IPhO 2015

Problem 1 (13 points)

Crazy Loop

An ambitious tinkerer builds the most varied tracks for toy cars. One of his favorite tracks contains a loop of radius $R=40\text{cm}$ that, as sketched in the adjacent figure, is interrupted. A very small car starts from a height $h=3R$, rolls down the slope, and then reaches the gap in the loop. The car jumps, flies, … lands gently at the start of the other part of the loop and continues its journey.

Calculate the length of the missing section of the loop.

Friction effects and the size of the toy car should be neglected.

loop track with height $3R$ and angle $\alpha$

Topic: Newtonian Mechanics, Oscillations & Waves, Thermodynamics Metodi: Physical Modeling, Free-Body Diagram, Conservation Laws Competenze: Physical Reasoning, Mathematical Modeling Objects: Cart Fonte: Testo (PDF) — p.2

Problema 1 (13 punti)

Loop di follia

An Ambitious Tinkerer costruisce i più vari tracciati per le auto da gioco. One of his favorite tracks contains a loop of radius $R=40\text{cm}$ that, as sketched in the adjacent figure, is interrupted. A very small car starts from a height $h=3R$, rolls down the slope, and then reaches the gap in the loop. La macchina salta, vola, Atterrà delicatamente all’inizio dell’altra parte del loop e continuerà il suo viaggio.

Calcolare la lunghezza della sezione mancante del loop.

Gli effetti di frattura e la dimensione della macchina da gioco dovrebbero essere trascurati.

loop track with height $3R$ and angle $\alpha$

Topic: Newtonian Mechanics, Oscillations & Waves, Thermodynamics Metodi: Physical Modeling, Free-Body Diagram, Conservation Laws Competenze: Physical Reasoning, Mathematical Modeling Objects: Cart Fonte: Testo (PDF) — p.2

Problem 1 (13 points)

Crazy loop

An ambitious tinkerer builds the most varied tracks for toy cars. One of his favorite tracks contains a loop of radius $R=40\text{cm}$ that, as sketched in the adjacent figure, is interrupted. A very small car starts from a height $h=3R$, rolls down the slope, and then reaches the gap in the loop. The car jumps, flies, lands gently at the start of the other part of the loop and continues its journey.

Calculate the length of the missing section of the loop.

Friction effects and the size of the toy car should be neglected.

loop track with height $3R$ and angle $\alpha$

Topic: Newtonian Mechanics, Oscillations & Waves, Thermodynamics Metodi: Physical Modeling, Free-Body Diagram, Conservation Laws Competenze: Physical Reasoning, Mathematical Modeling Objects: Cart Fonte: Testo (PDF) — p.2

Problem 2 (13 points)

Resistor Network

In addition to a voltage source, the circuit shown contains resistors with resistance values $R_1$ and $R_2$. Initially, identical resistors are used, i.e. $R_1=R_2$. The current $I$ flowing in this case is $1.0\text{A}$.

Determine the current $I$ when the two resistors labeled $R_2$ are each replaced by resistors with twice the resistance value.

Problem 4 (11 points)

The Fate of the Earth

Over the course of its evolution, the composition of the Sun changes due to the fusion processes taking place in its interior. By the end of its phase as a main-sequence star, the Sun’s radius will therefore increase to about 1.6 times its present value, while its surface temperature drops to about 96% of today’s value.

At the Sun’s current stage of development, neglecting the greenhouse effect, an equilibrium temperature of about $246\text{K}$ would be established on Earth.

Estimate by how much this temperature will shift due to the change in the Sun. Briefly explain what this could mean for life on Earth.

For the estimate, assume that the orbital radius and other relevant parameters of the Earth do not change and that the temperature is the same over the entire Earth.

Problem 3 (13 points)

Chaotic Physics

While sorting her physics notes, Hanna finds a graph that she cannot place. Unfortunately, she forgot to label the axes. In her notes, however, she finds three experiments to which she thinks the measured values might correspond.

• The observation of a bouncy ball after it impacts a flat, solid surface located $20\text{cm}$ above the floor: the graph shows the height of the bouncy ball (in cm) as a function of time (in ms) after the bounce. • The charging of a capacitor with a capacitance of $6.8\mu \text{F}$ through a $3.3\text{M}\Omega$ resistor. The capacitor is initially charged to a voltage of $20\text{V}$. The graph represents the capacitor voltage (in V) as a function of time (in s). • The heating of $100\text{ml}$ of water in a vessel with an electric heating device (power: $350\text{W}$). The water initially has ambient temperature. The graph shows the water temperature (in ${}^{\circ }\text{C}$) as a function of time (in s).

In fact, the graph corresponds to exactly one of the experiments. Find out which of the experiments it is and why the other two are not possible. Justify your answer physically for all three experiments.

The figure is available in higher resolution on the IPhO website.

resistor network $R_1$ $R_2$ with source $U$

Topic: Astrophysics, Thermodynamics, Circuits Metodi: Physical Modeling, Experimental Data Analysis, Ideal Gas Law Competenze: Physical Reasoning, Estimation & Approximation, Graph Linearization Objects: Resistor, Battery, Star, Planet, Ball, Capacitor Fonte: Testo (PDF) — p.2

Il problema 2 (13 punti)

Rete di resistore

In aggiunta a una fonte di voltage, il circuito mostrato contiene resistori con valori di resistenza $R_1$ e $R_2$. Inizialmente, sono utilizzati resistori identici, cioè: $R_1=R_2$. Il corrente $I$ che scorre in questo caso è $1.0\text{A}$.

Determine il corrente $I$ quando le due resistori etichettate $R_2$ sono ciascuna sostituita da resistori con doppio valore di resistenza.

Problema 4 (11 punti)

Il destino della terra

Nel corso della sua evoluzione, la composizione del Sole cambia a causa dei processi di fusione che si svolgono nel suo interno. Al termine della sua fase come stella di sequenza principale, il raggio del Sole aumenterà quindi a circa 1,6 volte il suo valore attuale, mentre la sua temperatura superficiale scenderà a circa il 96% del valore di oggi.

Al momento attuale dello sviluppo del Sole, trascurando l’effetto serra, si sarebbe stabilita sulla Terra una temperatura di equilibrio di circa $246\text{K}$.

Estimare quanto cambierà la temperatura a causa del cambiamento del sole. Spiegare brevemente cosa questo potrebbe significare per la vita sulla Terra.

Per la stima, supponiamo che il raggio orbitale e altri parametri pertinenti della Terra non cambiino e che la temperatura sia la stessa su tutta la Terra.

Problema 3 (13 punti)

Fisica caotica

Mentre sorta le sue note di fisica, Hanna trova un grafico che non può inserire. Sfortunatamente, ha dimenticato di etichettare gli assi. Nel suo libro, tuttavia, trova tre esperimenti ai quali pensa che i valori misurati possano corrispondere.

• L’osservazione di una bouncy ball after it impacts a flat, solid surface located $20\text{cm}$ above the floor: the graph shows the height of the bouncy ball (in cm) as a function of time (in ms) after the bounce. • Il carico di un condensatore con una capacità di $6.8\mu \text{F}$ attraverso un resistore $3.3\text{M}\Omega$. Il condensatore è inizialmente caricato a una voltage di $20\text{V}$. Il grafico rappresenta la tensione del condensatore (in V) come funzione di tempo (in s). • Il riscaldamento di $100\text{ml}$ di acqua in un recipiente con dispositivo di riscaldamento elettrico (potenza: $350\text{W}$). L’acqua inizia ad avere temperatura ambientale. Il grafico mostra la temperatura dell’acqua (in ${}^{\circ }\text{C}$) come funzione di tempo (in s).

Infatti, il grafico corrisponde esattamente a uno degli esperimenti. Scopri quale dei due esperimenti è possibile e perché gli altri due non sono possibili. Giustifica la tua risposta fisicamente per tutti e tre gli esperimenti.

The figure is available in higher resolution sul sito IPhO.

resistor network $R_1$ $R_2$ with source $U$

Topic: Astrophysics, Thermodynamics, Circuits Metodi: Physical Modeling, Experimental Data Analysis, Ideal Gas Law Competenze: Physical Reasoning, Estimation & Approximation, Graph Linearization Objects: Resistor, Battery, Star, Planet, Ball, Capacitor Fonte: Testo (PDF) — p.2

Problem 2 (13 points)

Resistor network

In addition to a voltage source, the circuit shown contains resistors with resistance values $R_1$ and $R_2$. Initially, identical resistors are used, i.e. $R_1=R_2$. The current $I$ flowing in this case is $1.0\text{A}$.

Determine the current $I$ when the two resistors labeled $R_2$ are each replaced by resistors with twice the resistance value.

Problem 4 (11 points)

The Fate of the Earth

Over the course of its evolution, the composition of the Sun changes due to the fusion processes taking place in its interior. By the end of its phase as a main-sequence star, the Sun’s radius will therefore increase to about 1.6 times its present value, while its surface temperature drops to about 96% of today’s value.

At the Sun’s current stage of development, neglecting the greenhouse effect, an equilibrium temperature of about $246\text{K}$ would be established on Earth.

Estimate by how much this temperature will shift due to the change in the sun. Briefly explain what this could mean for life on Earth.

For the estimate, assume that the orbital radius and other relevant parameters of the Earth do not change and that the temperature is the same over the entire Earth.

Problem 3 (13 points)

Chaotic physics

While sorting her physics notes, Hanna finds a graph that she cannot place. Unfortunately, she forgot to label the axes. In her notes, however, she finds three experiments to which she thinks the measured values might correspond.

• The observation of a bouncy ball after it impacts a flat, solid surface located $20\text{cm}$ above the floor: the graph shows the height of the bouncy ball (in cm) as a function of time (in ms) after the bounce. • The charging of a capacitor with a capacitance of $6.8\mu \text{F}$ through a $3.3\text{M}\Omega$ resistor. The capacitor is initially charged to a voltage of $20\text{V}$. The graph represents the capacitor voltage (in V) as a function of time (in s). • The heating of $100\text{ml}$ of water in a vessel with an electric heating device (power: $350\text{W}$). The water initially has ambient temperature. The graph shows the water temperature (in ${}^{\circ }\text{C}$) as a function of time (in s).

In fact, the graph corresponds to exactly one of the experiments. Find out which of the experiments it is and why the other two are not possible. Justify your answer physically for all three experiments.

The figure is available in higher resolution on the IPhO website.

resistor network $R_1$ $R_2$ with source $U$

Topic: Astrophysics, Thermodynamics, Circuits Metodi: Physical Modeling, Experimental Data Analysis, Ideal Gas Law Competenze: Physical Reasoning, Estimation & Approximation, Graph Linearization Objects: Resistor, Battery, Star, Planet, Ball, Capacitor Fonte: Testo (PDF) — p.2

Problem 3 (13 points)

Junior Problem (10 points)

Match Image

A real image of a match is produced with a thin lens. In the figure, the match, the optical axis of the lens, and the image of the match head are drawn.

Construct the resulting image of the entire match and mark the focal length of the lens in the figure. In doing so, explain your procedure.

The figure is available in higher resolution on the IPhO website.

graph of experiment data vs time

match, lens, optical axis, image

Topic: Geometric Optics Metodi: Ray Tracing, Thin Lens & Mirror Equation Competenze: Diagrammatic Reasoning, Physical Reasoning Objects: Lens Fonte: Testo (PDF) — p.3

Problema 3 (13 punti)

Problema junior (10 punti)

Immagine di match

Una vera immagine di una partita è prodotta con una lente sottile. Nella figura, la corrispondenza, l’asse ottico della lente, e l’immagine della testa di corrispondenza sono disegnati.

Costruire l’immagine risultante dell’intera partita e segnare la distanza focale della lente nella figura. In questo modo, spiega la tua procedura.

The figure is available in higher resolution sul sito IPhO.

graph of experiment data vs time

match, lens, optical axis, image

Topic: Geometric Optics Metodi: Ray Tracing, Thin Lens & Mirror Equation Competenze: Diagrammatic Reasoning, Physical Reasoning Objects: Lens Fonte: Testo (PDF) — p.3

Problem 3 (13 points)

The minimum number of points

Match image

A real image of a match is produced with a thin lens. In the figure, the match, the optical axis of the lens, and the image of the match head are drawn.

Construct the resulting image of the entire match and mark the focal length of the lens in the figure. In doing so, explain your procedure.

The figure is available in higher resolution on the IPhO website.

The following table shows the results of the experiment:

The following is the list of the types of equipment used:

Topic: Geometric Optics Metodi: Ray Tracing, Thin Lens & Mirror Equation Competenze: Diagrammatic Reasoning, Physical Reasoning Objects: Lens Fonte: Testo (PDF) — p.3