reasoning Objects: Conducting Sphere Fonte: Testo (PDF) — p.15

Problema 5 Forza di Coulomb (problema MC) 1o round towards IPhO 2017 Due sfere di metallo caricate di dimensioni uguali si trovano a una grande distanza l’una dall’altra. La carica di una sfera è tre volte più grande di quella dell’altra. La forza che le sfere esercitano l’una sull’altra è $F$ . Ora le sfere sono messe in contatto tra loro e poi posizionato a una distanza due volte maggiore dell’inizio. Cosa c’è ora circa la forza tra loro? A $0, 25 F$ B $0, 33 F$ C $0, 50 F$ D La forza rimane la stessa. Answer section Calcoli e spiegazioni Corretta risposta:

Topic: Electrostatics Metodi: Coulomb’s Law, Conservation Laws Competenze: Physical Reasoning Objects: Conducting Sphere Fonte: Testo (PDF) — p.15

The problem is that the force of the force is not the same as the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force of the force (first round towards IPhO 2017) Two equally sized charged metal spheres are located at a very large distance from each other. The charge of one sphere is three times as large as that of the other. The force that the spheres exert on each other is $F$ . Now the spheres are brought into contact with each other and then positioned at a distance that’s twice as large as the initial one. What is now approximately the force between them? A $0, 25 F$ B $0, 33 F$ C $0, 50 F$ D The force remains the same. Answer section Calculations and explanations Correct answer:

Topic: Electrostatics Metodi: Coulomb’s Law, Conservation Laws Competenze: Physical Reasoning Objects: Conducting Sphere Fonte: Testo (PDF) — p.15

Problem 6 Alternating-current circuit (MC problem) (2nd round towards IPhO 2019, problem group of the PhysicsOlympiad - Stefan Petersen) A resistor with resistance value $R$ , a capacitor of capacitance $C$ and a coil of inductance $L$ are connected to an alternating-voltage source. The amplitude of the alternating voltage is $U$ and the components can be assumed to be ideal. The following graph shows the amplitude $I$ of the current in the circuit as a function of the frequency $f$ of the sinusoidal alternating voltage. Which of the following circuit diagrams correctly represents the circuit used? 100 200 300 400 50 100 150 200 $f$ / kHz $I$ / mA $R$ $C$ $L$ $U$ A $R$ $L$ $C$ $U$ B $L$ $C$ $R$ $U$ C $R$ $C$ $L$ $U$ D Answer section Calculations and explanations Correct answer:

Topic: Circuits, Oscillations & Waves Metodi: Kirchhoff’s Laws, Equivalent Circuit Reduction, Simple Harmonic Motion Analysis Competenze: Diagrammatic Reasoning, Mathematical Modeling Objects: Resistor, Capacitor, Inductor Fonte: Testo (PDF) — p.16

Problema 6 Circuito alternativo corrente (problema MC) (II round towards IPhO 2019, problem group of the PhysicsOlympiad - Stefan Petersen) Un resistore con valore di resistenza $R$ , un condensatore di capacità $C$ e una bobina di induttanza $L$ sono collegati a una fonte di voltaggio alternativo. L’ampiezza della volta alternata è $U$ e i componenti possono essere presunti Perfetto. Il grafico seguente mostra l’amplitude $I$ del corrente nel circuito come a funzione della frequenza $f$ della sinusoidale volta alternata. Which of the following circuit diagrams correctly represents Il circuito usato? 100 200 300 400 50 100 150 200 $f$ / kHz $I$ / mA $R$ $C$ $L$ $U$ A $R$ $L$ $C$ $U$ B $L$ $C$ $R$ $U$ C $R$ $C$ $L$ $U$ D Answer section Calcoli e spiegazioni Corretta risposta:

The following is the list of the types of electrical power generated by the electrical power generated by the electrical power generated by the electrical power generated by the electrical power generated by the electrical power generated by the electrical power generated by the electrical power generated by the electrical power generated by the electrical power generated by the electrical power generated by the electrical power generated by the electrical power generated by the electrical power generated by the electrical power generated by the electrical power generated by the electrical power generated by the electrical power generated by the electrical power generated by the electrical power generated electrical power. (second round towards IPhO 2019, problem group of the PhysicsOlympiad - Stefan Petersen) A resistor with resistance value $R$ , a capacitor of capacitance $C$ and a coil of inductance $L$ are connected to an alternating-voltage source. The amplitude of the alternating voltage is $U$ and the components can be assumed to be It’s perfect. The following graph shows the amplitude $I$ of the current in the circuit as a function of the frequency $f$ of the sinusoidal alternating voltage. Which of the following circuit diagrams correctly represents The circuit used? 100 200 300 400 50 100 150 200 $f$ / kHz $I$ / mA $R$ $C$ $L$ $U$ A $R$ $L$ $C$ $U$ B $L$ $C$ $R$ $U$ C $R$ $C$ $L$ $U$ D Answer section Calculations and explanations Correct answer:

Problem 7 Thermodynamic cycle (MC problem) (2nd round towards IPhO 2019) An ideal gas undergoes a cyclic process. Starting from state A, it is first heated at constant volume to a state B, then it expands without a change in temperature to a state C and is finally compressed isobarically back to the initial state A. Let $p$ , $V$ and $T$ denote the pressure, the volume and the temperature of the gas. Which of the following graphs correctly represent the cyclic process? I 0 A B C $V$ $p$ II 0 A B C $p$ $T$ III 0 A B C $T$ $V$ A Only graphs I and II. B Only graphs I and III. C Only graphs II and III. D All three graphs. Answer section Calculations and explanations Correct answer:

Topic: Thermodynamics Metodi: Ideal Gas Law, Thermodynamic Cycle Analysis, First Law of Thermodynamics Competenze: Diagrammatic Reasoning, Mathematical Modeling Objects: Gas Fonte: Testo (PDF) — p.18

Problema 7 Ciclo termodinamico (problema MC) (II round towards IPhO 2019) Il gas ideale subisce un processo ciclico. Partendo dallo stato A, è stato riscaldato a volume costante a uno stato B, poi si espandono senza un cambiamento di temperatura a uno stato C e finalmente vengono compresse isobaricamente indietro allo stato iniziale A. Let $p$ , $V$ and $T$ denotano la pressione, il volume e la temperatura del gas. Quali dei seguenti grafici rappresentano correttamente il processo ciclico? I 0 A B C $V$ $p$ II 0 A B C $p$ $T$ III 0 A B C $T$ $V$ A Only grafici I e II. B Only grafici I e III. C Only grafici II e III. D Tutti e tre i grafici. Answer section Calcoli e spiegazioni Corretta risposta:

The thermodynamic cycle (MC problem) (second round towards IPhO 2019) The ideal gas undergoes a cyclic process. Starting from state A, it’s first heated at It expands without a change in temperature to a state C and is finally compressed isobarically back to the initial state A. Let $p$ , $V$ and $T$ denote the pressure, the volume and the temperature of the gas. Which of the following graphs correctly represents the cyclic process? I 0 A B C $V$ $p$ II 0 A B C $p$ $T$ The Commission 0 A B C $T$ $V$ A only graphs I and II. B Only graphs I and III. C Only graphs II and III. D all three graphs. Answer section Calculations and explanations Correct answer:

Problem 8 Shifted image (1st round towards IPhO 2017) A thin lens produces, as shown in the figure, an image of an object. Lens Object Image -5 -4 -3 -2 -1 0 1 2 3 4 5 Ruler Determine the focal length of the lens. Use the drawn ruler as a scale for this. In addition, draw the resulting image when a second, identical lens is placed directly behind the first one. Answer section Calculations and explanations Result for the focal length of the lens:

Topic: Geometric Optics Metodi: Thin Lens & Mirror Equation, Ray Tracing Competenze: Diagrammatic Reasoning, Physical Reasoning Objects: Lens Fonte: Testo (PDF) — p.19

Problema 8 Immagine spostata 1o round towards IPhO 2017 Una lente sottile produce, come mostrato nella figura, un’immagine di un oggetto. Lenti Object Immagine -5 -4 -3 -2 -1 0 1 2 3 4 5 Ruler Determina la lunghezza focale della lente. Usate il rullo disegnato come scala per questo. Inoltre, draw the resulting image quando una seconda lente identica viene posta direttamente dietro la prima. Answer section Calcoli e spiegazioni Result for the focal length of the lens:

Topic: Geometric Optics Metodi: Thin Lens & Mirror Equation, Ray Tracing Competenze: Diagrammatic Reasoning, Physical Reasoning Objects: Lens Fonte: Testo (PDF) — p.19

Problem 8 Shifted image (first round towards IPhO 2017) A thin lens produces, as shown in the figure, an image of an object. Lens The object Image of the -5 -4 -3 -2 -1 0 1 2 3 4 5 Other vehicles Determine the focal length of the lens. Use the drawn ruler as a scale for this. In addition, draw the resulting image when a second, identical lens is placed directly behind the first one. Answer section Calculations and explanations Result for the focal length of the lens:

Topic: Geometric Optics Metodi: Thin Lens & Mirror Equation, Ray Tracing Competenze: Diagrammatic Reasoning, Physical Reasoning Objects: Lens Fonte: Testo (PDF) — p.19

Problem 9 Somehow shifted (1st round towards IPhO 2023, problem group of the PhysicsOlympiad) A thin converging lens forms a sharp image of an object $140, 0$ cm away on a screen positioned at a distance of $16, 8$ cm behind the lens. 9.a) Determine the focal length of the lens. Now, as sketched in the figure, a $3, 0$ cm thick, plane-parallel glass plate with refractive index $n = 1, 50$ is placed between the lens and the screen. To produce a sharp image on the screen again, the screen is shifted by a distance $Δ b$ . ? $3, 0$ cm $140, 0$ cm Screen Glass plate Lens Object Fig. 1. Sketch of the setup, not to scale. 9.b) Explain what effect the glass plate has on a light ray that does not strike it perpendicularly. Use this to argue whether, in order to produce a sharp image, the screen must be moved closer to the lens or farther away from it. 9.c) Determine the magnitude $Δ b$ of the necessary shift of the screen. For simplicity, you may assume that only paraxial rays are involved in the imaging process. Answer section 9.a) Calculations and explanations Result for the focal length of the lens: 9.b) Calculations and explanations 9.c) Calculations and explanations Result for the shift of the screen:

Topic: Geometric Optics Metodi: Thin Lens & Mirror Equation, Snell’s Law, Ray Tracing Competenze: Mathematical Modeling, Physical Reasoning Objects: Lens, Screen Fonte: Testo (PDF) — p.20

Problema 9 In qualche modo spostato (Primo round verso l’IPhO 2023, gruppo problematico della PhysicsOlympiad) A thin converging lens forms a sharp image of an object $140, 0$ cm away on a screen posizionato a una distanza di $16, 8$ cm dietro il lente. 9. (a) Determina la lunghezza focale del lente. Ora, come disegnato nella figura, un $3, 0$ cm spessore, piano parallelo la piastra di cristallo con indice di refraczione $n = 1, 50$ è collocata tra il lente e lo schermo. Per produrre un’immagine acuta sullo schermo di nuovo, il schermo è spostato da una distanza $Δ b$ . ? $3, 0$ cm $140, 0$ cm Scatto Dischi di vetro Lenti Object Fig. 1. Sketch of the setup, non a scala. 9.b) Spiegare l’effetto che la piastra di vetro ha su un raggio di luce che non lo colpisce perpendicularmente. Usare questo per discutere se, per produrre un’immagine acuta, lo schermo deve essere spostato più vicino alla lente o più lontano da essa. 9.c) Determina la magnitudo $Δ b$ del necessario spostamento dello schermo. Per semplicità, si può presumere che solo i raggi parazziali sono coinvolti nel processo di imaging. Answer section 9.a) Calcoli e spiegazioni Result for the focal length of the lens: 9.b) Calcoli e spiegazioni 9.c) Calcoli e spiegazioni Result for the shift of the screen:

Problem 9 Somehow shifted (first round towards IPhO 2023, problem group of the PhysicsOlympiad) A thin converging lens forms a sharp image of an object $140, 0$ cm away on a screen positioned at a distance of $16, 8$ cm behind the lens. 9. (a) Determine the focal length of the lens. Now, as sketched in the figure, a $3, 0$ cm thick, plane parallel glass plate with refractive index $n = 1, 50$ is placed between the lens and the screen. To produce a sharp image on the screen again, the screen is shifted by a distance $Δ b$ . ? $3, 0$ cm $140, 0$ cm Screen Glass plate Lens The object Fig. 1. Sketch of the setup, not to scale. 9. (b) Explain what effect the glass plate has on a light ray that does not strike it perpendicularly. Use this to argue whether, in order to produce a sharp image, the screen must be moved closer to the lens or farther away from it. 9. (c) Determine the magnitude $Δ b$ of the necessary shift of the screen. For simplicity, you may assume that only paraxial rays are involved in the The imaging process. Answer section 9.a) Calculations and explanations Result for the focal length of the lens: 9.b) Calculations and explanations 9.c) Calculations and explanations Result for the shift of the screen:

Problem 10 Race between photon and proton (2nd round towards IPhO 2019, problem group of the PhysicsOlympiad - Richard Reindl & Thomas Hellerl) In a supernova in Barnard’s Galaxy, a neighboring galaxy of our Milky Way, a photon and a proton set off on the journey to Earth at the same time. There the proton is registered 72 hours later than the photon. The total energy of the proton is $9, 38$ TeV $= 9, 38 \cdot 1 0^{12}$ eV. 10.a) Show that the total energy of the proton is about 10,000 times its rest energy. (3.0 pts.) 10.b) Calculate at what distance from Earth the supernova took place. Give your result in light-years. (5.0 pts.) 10.c) Determine how long the journey of the proton lasted in its reference frame. (2.0 pts.) Answer section 10.a) Calculations and explanations 10.b) Calculations and explanations Result for the distance from Earth at which the supernova took place: 10.c) Calculations and explanations Result for the duration of the proton’s journey in its reference frame:

Topic: Special Relativity, Astrophysics Metodi: Relativistic Energy-Momentum, Lorentz Transformation, Mass-Energy Equivalence Competenze: Mathematical Modeling, Physical Reasoning Objects: Photon Fonte: Testo (PDF) — p.22

Il problema 10 Race between photon and proton (II round towards IPhO 2019, problem group of the PhysicsOlympiad - Richard Reindl & Thomas Hellerl) In una supernova nella galassia di Barnard, una galassia vicina della nostra Via Lattea, un fotone e un protone partono nel viaggio verso la Terra allo stesso tempo. Il protone è registrato 72 ore dopo che il fotone. La energia totale del protone è $9, 38$ TeV $= 9, 38 \cdot 1 0^{12}$ eV. 10. (a) Mostra che l’energia totale del protone è circa 10.000 volte la sua energia restante. (Punto di riferimento) 10. (b) Calcolare a che distanza dalla Terra si è verificata la supernova. Give your result

In anni luce. (5,0 p. d.) 10.c) Determina quanto tempo il viaggio del protone è durato nel suo frame di riferimento. (punto 2.0) Answer section 10.a) Calcoli e spiegazioni 10.b) Calcoli e spiegazioni Risultato per la distanza dalla Terra a cui si è verificata la supernova: 10.c) Calcoli e spiegazioni Risultato per la durata del viaggio del protone nel suo frame di riferimento:

Problem 10 Race between photon and proton (second round towards IPhO 2019, problem group of the PhysicsOlympiad - Richard Reindl & Thomas Hellerl) In a supernova in Barnard’s Galaxy, a neighboring galaxy of our Milky Way, a photon and a proton set off on the journey to Earth at the same time. There the proton is registered 72 hours later than the photon. The total energy of the proton is $9, 38$ TeV $= 9, 38 \cdot 1 0^{12}$ eV. 10.a) Show that the total energy of the proton is about 10,000 times its rest energy. (including the following) 10. (b) Calculate at what distance from Earth the supernova took place. Give your result In light-years. (5.0 p.p.) 10.c) Determine how long the journey of the proton lasted in its reference frame. (b) the number of persons who are not members of the Answer section 10.a) Calculations and explanations 10.b) Calculations and explanations Result for the distance from Earth at which the supernova took place: 10.c) Calculations and explanations Result for the duration of the proton’s journey in its reference frame:

Problem 11 Fundamentals of laser cooling (3rd round towards IPhO 2017, based on a problem from IPhO 2009) In this problem the basis of cooling a gas with the help of lasers is investigated. For this, consider an atom that moves with a velocity $v$ in one direction. For simplicity, in the following discussion we also restrict ourselves to this one dimension. The atom has two internal energy states and is initially in the ground state. The energy difference between the two states is $h f_{0}$ . $v$ $f_{P h}$ A laser beam of frequency $f_{P h}$ travels opposite to the direction of motion of the atom. With a suitable choice of the frequency $f_{P h}$ , the atom can absorb a photon of the laser beam and then spontaneously emit it. The emission then occurs with equal probability in or against the direction of motion of the atom. In the following, assume that the velocity $v$ is much smaller than the speed of light $c$ and that the momentum of the atom is much larger than the momentum of a single photon. 11.a) Determine the average change in the kinetic energy of the atom in the laboratory frame when it absorbs a photon and then emits it again. Also calculate the average change in the velocity of the atom in such a process. In laser cooling of atoms, an additional laser beam is now used in the direction of the velocity of the atom, whose photons likewise have the frequency $f_{P h}$ . A detailed investigation of the cooling mechanism is more involved. The preceding investigation, however, allows you to understand the basic principle of laser cooling. 11.b) Explain qualitatively how the described arrangement can be used for cooling, that is, for slowing down, atoms. In doing so, also state whether the frequency $f_{P h}$ of the photons must be chosen larger than, smaller than, or equal to the excitation frequency $f_{0}$ of the atoms. Answer section 11.a) Calculations and explanations 11.b) Calculations and explanations (continued) Result for the changes in kinetic energy and velocity: 11.c) Calculations and explanations

Topic: Modern-Quantum Physics, Oscillations & Waves Metodi: Photon Energy Relation, Conservation of Momentum, Conservation of Energy Competenze: Physical Reasoning, Mathematical Modeling Objects: Atom, Photon Fonte: Testo (PDF) — p.24

Problema 11 Fondamenti del laser cooling (3rd round towards IPhO 2017, based on a problem from IPhO 2009) In questo problema è stata studiata la base del raffreddamento di un gas con l’aiuto di laser. Per questo, considerate un atomo che si muove con una velocità $v$ in una direzione. Per semplicità, in Dopo la discussione ci limitiamo anche a questa dimensione. L’atomo ha due stati di energia interna ed è inizialmente in stato di base. La differenza di energia tra i due stati è $h f_{0}$ . $v$ $f_{P h}$ Un fascio laser di frequenza $f_{P h}$ viaggia opposto alla direzione di movimento dell’atomo. Con un suitable choice of the frequency $f_{P h}$ , the atom can absorb a photon of the laser beam and E poi lo emettono spontaneamente. L’emissione allora si verifica con uguale probabilità in Oppure contro la direzione del movimento dell’atomo. In the following, supponiamo che la velocità $v$ è molto inferiore alla velocità di luce $c$ e che l’impulso dell’atomo è molto più grande dell’impulso di un singolo fotone. 11. (a) Determina il cambiamento medio nell’energia cinetica dell’atomo nel laboratorio quando Assorbe un fotone e lo emette di nuovo. Quindi calcola il medio cambiamento della velocità dell’atomo in un processo simile. In laser cooling of atoms, un ulteriore raggio laser è ora usato nella direzione della velocità dell’atomo, i cui fotoni hanno anche la frequenza $f_{P h}$ . A dettaglio L’investigazione del meccanismo di raffreddamento è più coinvolta. L’investigazione precedente, tuttavia, consente di Per capire il principio di base del raffreddamento laser. 11.b) Spiegare qualitativamente come l’arrangimento descritto possa essere utilizzato per il raffreddamento, cioè per il rallentamento,

Gli atomi. In questo modo, anche indicare se la frequenza $f_{P h}$ dei fotoni deve essere scelto più grande di, più piccolo di, o uguale alla frequenza di eccitazione $f_{0}$ degli atomi. Answer section 11.a) Calcoli e spiegazioni 11.b) Calcoli e spiegazioni (continuato) Risultato per i cambiamenti di energia e velocità cinetica: 11.c) Calcoli e spiegazioni

Problem 11 Fundamentals of laser cooling (third round towards IPhO 2017, based on a problem from IPhO 2009) In this problem the basis of cooling a gas with the help of lasers is investigated. For this, consider an atom that moves with a velocity $v$ in one direction. For simplicity, in the We also restrict ourselves to this one dimension. The atom has two internal energy states and is initially in the ground state. The energy difference between the two states is $h f_{0}$ . $v$ $f_{P h}$ A laser beam of frequency $f_{P h}$ travels opposite to the direction of motion of the atom. With a suitable choice of the frequency $f_{P h}$ , the atom can absorb a photon of the laser beam and Then spontaneously emit it. The emission then occurs with equal probability in or against the direction of the atom’s motion. In the following, assume that the velocity $v$ is much smaller than the speed of light $c$ And that the momentum of the atom is much greater than the momentum of a single photon. 11. (a) Determine the average change in the kinetic energy of the atom in the laboratory frame when It absorbs a photon and then emits it again. So calculate the average change in the velocity of the atom in such a process. In laser cooling of atoms, an additional laser beam is now used in the direction of the velocity of the atom, whose photons likewise have the frequency $f_{P h}$ . A detailed The investigation of the cooling mechanism is more involved. The preceding investigation, however, allows You understand the basic principle of laser cooling. 11. (b) Explain qualitatively how the described arrangement can be used for cooling, that is, for slowing down, The atomic energy of the atom. In doing so, also state whether the frequency $f_{P h}$ of the photons must be chosen larger than, smaller than, or equal to the excitation frequency $f_{0}$ of the atoms. Answer section 11.a) Calculations and explanations 11.b) Calculations and explanations (continued) Result for the changes in kinetic energy and velocity: 11.c) Calculations and explanations

Problem 12 Simple pendulum (Companion booklet of the 1st round towards the 50th IPhO 2019) From a thin thread and a small weight, such as a screw or nut, a simple pendulum can be built. If the extent of the weight is very small compared to the thread length $ℓ$ , then for the period $T$ of the pendulum at small displacements $T = 2 π \frac{ℓ}{g} .$ Here $g$ denotes the gravitational acceleration on Earth. Theoretically, $T^{2}$ should therefore be a linear function of the thread length $ℓ$ . The following table presents values of the oscillation periods $T$ measured in an experiment, together with the averaged oscillation period $\overset{ˉ}{T}$ and the square of this quantity. Thread length Time for 10 oscillation periods Mean value $ℓ$ / cm $10 T$ / s $\overset{ˉ}{T}$ / s $\overset{ˉ}{T}^{2}$ / s $^{2}$ 67.2 16.62 16.87 15.43 17.50 17.61 1.68 2.82 55.5 15.12 13.94 16.18 15.04 15.53 1.51 2.29 47.0 13.79 12.60 13.37 14.41 14.80 1.38 1.90 34.5 11.93 13.02 10.77 12.18 11.72 1.19 1.42 22.0 9.50 11.44 9.24 9.59 8.73 0.97 0.94 13.4 7.91 6.38 8.32 8.91 7.89 0.79 0.62 Using a suitable graph, check whether the experimental data fit the theoretically expected behavior, and determine the value of the gravitational acceleration $g$ . Answer section Calculations and explanations Graph Calculations and explanations (continued) Result:

Topic: Oscillations & Waves, Newtonian Mechanics Metodi: Simple Harmonic Motion Analysis, Graph Linearization, Experimental Data Analysis Competenze: Graph Linearization, Experimental Data Analysis, Mathematical Modeling Objects: Pendulum, String Fonte: Testo (PDF) — p.26

Problema 12 PENDULO Semplice (Companion booklet of the 1st round towards the 50th IPhO 2019) Da un filo sottile e un peso piccolo, come una vite o una nut, Un semplice pendolo può essere costruito. Se l’estensione del peso è molto piccola rispetto alla lunghezza del filo $ℓ$ , allora per il periodo $T$ del pendolo a piccoli spostamenti $T = 2 π \frac{ℓ}{g} .$ Qui $g$ indica l’accelerazione gravitazionale sulla Terra. In teoria, il $T^{2}$ dovrebbe quindi essere un funzione lineare della lunghezza del filo $ℓ$ . La tabella seguente presenta i valori dei periodi di oscillazione $T$ misurati in un esperimento, insieme al periodo di oscillazione medio $\overset{ˉ}{T}$ e al quadrato di questa quantità. Thread length Time for 10 oscillation periods Valore medio $ℓ$ / cm $10 T$ / s $\overset{ˉ}{T}$ / s $\overset{ˉ}{T}^{2}$ / s $^{2}$ 67.2 16.62 16.87 15.43 17.50 17.61 1.68 2.82 55.5 15.12 13.94 16.18 15.04 15.53 1.51 2.29 47.0 13.79 12.60 13.37 14.41 14.80 1.38 1.90 34.5 11.93 13.02 10.77 12.18 11.72 1.19 1.42 22.0 9.50 11.44 9.24 9.59 8.73 0.97 0.94 13.4 7.91 6.38 8.32 8.91 7.89 0.79 0.62 Usando un grafico appropriato, controllare se i dati sperimentali si adattano teoricamente atteso comportamento, e determinare il valore dell’accelerazione gravitazionale $g$ . Answer section Calcoli e spiegazioni Grafico Calcoli e spiegazioni (continuato) Risultato:

Problem 12 Simple pendulum is not a problem (Companion booklet of the 1st round towards the 50th IPhO 2019) From a thin thread and a small weight, such as a screw or nut, A simple pendulum can be built. If the extent of the weight is very small compared to the thread length $ℓ$ , then for the period $T$ of the pendulum at small displacements $T = 2 π \frac{ℓ}{g} .$ Here $g$ denotes the gravitational acceleration on Earth. Theoretically, $T^{2}$ should therefore be a linear function of the thread length $ℓ$ . The following table presents values of the oscillation periods $T$ measured in an experiment, together with the average oscillation period $\overset{ˉ}{T}$ and the square of this quantity. Thread length Time for 10 oscillation periods Mean value $ℓ$ / cm $10 T$ / s $\overset{ˉ}{T}$ / s $\overset{ˉ}{T}^{2}$ / s $^{2}$ 67.2 16.62 16.87 15.43 17.50 17.61 1.68 2.82 55.5 15.12 13.94 16.18 15.04 15.53 1.51 2.29 47.0 13.79 12.60 13.37 14.41 14.80 1.38 1.90 34.5 11.93 13.02 10.77 12.18 11.72 1.19 1.42 22.0 9.50 11.44 9.24 9.59 8.73 0.97 0.94 13.4 7.91 6.38 8.32 8.91 7.89 0.79 0.62 Using a suitable graph, check whether the experimental data fit the theoretically expected behavior, and determine the value of the gravitational acceleration $g$ . Answer section Calculations and explanations Graph Calculations and explanations (continued) The result:

Raccolta Gare di Fisica

Esplora

IPhO 2026

Vista grafico