CAP 2024 Nazionale — 2024_CAP_Exam_with_solutions

Fonte: gare di altri paesi/Canada/all/2024_CAP_Exam_with_solutions_May62025.pdf · Apri PDF · apri PDF p.2 Cluster: Meccanica

Problema 1

Schwarzschild radius of the Sun

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

Excess pressure in a soap bubble

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.3

Problema 3

Atwood-like machine constraint relation

Consider the setup below (a pulley with masses $m_{1}$ , $m_{2}$ , $m_{3}$ connected by strings over a movable pulley).

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. $a_{1} = - (a_{2} + a_{3})$

Puleggia mobile 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

Rolling ring 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, Conservation of Momentum Competenze: Physical Reasoning Fonte: Testo (PDF) — p.3

Problema 5

Force of rain on an 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 Metodi: Impulse-Momentum Theorem, Physical Modeling Competenze: Estimation & Approximation, Physical Reasoning Fonte: Testo (PDF) — p.3

Problema 6

Hanging towel friction limit

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

Projectile range 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 comparison

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 Competenze: Physical Reasoning Fonte: Testo (PDF) — p.5

Problema 9

Two springs in series: SHM speed scaling

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, Newtonian Mechanics Metodi: Simple Harmonic Motion Analysis, Energy Conservation Method Competenze: Physical Reasoning 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 the 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: Superposition Principle, Coulomb’s Law, Symmetry Argument Competenze: Physical Reasoning, Diagrammatic Reasoning Fonte: Testo (PDF) — p.6

Problema 12

Battery placement for equal resistor voltage

In the circuit below 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 ABCD

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

Problema 13

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

Problema 14

Equivalent resistance of symmetric circuit

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.

Circuito simmetrico resistenze tra A e B

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

Problema 15

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

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.)

[Four graphs A–D showing different current vs. time profiles are provided in the exam.]

Topic: Electromagnetic Induction Metodi: Faraday’s Law of Induction, Lenz’s Law Competenze: Diagrammatic Reasoning, Physical 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 Competenze: Physical Reasoning Fonte: Testo (PDF) — p.8

Problema 19

RC circuit just after switch is closed

In the circuit below, 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 switch batteria V

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

Problema 20

RC circuit long after switch is closed

In the circuit below, 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 switch batteria V

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

Problema 21

Ideal gas V-T cycle: work and pressure

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 below.

Consider the following statements:

i. Work done by the gas during $2 \to 3$ is zero.

ii. The pressure during $3 \to 1$ is constant.

iii. The total work done by the gas during $1 \to 2 \to 3 \to 1$ 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 V-T ciclo termico 1-2-3-1

Topic: Thermodynamics Metodi: Ideal Gas Law, Thermodynamic Cycle Analysis, First Law of Thermodynamics Competenze: Diagrammatic Reasoning, Physical Reasoning Fonte: Testo (PDF) — p.9

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 isn’t 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: Wave Optics, Geometric Optics Metodi: Snell’s Law, Physical Modeling Competenze: Physical Reasoning Fonte: Testo (PDF) — p.9

Problema 24

Magnifying glass: object placement

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

Two very large mirrors intersect each other at right angles. 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?

Foto giocoliere con ombrello e disco rotante

Ingrandimento disco su ombrello rotante

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

Problema 27

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:

$φ = \frac{2 π n d}{λ}$

a. Suppose we have a birefringent crystal with thickness $d$ . If a beam of light with a wavelength of $λ$ 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 colours 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. Our 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 assi ne no e campo E0

Polarizzatore con asse polarizzazione θ

Sandwitch polarizzatore cristallo polarizzatore

Diagramma vettore campo E polarizzazione massima

Diagramma vettore campo E polarizzazione minima

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

Problema 28

Water rising between charged plates (electrostriction/capacitor energy)

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 verticali in acqua tensione V

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

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CAP 2024 Nazionale — 2024_CAP_Exam_with_solutions_May62025.pdf