CAP 2024 Nazionale — 2024_CAP_Exam_with

Fonte: gare di altri paesi/Canada/all/2024_CAP_Exam_with_solutions.pdf · Apri PDF · apri PDF p.2 Cluster: Elettromagnetismo

Problema 1

Black hole radius

Black holes have a very strong gravitational field from which nothing can escape, including light. Approximately, what is the radius of a black hole having mass equal to the sun?

Hint: Use Newtonian mechanics.

A. 1.5 km B. 3 km C. 15 km D. 30 km E. The mass is too small to be a black hole.

Topic: Gravitation, Newtonian Mechanics Metodi: Newton’s Law of Gravitation, Energy Conservation Method Competenze: Physical Reasoning, Estimation & Approximation Fonte: Testo (PDF) — p.2

Problema 2

Soap bubble pressure difference

Surface tension is defined as the force per unit length acting on the surface of a liquid. Due to the presence of it, the surface of a liquid can be thought of as having a “stretchy” property like a piece of rubber. Assume that a soap bubble of radius $R$ has a surface tension $T$ . What is the difference in pressure between the pressure inside the bubble and the atmospheric pressure? Note that a soap bubble has both an inner and an outer surface.

A. $\frac{T}{2 R}$ B. $\frac{T}{R}$ C. $\frac{2 T}{R}$ D. $\frac{4 T}{R}$ E. $\frac{8 T}{R}$

Topic: Fluid Mechanics Metodi: Physical Modeling, Free-Body Diagram Competenze: Physical Reasoning Fonte: Testo (PDF) — p.2

Problema 3

Pulley system acceleration constraint

Consider the setup shown (a pulley system with masses $m_{1}$ , $m_{2}$ , $m_{3}$ ). Let $a_{1}, a_{2}, a_{3}$ be the accelerations of $m_{1}, m_{2}, m_{3}$ respectively. Which of the following is true?

A. $a_{1} = - \frac{1}{2} (a_{2} + a_{3})$ B. $a_{1} = - \frac{1}{2} (a_{2} - a_{3})$ C. $a_{1} = \frac{1}{2} (a_{2} + a_{3})$ D. $a_{1} = a_{2} + a_{3}$ E. $a_{1} = a_{2} - a_{3}$ F. $a_{1} = - (a_{2} + a_{3})$

sistema di carrucole con masse m1 m2 m3

Topic: Newtonian Mechanics Metodi: Free-Body Diagram, Kinematic Equations Competenze: Physical Reasoning, Diagrammatic Reasoning Fonte: Testo (PDF) — p.3

Problema 4

Ring rolling without slipping

A ring of radius $r$ (and moment of inertia with respect to an axis going through its center of mass $I = m r^{2}$ ) is given an initial velocity $v_{0}$ . The ring slides for a little while and then starts to roll. What is the velocity of the ring when it rolls without slipping?

A. $0.5 v_{0}$ B. $v_{0}$ C. $1.5 v_{0}$ D. $2.5 v_{0}$ E. $3.5 v_{0}$

Topic: Rotational Dynamics, Newtonian Mechanics Metodi: Torque & Angular Momentum Analysis Competenze: Physical Reasoning, Mathematical Modeling Fonte: Testo (PDF) — p.3

Problema 5

Force of rain on umbrella

The speed of a falling raindrop is 10 m/s. The weather forecast predicts 10 mm of rain in the next hour. If you go out with an umbrella of radius 70 cm, approximately what is the force of the rain hitting your umbrella?

A. 0.0002 N B. 0.0004 N C. 0.002 N D. 0.004 N E. 0.02 N F. 0.04 N

Topic: Newtonian Mechanics, Order-of-Magnitude Estimation Metodi: Impulse-Momentum Theorem, Order-of-Magnitude Estimation Competenze: Estimation & Approximation, Physical Reasoning Fonte: Testo (PDF) — p.3

Problema 6

Towel hanging off table

A kitchen towel hangs off a table. The coefficient of friction between towel and table is $μ$ . What is the maximum fraction of the towel that can hang before the towel falls?

A. $\frac{μ}{1 + μ}$ B. $\frac{μ}{1 - μ}$ C. $\frac{1 - μ}{μ}$ D. $\frac{μ}{2 + μ}$

Topic: Newtonian Mechanics, Rigid Body Statics Metodi: Free-Body Diagram, Conservation Laws Competenze: Physical Reasoning, Mathematical Modeling Fonte: Testo (PDF) — p.3

Problema 7

Baseball throw on the Moon

Neglecting air resistance, a baseball player on Earth can throw a baseball a distance of 100 m. If gravity is 5 times weaker on the Moon, how far could he throw the ball if he were on the Moon?

A. 20 m B. 100 m C. 500 m D. 2500 m

Topic: Newtonian Mechanics Metodi: Kinematic Equations, Dimensional Analysis Competenze: Physical Reasoning, Estimation & Approximation Fonte: Testo (PDF) — p.5

Problema 8

Elevator energy consumption

You live on the 20th floor of a building with two elevators, A and B, of the same mass. To go to floor 20, elevator A accelerates for the first 10 floors, then decelerates for the last 10. Meanwhile elevator B accelerates for 5 floors, moves at a constant speed for 10 floors, then decelerates for 5 floors. Both elevators complete the trip in the same amount of time. You are environmentally conscious, so you want to take the elevator which uses the least electricity. Assume friction in the motors and air drag are negligible. Which elevator should you take?

A. Elevator A. B. Elevator B. C. Both elevators use the same amount of electricity. D. Can’t say without more information.

Topic: Conservation of Energy, Newtonian Mechanics Metodi: Energy Conservation Method, Physical Modeling Competenze: Physical Reasoning Fonte: Testo (PDF) — p.5

Problema 9

Springs in series oscillation

A mass hangs from the ceiling, held up by two springs in series. The upper spring has spring constant $k_{1}$ , and the bottom spring has spring constant $k_{2}$ . If the mass is displaced from equilibrium by a small amount $x$ , it has a maximum speed $v$ . If instead the mass is displaced by $x /2$ , what is the maximum speed?

A. $\frac{1}{4} v$ B. $\frac{1}{2} v$ C. $v$ D. $\frac{k _{1} + k _{2}}{2 k _{1} k _{2}} v$ E. $\frac{k _{1} - k _{2}}{2 k _{1} k _{2}} v$ F. $\frac{∣ k _{1} - k _{2} ∣}{2 k _{1} k _{2}} v$

Topic: Oscillations & Waves Metodi: Simple Harmonic Motion Analysis, Energy Conservation Method Competenze: Physical Reasoning, Mathematical Modeling Fonte: Testo (PDF) — p.5

Problema 10

A car is speeding up as it goes in a circle. When it is at the northernmost point of the circle, what are the possible directions for its acceleration?

A. Exactly north B. Between north and west C. Between north and east D. Exactly south E. Between south and west F. Between south and east

Topic: Newtonian Mechanics, Rotational Dynamics Metodi: Vector Decomposition, Free-Body Diagram Competenze: Physical Reasoning, Diagrammatic Reasoning Fonte: Testo (PDF) — p.5

Problema 11

A regular $n$ -gon is a polygon with $n$ equal edges as well as equal angles. Positive charges each of magnitude $q$ sit on the corners of a regular 2024-gon. The distance from each charge to the center of the 2024-gon is $a$ . If one of the charges is removed, what is the magnitude and direction of the electric field at the center?

A. $\frac{q}{4 π ϵ _{0} a ^{2}}$ , directed away from the removed charge. B. $\frac{q}{4 π ϵ _{0} a ^{2}}$ , directed towards the removed charge. C. $\frac{1}{2024} \frac{q}{4 π ϵ _{0} a ^{2}}$ , directed away from the removed charge. D. $\frac{1}{2024} \frac{q}{4 π ϵ _{0} a ^{2}}$ , directed towards the removed charge. E. $\frac{1}{2023} \frac{q}{4 π ϵ _{0} a ^{2}}$ , directed away from the removed charge. F. $\frac{1}{2023} \frac{q}{4 π ϵ _{0} a ^{2}}$ , directed towards the removed charge.

Topic: Electrostatics Metodi: Coulomb’s Law, Superposition Principle, Symmetry Argument Competenze: Physical Reasoning, Mathematical Modeling Fonte: Testo (PDF) — p.5

Problema 12

In the circuit shown, each resistor has equal resistance. Where should a battery be connected so that each resistor has voltage equal to the emf of the battery?

A. Between A and B B. Between B and C C. Between C and D D. Between A and C E. Between B and D F. Between A and D

circuito con resistori uguali nodi A-D

Topic: Circuits Metodi: Kirchhoff’s Laws, Equivalent Circuit Reduction Competenze: Diagrammatic Reasoning, Physical Reasoning Fonte: Testo (PDF) — p.6

Problema 13

Electron in magnetic field hitting screen

An electron with an initial velocity $v$ enters a uniform magnetic field making an angle $α$ with the magnetic field $B$ . It then hits a screen located at a distance $D$ from the entrance point. Assuming $0 < α < 9 0^{\circ}$ , which of the following statements is true?

A. The time to hit the screen increases if $B$ increases. B. The time to hit the screen increases if $B$ decreases. C. The time to hit the screen increases if $α$ increases. D. The time to hit the screen increases if $α$ decreases. E. The time to hit the screen is independent of $α$ .

Topic: Magnetism, Electromagnetism Metodi: Lorentz Force Analysis, Kinematic Equations Competenze: Physical Reasoning, Mathematical Modeling Fonte: Testo (PDF) — p.6

Problema 14

Equivalent resistance of resistor network

In the following circuit, each resistor has resistance $1 Ω$ . Find the equivalent resistance of the circuit between points $A$ and $B$ .

A. $\frac{3}{2} Ω$ B. $\frac{6}{2} Ω$ C. $\frac{4}{3} Ω$ D. $\frac{8}{3} Ω$ E. None of these.

rete resistori a diamante tra punti A e B

Topic: Circuits Metodi: Equivalent Circuit Reduction, Symmetry Argument, Kirchhoff’s Laws Competenze: Diagrammatic Reasoning, Mathematical Modeling Fonte: Testo (PDF) — p.6

Problema 15

Charged particle between parallel wires

Wires A and B lie in the $x$ - $y$ plane. The wires carry equal currents in the same direction. A charged particle passes precisely between the two wires, heading in the $+ \overset{z}{^}$ direction. What happens to the velocity $v$ of the particle over time?

A. The direction and magnitude change. B. The direction changes but the magnitude remains constant. C. The direction remains constant but the magnitude changes. D. The direction and magnitude both remain constant.

Topic: Magnetism, Electromagnetism Metodi: Lorentz Force Analysis, Biot-Savart Law Competenze: Physical Reasoning, Diagrammatic Reasoning Fonte: Testo (PDF) — p.7

Problema 16

Magnet dropped through conducting loop

A permanent magnet is dropped north-first through a conducting loop. Which diagram could represent the current in the loop as a function of time? (A positive value indicates a clockwise current, and a negative value a counterclockwise current, when viewed from above.)

A. [graph option A] B. [graph option B] C. [graph option C] D. [graph option D]

Topic: Electromagnetic Induction Metodi: Faraday’s Law of Induction, Lenz’s Law Competenze: Physical Reasoning, Diagrammatic Reasoning Fonte: Testo (PDF) — p.7

Problema 17

Two charges A and B are held at a fixed distance apart. A material M is placed between the charges, where M is either a block of aluminum or a block of rubber. M does not touch the charges. What happens to the magnitude of the electromagnetic force on charge A as a result of the presence of the material?

A. Decreases substantially for both materials. B. Increases substantially for both materials. C. Remains approximately the same for both materials. D. Increases for one material, decreases for the other. E. Increases for one material, remains same for the other. F. Decreases for one material, remains same for the other.

Topic: Electrostatics, Electromagnetism Metodi: Coulomb’s Law, Physical Modeling Competenze: Physical Reasoning Fonte: Testo (PDF) — p.7

Problema 18

Two permanent magnets A and B are held at a fixed distance apart. A material M is placed between the magnets, where M is either a block of aluminum or a block of rubber. M does not touch the magnets. What happens to the magnitude of the electromagnetic force on magnet A as a result of the presence of the material?

Topic: Magnetism, Electromagnetism Metodi: Physical Modeling, Symmetry Argument Competenze: Physical Reasoning Fonte: Testo (PDF) — p.7

Problema 19

RC circuit — right after switch closed

In the circuit shown, all resistors are identical with resistance $R$ and all capacitors are identical with capacitance $C$ . A battery with voltage $V$ is connected between points A and D. Initially, the switch is opened and all of the capacitors are uncharged. Which of the following options is true right after the switch is closed? Assume the internal resistance of the battery is negligible.

A. Most of the current only flows through all resistors. B. Most of the current only flows through resistors AB, FE, and CD. C. Most of the current only flows through resistors OB, OF, and OD. D. Almost no current flows through any resistor.

circuito RC con interruttore punti A-D

Topic: Circuits Metodi: Kirchhoff’s Laws, Physical Modeling Competenze: Physical Reasoning, Diagrammatic Reasoning Fonte: Testo (PDF) — p.8

Problema 20

RC circuit — long time after switch opened

In the circuit shown, all resistors are identical with resistance $R$ and all capacitors are identical with capacitance $C$ . A battery with voltage $V$ is connected between points A and D. Initially, the switch is opened and all of the capacitors are uncharged. Which of the following options is true a long time after the switch is opened? Assume the internal resistance of the battery is negligible.

A. Most of the current only flows through all capacitors. B. Most of the current only flows through resistors OF, FE, and OD. C. Most of the current only flows through resistors AB, OB, and OD. D. Almost no current flows through any capacitors.

circuito RC con interruttore punti A-D

Topic: Circuits Metodi: Kirchhoff’s Laws, Physical Modeling Competenze: Physical Reasoning, Diagrammatic Reasoning Fonte: Testo (PDF) — p.8

Problema 21

Ideal gas thermodynamic cycle on V-T diagram

One mole of an ideal gas inside an engine goes through a process labelled by $1 \to 2 \to 3 \to 1$ . The Volume vs Temperature graph during this process is shown.

Consider the following statements:

i. Work done by the gas during $2 - 3$ is zero. ii. The pressure during $3 - 1$ is constant. iii. The total work done by the gas during $1 - 2 - 3 - 1$ is zero.

Which of these statements is/are true?

A. i B. ii C. iii D. i and ii E. ii and iii F. i, ii and iii

grafico Volume vs Temperatura ciclo 1-2-3

Topic: Thermodynamics Metodi: Thermodynamic Cycle Analysis, Ideal Gas Law, First Law of Thermodynamics Competenze: Physical Reasoning, Diagrammatic Reasoning Fonte: Testo (PDF) — p.8

Problema 22

How does increasing the pressure affect the freezing temperature $T_{f}$ and boiling temperature $T_{b}$ of most liquids?

A. It decreases $T_{f}$ and $T_{b}$ . B. It increases $T_{f}$ and $T_{b}$ . C. It decreases $T_{f}$ and increases $T_{b}$ . D. It increases $T_{f}$ and decreases $T_{b}$ .

Topic: Thermodynamics Metodi: Physical Modeling, Thermodynamic Cycle Analysis Competenze: Physical Reasoning Fonte: Testo (PDF) — p.9

Problema 23

The gas in the Earth’s atmosphere is more dense closer to the Earth’s surface. Based on this fact, where do you expect the sun to be when you look directly at it? (Assume the sun is not directly above you.)

A. Slightly higher than it appears B. Slightly lower than it appears C. Slightly right of where it appears D. Slightly left of where it appears

Topic: Geometric Optics, Wave Optics Metodi: Snell’s Law, Ray Tracing, Physical Modeling Competenze: Physical Reasoning Fonte: Testo (PDF) — p.9

Problema 24

Distance for magnifying glass

You want to magnify an ant using a magnifying glass with a focal length of 20 cm. How far should you place your magnifying glass from the ant?

A. Less than 20 cm B. Approximately 20 cm C. More than 20 cm D. It depends on other factors

Topic: Geometric Optics Metodi: Thin Lens & Mirror Equation, Ray Tracing Competenze: Physical Reasoning Fonte: Testo (PDF) — p.9

Problema 25

Images in two perpendicular mirrors

Two very large mirrors intersect each other at a right angle. Suppose you are standing at an equal distance from each mirror. How many images of yourself will you see in the mirrors?

A. 2 B. 3 C. 4 D. 5 E. More than 5

Topic: Geometric Optics Metodi: Ray Tracing, Superposition Principle Competenze: Physical Reasoning, Diagrammatic Reasoning Fonte: Testo (PDF) — p.9

Problema 26

The juggler in the photograph rotates the umbrella in such a way that the disk rolling on the umbrella stays in the same place in the room reference frame.

(a) The coefficient of rolling friction $C$ is a property of a rolling object and the surface on which it rolls. It is defined by the equation $F = C N$ , where $F$ is the magnitude of the applied force required for the object to roll at a constant velocity, and $N$ is the magnitude of the normal force of the object relative to the surface. Make appropriate measurements on the photo and some approximations to figure out what is the coefficient of rolling friction of the disk over the umbrella material.

(b) Using the photo can you determine the angular velocity of the rotation of the umbrella?

giocoliere con ombrello e disco rotante

Topic: Rotational Dynamics, Newtonian Mechanics Metodi: Free-Body Diagram, Physical Modeling, Experimental Data Analysis Competenze: Experimental Data Analysis, Estimation & Approximation, Measurement & Instrumentation Fonte: Testo (PDF) — p.10

Problema 27

Birefringent crystal optics

Birefringence is an optical phenomenon where a birefringent crystal refracts light in two different directions, resulting in the separation of a single incident light beam into two polarized rays with different refractive indices ( $n_{o}$ and $n_{e}$ ). These two rays are often called the ordinary ray (o-ray) and the extraordinary ray (e-ray). Each ray travels through the crystal at a different speed.

Part 1: Refraction from a birefringent crystal

When an electromagnetic wave propagates in a medium with a refractive index $n$ , the change in oscillation phase after going through a distance $d$ can be expressed as follows:

$φ = \frac{2 π n d}{λ}$

(a) Suppose we have a birefringent crystal with thickness $d$ . If a beam of light with a wavelength $λ$ propagates through the crystal, express the phase difference between the ordinary and extraordinary ray, $φ$ , in terms of $λ$ , $n_{e}$ , $n_{o}$ , and $d$ .

A beam of linearly polarized polychromatic light, whose electric field vector makes an angle $θ = 4 5^{\circ}$ with the $x$ -axis, falls normally on a crystal. The refractive index for the electric field component in the $x$ -direction is $n_{o} = 2.647$ , and the refractive index for the electric field component in the $y$ -direction is $n_{e} = 2.693$ .

(b) Write down the conditions for the phase difference $φ$ that causes the transmitted light to have:

i. Maximum intensity ii. Minimum intensity

(c) It is known that the wavelengths for three primary colors are:

$λ_{red} = 680 nm$
$λ_{green} = 530 nm$
$λ_{blue} = 470 nm$

Find the minimum thickness of the crystal so that the transmitted light:

i. Appears blue. ii. Appears yellow.

Part 2: Birefringent crystal and polarizers

A polarizer is an optical device that transmits electromagnetic waves with a particular orientation. Now, the birefringent crystal is placed between two linear polarizers. The first (front) polarizer is aligned $θ = 4 5^{\circ}$ clockwise with respect to the $y$ -axis, while the second (rear) one is aligned $θ = 4 5^{\circ}$ counterclockwise with respect to the $y$ -axis. An unpolarized light propagates normally to the first polarizer.

(d) Write down the conditions for the phase difference $φ$ that causes the transmitted light to have:

i. Maximum intensity ii. Minimum intensity

(e) Find the minimum thickness of the crystal so that the transmitted light:

i. Appears blue. ii. Appears yellow.

cristallo birifrangente raggi o-ray e e-ray

cristallo con luce incidente polarizzata angolo theta

polarizzatore lineare asse di polarizzazione

cristallo tra due polarizzatori con luce incidente

vettori campo elettrico intensita massima

vettori campo elettrico intensita minima

Topic: Wave Optics, Electromagnetism Metodi: Interference & Diffraction Analysis, Superposition Principle, Physical Modeling Competenze: Mathematical Modeling, Physical Reasoning, Diagrammatic Reasoning Fonte: Testo (PDF) — p.12

Problema 28

Water rising between capacitor plates

Two uncharged $10 cm \times 10 cm$ metal plates, separated by a distance $d = 1 mm$ , are just touching the surface of water as shown in the figure. The electric permittivity of water is 10 times that of air. If a constant voltage difference of $V = 600 V$ is applied to the plates, determine how much the water rises inside the plates.

lastre metalliche a contatto superficie acqua

Topic: Electrostatics, Fluid Mechanics Metodi: Energy Conservation Method, Physical Modeling, Gauss’s Law Competenze: Mathematical Modeling, Physical Reasoning Fonte: Testo (PDF) — p.17

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CAP 2024 Nazionale — 2024_CAP_Exam_with_solutions.pdf