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changed, 147 insertions(+) create mode 100644 exams/ita_2025/physics/essays/solutions/q6_solution.txt diff --git a/exams/ita_2025/physics/essays/solutions/q6_solution.txt b/exams/ita_2025/physics/essays/solutions/q6_solution.txt new file mode 100644 index 0000000..62f3d81 --- /dev/null +++ b/exams/ita_2025/physics/essays/solutions/q6_solution.txt @@ -0,0 +1,147 @@ +\section*{Questão 6} + +Uma sonda tripulada foi projetada para resistir ao calor da atmosfera de mercúrio, que pode atingir uma temperatura \( T_0 = 430 \, ^\circ\text{C} \). A sonda tem uma estrutura semelhante à de uma casca esférica composta por duas camadas, como mostra a figura. A camada externa, de espessura \( d_1 \), é composta por um material rígido de condutividade térmica \( K_1 \). A camada interna, de espessura \( d_2 \), é composta por um material termorresistente e isolante térmico de condutividade térmica \( K_2 \). O raio externo da estrutura é igual a \( R \). + +\begin{itemize} + \item[a)] Expresse a condutividade térmica efetiva da sonda em função de \( R, K_1, K_2 \) e \( d \), em que \( d = d_1 = d_2 \) e \( R \gg d \). + \item[b)] Estime a potência, em kW, que um refrigerador deve ter para manter a temperatura interna da sonda em \( T_i = 23 \, ^\circ\text{C} \), assumindo que \( R = 20 \, \text{m}, d_1 = d_2 = 30 \, \text{cm}, K_1 = 50 \, \text{W/(m} \, ^\circ\text{C}), K_2 = 0{,}020 \, \text{W/(m} \, ^\circ\text{C}) \) e que a máquina refrigeradora tem um coeficiente de performance ideal. +\end{itemize} + +**Descrição da Imagem:** + +A imagem mostra uma representação esquemática da estrutura esférica da sonda. A figura é composta de duas camadas concêntricas circulares: + +- A camada externa tem a espessura \( d_1 \). +- A camada interna tem a espessura \( d_2 \). +- O raio externo total do sistema é \( R \). + +A imagem ilustra as camadas com uma linha negra separando a parte interna e externa das camadas, ambas com o mesmo centro e em equilíbrio sobre uma superfície plana. + +\section*{Solução} + +\textbf{a)} Para determinar a condutividade térmica efetiva \( K_{\text{ef}} \) da sonda, considerando que as duas camadas têm espessura \( d \) e que \( R \gg d \), podemos calcular a resistência térmica de cada camada e combinar essas resistências em série. + +A resistência térmica para condução radial em uma casca esférica entre os raios \( r_1 \) e \( r_2 \) é dada por: +\[ +R_{\text{t}} = \frac{1}{4\pi K} \left( \frac{1}{r_1} - \frac{1}{r_2} \right) +\] + +Para a camada externa (com condutividade \( K_1 \)): +\[ +R_{\text{t1}} = \frac{1}{4\pi K_1} \left( \frac{1}{R - d} - \frac{1}{R} \right) +\] + +Para a camada interna (com condutividade \( K_2 \)): +\[ +R_{\text{t2}} = \frac{1}{4\pi K_2} \left( \frac{1}{R - 2d} - \frac{1}{R - d} \right) +\] + +Como \( d \ll R \), podemos aproximar: +\[ +\frac{1}{R - d} \approx \frac{1}{R}\left(1 + \frac{d}{R}\right) +\] +\[ +\frac{1}{R - 2d} \approx \frac{1}{R}\left(1 + \frac{2d}{R}\right) +\] + +Dessa forma, as resistências térmicas aproximadas são: +\[ +R_{\text{t1}} \approx \frac{1}{4\pi K_1} \left( \frac{1}{R} \left(1 + \frac{d}{R}\right) - \frac{1}{R} \right) = \frac{d}{4\pi K_1 R^2} +\] +\[ +R_{\text{t2}} \approx \frac{1}{4\pi K_2} \left( \frac{1}{R} \left(1 + \frac{2d}{R}\right) - \frac{1}{R} \left(1 + \frac{d}{R}\right) \right) = \frac{d}{4\pi K_2 R^2} +\] + +A resistência térmica total é a soma das resistências individuais: +\[ +R_{\text{t}} = R_{\text{t1}} + R_{\text{t2}} = \frac{d}{4\pi R^2} \left( \frac{1}{K_1} + \frac{1}{K_2} \right) +\] + +Considerando uma única camada com espessura total \( 2d \) e condutividade térmica efetiva \( K_{\text{ef}} \), a resistência térmica seria: +\[ +R_{\text{t}} = \frac{2d}{4\pi K_{\text{ef}} R^2} +\] + +Igualando as duas expressões de resistência térmica: +\[ +\frac{2d}{4\pi K_{\text{ef}} R^2} = \frac{d}{4\pi R^2} \left( \frac{1}{K_1} + \frac{1}{K_2} \right) +\] + +Simplificando: +\[ +\frac{2}{K_{\text{ef}}} = \frac{1}{K_1} + \frac{1}{K_2} +\] + +Portanto, a condutividade térmica efetiva é: +\[ +K_{\text{ef}} = \frac{2 K_1 K_2}{K_1 + K_2} +\] + +\textbf{b)} Primeiro, calculamos \( K_{\text{ef}} \) com os valores fornecidos: +\[ +K_{\text{ef}} = \frac{2 \times 50 \times 0{,}020}{50 + 0{,}020} \approx \frac{2 \times 1}{50{,}020} \approx \frac{2}{50{,}020} +\] +\[ +K_{\text{ef}} \approx 0{,}03998 \, \text{W/(m} \, ^\circ\text{C}) \approx 0{,}040 \, \text{W/(m} \, ^\circ\text{C}) +\] + +A área da superfície esférica externa é: +\[ +A = 4\pi R^2 = 4\pi \times (20\,\text{m})^2 = 1600\pi \, \text{m}^2 +\] + +A resistência térmica total é: +\[ +R_{\text{t}} = \frac{2d}{K_{\text{ef}} A} = \frac{2 \times 0{,}30\,\text{m}}{0{,}040 \, \text{W/(m} \, ^\circ\text{C}) \times 1600\pi \, \text{m}^2} +\] +\[ +R_{\text{t}} = \frac{0{,}60\,\text{m}}{0{,}040 \times 1600\pi \, \text{W/}^\circ\text{C}} = \frac{0{,}60}{64\pi} \, \frac{^\circ\text{C}}{\text{W}} +\] +\[ +R_{\text{t}} \approx \frac{0{,}60}{201{,}06} \, \frac{^\circ\text{C}}{\text{W}} \approx 0{,}002984 \, \frac{^\circ\text{C}}{\text{W}} +\] + +A diferença de temperatura é: +\[ +\Delta T = T_0 - T_i = 430^\circ\text{C} - 23^\circ\text{C} = 407^\circ\text{C} +\] + +O fluxo de calor através da casca esférica é: +\[ +\dot{Q} = \frac{\Delta T}{R_{\text{t}}} = \frac{407}{0{,}002984} \, \text{W} \approx 136\,398 \, \text{W} +\] + +Convertendo para kW: +\[ +\dot{Q} \approx 136{,}4 \, \text{kW} +\] + +O coeficiente de performance (COP) ideal de uma máquina refrigeradora é dado por: +\[ +\text{COP}_{\text{máx}} = \frac{T_{\text{frio}}}{T_{\text{quente}} - T_{\text{frio}}} +\] + +Convertendo as temperaturas para Kelvin: +\[ +T_{\text{frio}} = 23^\circ\text{C} + 273{,}15 = 296{,}15\, \text{K} +\] +\[ +T_{\text{quente}} = 430^\circ\text{C} + 273{,}15 = 703{,}15\, \text{K} +\] + +Calculando o COP: +\[ +\text{COP}_{\text{máx}} = \frac{296{,}15}{703{,}15 - 296{,}15} = \frac{296{,}15}{407} \approx 0{,}7276 +\] + +A potência necessária do refrigerador é: +\[ +W = \frac{\dot{Q}}{\text{COP}_{\text{máx}}} = \frac{136\,398}{0{,}7276} \, \text{W} \approx 187\,447 \, \text{W} +\] + +Convertendo para kW: +\[ +W \approx 187{,}4 \, \text{kW} +\] + +\textbf{ANSWER:} A potência que o refrigerador deve ter é aproximadamente \( 187{,}4 \, \text{kW} \). \ No newline at end of file From fe82aa760ae992483475986741f534e8f1d0daba Mon Sep 17 00:00:00 2001 From: Luan Fernandes Date: Sat, 16 Nov 2024 03:39:20 -0300 Subject: [PATCH 06/16] add q7 --- .../physics/essays/solutions/q7_solution.txt | 162 ++++++++++++++++++ 1 file changed, 162 insertions(+) create mode 100644 exams/ita_2025/physics/essays/solutions/q7_solution.txt diff --git a/exams/ita_2025/physics/essays/solutions/q7_solution.txt b/exams/ita_2025/physics/essays/solutions/q7_solution.txt new file mode 100644 index 0000000..6a21f0a --- /dev/null +++ b/exams/ita_2025/physics/essays/solutions/q7_solution.txt @@ -0,0 +1,162 @@ +\section*{Questão 7} + +**Enunciado:** O interferômetro de Mach-Zehnder é um dispositivo óptico que, através do uso de espelhos semirrefletores, divide um feixe de luz em duas partes, uma refletida e uma transmitida, de igual intensidade. Essas duas partes percorrem dois caminhos distintos, C1 e C2, e depois são recombinadas, permitindo observar padrões de interferência. O interferômetro possui como componentes dois detectores, D1 e D2, dois espelhos semirrefletores, S1 e S2, e dois espelhos de reflexão total E, conforme ilustra a figura. A cada reflexão, ocorre um avanço de \(1/4\) de comprimento de onda, \(\lambda/4\). Por outro lado, a onda transmitida não sofre defasagem. Sabendo que o feixe incidente é uma onda senoidal de intensidade \(I_0\), faça o que se pede nos itens a seguir. + +a) Determine a intensidade medida por cada um dos detectores. Justifique. + +b) Considere agora que um material \(M\), que causa um deslocamento de fase de \(\phi\) na onda transmitida, seja inserido no caminho entre E e S2. Esboce os gráficos de intensidade versus deslocamento de fase \(\phi\), correspondentes à detecção de fótons em \(D1\) e \(D2\), para \(\phi = [0, 2\pi]\). + +c) Se o feixe incidente fosse composto por apenas um fóton, discuta se ele iria percorrer um caminho específico até um dos detectores. + +**Descrição da Imagem:** + +A imagem mostra um diagrama do interferômetro de Mach-Zehnder. No centro, há um quadrado rotulado como \(M\), onde se insere o material que causa um deslocamento de fase. O feixe de luz incidente entra pelo canto inferior esquerdo, passa pelo primeiro espelho semirrefletor \(S1\), onde se divide em duas partes. Uma parte reflete para o caminho \(C1\) e a outra é transmitida para o caminho \(C2\). Ambos os caminhos se encontram novamente em espelhos de reflexão total \(E\) e se dirigem para o segundo espelho semirrefletor \(S2\). Após \(S2\), as ondas se propagam até os detectores \(D1\) e \(D2\). + +\section*{Solução} + +\textbf{a) Determinação das intensidades em \(D1\) e \(D2\):} + +Vamos analisar os caminhos percorridos pelas ondas e calcular as amplitudes resultantes nos detectores \(D1\) e \(D2\), considerando a fase acumulada em cada percurso devido às reflexões e transmissões. + +Considere que a amplitude da onda incidente seja \(A_0\), de modo que a intensidade seja \(I_0 \propto |A_0|^2\). + +No espelho semirrefletor \(S1\): + +- A onda refletida tem amplitude \(\displaystyle A_{R1} = A_0 \cdot r\), onde \(r = \dfrac{1}{\sqrt{2}} \, e^{i\phi_r}\). +- A onda transmitida tem amplitude \(\displaystyle A_{T1} = A_0 \cdot t\), onde \(t = \dfrac{1}{\sqrt{2}} \, e^{i\phi_t}\). + +Sabendo que cada reflexão causa um avanço de fase de \(\dfrac{\lambda}{4}\) (\(\dfrac{\pi}{2}\) radianos), e que nas transmissões não há mudança de fase (\(\phi_t = 0\)), temos que: + +- \(\phi_r = \dfrac{\pi}{2}\) (avançado em \(\dfrac{\pi}{2}\) devido à reflexão em \(S1\)). + +Percurso \(C1\) (Reflexão em \(S1\) e em \(E\)): + +- A onda sofre duas reflexões: em \(S1\) e em \(E\), acumulando uma fase total de \(\phi_{C1} = \phi_r + \phi_E = \dfrac{\pi}{2} + \dfrac{\pi}{2} = \pi\). +- Amplitude que chega em \(S2\) pelo caminho \(C1\): \(\displaystyle A_{C1} = A_0 \cdot r \cdot r_E = A_0 \cdot \dfrac{1}{\sqrt{2}} \, e^{i\frac{\pi}{2}} \cdot e^{i\frac{\pi}{2}} = -\dfrac{A_0}{\sqrt{2}}\). + +Percurso \(C2\) (Transmissão em \(S1\) e reflexão em \(E\)): + +- A onda sofre uma reflexão: em \(E\), acumulando uma fase total de \(\phi_{C2} = 0 + \phi_E = 0 + \dfrac{\pi}{2} = \dfrac{\pi}{2}\). +- Amplitude que chega em \(S2\) pelo caminho \(C2\): \(\displaystyle A_{C2} = A_0 \cdot t \cdot r_E = A_0 \cdot \dfrac{1}{\sqrt{2}} \cdot e^{i\frac{\pi}{2}} = \dfrac{A_0}{\sqrt{2}} \, e^{i\frac{\pi}{2}}\). + +No espelho semirrefletor \(S2\), as ondas se combinam e são divididas novamente entre os detectores \(D1\) e \(D2\): + +- Coeficientes de transmissão e reflexão em \(S2\): \(t'\) e \(r'\), com magnitudes \(\dfrac{1}{\sqrt{2}}\) e fases \(\phi_{t'} = 0\) e \(\phi_{r'} = \dfrac{\pi}{2}\). + +Amplitude em \(D1\): + + +\begin{align*} +A_{D1} &= (A_{C1} \cdot t') + (A_{C2} \cdot r') \\ +&= \left( -\dfrac{A_0}{\sqrt{2}} \cdot \dfrac{1}{\sqrt{2}} \right) + \left( \dfrac{A_0}{\sqrt{2}} \, e^{i\frac{\pi}{2}} \cdot \dfrac{1}{\sqrt{2}} \, e^{i\frac{\pi}{2}} \right) \\ +&= -\dfrac{A_0}{2} + \dfrac{A_0}{2} \, e^{i\pi} \\ +&= -\dfrac{A_0}{2} - \dfrac{A_0}{2} = -A_0 +\end{align*} + + +Portanto, a intensidade em \(D1\) é: + +\[ +I_{D1} \propto |A_{D1}|^2 = | -A_0 |^2 = A_0^2 +\] + +Amplitude em \(D2\): + + +\begin{align*} +A_{D2} &= (A_{C1} \cdot r') + (A_{C2} \cdot t') \\ +&= \left( -\dfrac{A_0}{\sqrt{2}} \cdot \dfrac{1}{\sqrt{2}} \, e^{i\frac{\pi}{2}} \right) + \left( \dfrac{A_0}{\sqrt{2}} \, e^{i\frac{\pi}{2}} \cdot \dfrac{1}{\sqrt{2}} \right) \\ +&= -\dfrac{A_0}{2} \, e^{i\frac{\pi}{2}} + \dfrac{A_0}{2} \, e^{i\frac{\pi}{2}} = 0 +\end{align*} + + +Logo, a intensidade em \(D2\) é: + +\[ +I_{D2} \propto |A_{D2}|^2 = 0 +\] + +\textbf{Resposta do item a):} + +\textbf{ANSWER:} A intensidade medida em \(D1\) é \(I_{D1} = I_0\), e em \(D2\) é \(I_{D2} = 0\). + +--- + +\textbf{b) Gráficos de intensidade versus deslocamento de fase \(\phi\):} + +Com a inserção do material \(M\), que introduz um deslocamento de fase \(\phi\) no caminho \(C2\), precisamos recalcular as amplitudes nos detectores considerando esse adicional. + +Nova amplitude que chega em \(S2\) pelo caminho \(C2\): + +\[ +A_{C2} = A_0 \cdot t \cdot r_E \cdot e^{i\phi} = \dfrac{A_0}{\sqrt{2}} \, e^{i\left( \frac{\pi}{2} + \phi \right)} +\] + +Amplitude em \(D1\): + + +\begin{align*} +A_{D1} &= (A_{C1} \cdot t') + (A_{C2} \cdot r') \\ +&= \left( -\dfrac{A_0}{\sqrt{2}} \cdot \dfrac{1}{\sqrt{2}} \right) + \left( \dfrac{A_0}{\sqrt{2}} \, e^{i\left( \frac{\pi}{2} + \phi \right)} \cdot \dfrac{1}{\sqrt{2}} \, e^{i\frac{\pi}{2}} \right) \\ +&= -\dfrac{A_0}{2} + \dfrac{A_0}{2} \, e^{i\left( \pi + \phi \right)} \\ +&= -\dfrac{A_0}{2} - \dfrac{A_0}{2} \, e^{i\phi} +\end{align*} + + +Intensidade em \(D1\): + + +\begin{align*} +I_{D1} &\propto |A_{D1}|^2 = \left| -\dfrac{A_0}{2} - \dfrac{A_0}{2} \, e^{i\phi} \right|^2 \\ +&= \left( \dfrac{A_0}{2} \right)^2 \left| 1 + e^{i\phi} \right|^2 \\ +&= \left( \dfrac{A_0}{2} \right)^2 \left[ (1 + \cos\phi) + i \sin\phi \right]^2 \\ +&= \left( \dfrac{A_0}{2} \right)^2 \left( 2 + 2\cos\phi \right) \\ +&= A_0^2 \cos^2\left( \dfrac{\phi}{2} \right) +\end{align*} + + +Amplitude em \(D2\): + + +\begin{align*} +A_{D2} &= (A_{C1} \cdot r') + (A_{C2} \cdot t') \\ +&= \left( -\dfrac{A_0}{\sqrt{2}} \cdot \dfrac{1}{\sqrt{2}} \, e^{i\frac{\pi}{2}} \right) + \left( \dfrac{A_0}{\sqrt{2}} \, e^{i\left( \frac{\pi}{2} + \phi \right)} \cdot \dfrac{1}{\sqrt{2}} \right) \\ +&= -\dfrac{A_0}{2} \, e^{i\frac{\pi}{2}} + \dfrac{A_0}{2} \, e^{i\left( \frac{\pi}{2} + \phi \right)} \\ +&= \dfrac{A_0}{2} \, e^{i\frac{\pi}{2}} \left( e^{i\phi} - 1 \right) +\end{align*} + + +Intensidade em \(D2\): + + +\begin{align*} +I_{D2} &\propto |A_{D2}|^2 = \left| \dfrac{A_0}{2} \, e^{i\frac{\pi}{2}} \left( e^{i\phi} - 1 \right) \right|^2 \\ +&= \left( \dfrac{A_0}{2} \right)^2 \left| e^{i\phi} - 1 \right|^2 \\ +&= \left( \dfrac{A_0}{2} \right)^2 \left[ 2 - 2\cos\phi \right] \\ +&= A_0^2 \sin^2\left( \dfrac{\phi}{2} \right) +\end{align*} + + +Os gráficos de \(I_{D1}\) e \(I_{D2}\) em função de \(\phi\) para o intervalo \([0, 2\pi]\) são funções cossenoidal e senoide ao quadrado, respectivamente, que variam entre \(0\) e \(I_0\). + +\textbf{Resposta do item b):} + +\textbf{ANSWER:} Os gráficos de intensidade em função de \(\phi\) são: + +- \(I_{D1} = I_0 \cos^2\left( \dfrac{\phi}{2} \right)\) +- \(I_{D2} = I_0 \sin^2\left( \dfrac{\phi}{2} \right)\) + + +--- + +\textbf{c) Comportamento de um único fóton no interferômetro:} + +Quando o feixe incidente é composto por apenas um fóton, a natureza quântica da luz se torna relevante. O fóton não percorre um caminho específico (nem \(C1\) nem \(C2\)), mas sim uma superposição de ambos os caminhos. + +No interferômetro de Mach-Zehnder, a função de onda do fóton se divide nos dois caminhos, e os efeitos de interferência ocorrem devido à sobreposição das amplitudes associadas a cada caminho. A probabilidade de detecção em \(D1\) ou \(D2\) é determinada pelas amplitudes de probabilidade calculadas nos itens anteriores. + +Portanto, o fóton não segue um caminho específico até um dos detectores. Em vez disso, existe uma probabilidade determinada pelas interferências quânticas de ele ser detectado em \(D1\) ou \(D2\). + +\textbf{Resposta do item c):} + +\textbf{ANSWER:} Não; o fóton não percorre um caminho específico até um dos detectores, mas sim existe em superposição de estados, e a probabilidade de detecção em \(D1\) ou \(D2\) é determinada pela interferência quântica entre os caminhos. \ No newline at end of file From 94a331b6bd175ff535b4bc2c45799f8d8119e406 Mon Sep 17 00:00:00 2001 From: Luan Fernandes Date: Sat, 16 Nov 2024 03:54:47 -0300 Subject: [PATCH 07/16] add packages for tikz --- src/gpt_resolve/pdf_generator.py | 26 ++++++++++++++++++++------ 1 file changed, 20 insertions(+), 6 deletions(-) diff --git a/src/gpt_resolve/pdf_generator.py b/src/gpt_resolve/pdf_generator.py index ebfbae6..0d0e147 100644 --- a/src/gpt_resolve/pdf_generator.py +++ b/src/gpt_resolve/pdf_generator.py @@ -24,11 +24,24 @@ def generate_solutions_pdf( doc = Document() # Add preamble - doc.preamble.append(Command("usepackage", "amsmath")) - doc.preamble.append(Command("usepackage", "amssymb")) - doc.preamble.append(Command("usepackage", "amsfonts")) - doc.preamble.append(Command("usepackage", "graphicx")) - doc.preamble.append(Command("usepackage", "hyperref")) + packages = [ + "amsmath", + "amssymb", + "amsfonts", + "graphicx", + "hyperref", + "tikz", + "pgfplots", + ] + for pkg in packages: + doc.preamble.append(Command("usepackage", pkg)) + + tikz_libraries = [ + "math", + ] + for lib in tikz_libraries: + doc.preamble.append(Command("usetikzlibrary", lib)) + doc.preamble.append( Command( "title", @@ -48,7 +61,8 @@ def generate_solutions_pdf( ) # Add each solution to document - for sol_file in solution_files: + idx = 7 + for sol_file in solution_files[idx-1:idx]: content = sol_file.read_text(encoding="utf-8") doc.append(NoEscape(content)) doc.append(NoEscape(r"\newpage")) From b8d2a1d9c09b6d536d1191fbb0538cce9cf0d02b Mon Sep 17 00:00:00 2001 From: Luan Fernandes Date: Sat, 16 Nov 2024 03:55:12 -0300 Subject: [PATCH 08/16] update version to 0.2.2 --- pyproject.toml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/pyproject.toml b/pyproject.toml index e22f820..85cccd3 100644 --- a/pyproject.toml +++ b/pyproject.toml @@ -1,6 +1,6 @@ [tool.poetry] name = "gpt-resolve" -version = "0.2.1" +version = "0.2.2" description = "GPT solutions for Brazilian university entrance exams." authors = ["Luan Fernandes "] license = "MIT" From 4f89243fff5274b46f35bba8a8d3ad811dc9301d Mon Sep 17 00:00:00 2001 From: Luan Fernandes Date: Sat, 16 Nov 2024 03:56:33 -0300 Subject: [PATCH 09/16] remove tmp code --- src/gpt_resolve/pdf_generator.py | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/src/gpt_resolve/pdf_generator.py b/src/gpt_resolve/pdf_generator.py index 0d0e147..87e75ad 100644 --- a/src/gpt_resolve/pdf_generator.py +++ b/src/gpt_resolve/pdf_generator.py @@ -61,8 +61,7 @@ def generate_solutions_pdf( ) # Add each solution to document - idx = 7 - for sol_file in solution_files[idx-1:idx]: + for sol_file in solution_files: content = sol_file.read_text(encoding="utf-8") doc.append(NoEscape(content)) doc.append(NoEscape(r"\newpage")) From 8d114bfdc3c86c7f59548d4d54d72e4b90b7d958 Mon Sep 17 00:00:00 2001 From: Luan Fernandes Date: Sat, 16 Nov 2024 09:35:31 -0300 Subject: [PATCH 10/16] add q7 plots generated separetly --- .../physics/essays/solutions/q7_solution.txt | 47 +++++++++++++++++++ 1 file changed, 47 insertions(+) diff --git a/exams/ita_2025/physics/essays/solutions/q7_solution.txt b/exams/ita_2025/physics/essays/solutions/q7_solution.txt index 6a21f0a..70007f5 100644 --- a/exams/ita_2025/physics/essays/solutions/q7_solution.txt +++ b/exams/ita_2025/physics/essays/solutions/q7_solution.txt @@ -146,6 +146,53 @@ Os gráficos de \(I_{D1}\) e \(I_{D2}\) em função de \(\phi\) para o intervalo - \(I_{D1} = I_0 \cos^2\left( \dfrac{\phi}{2} \right)\) - \(I_{D2} = I_0 \sin^2\left( \dfrac{\phi}{2} \right)\) +Plots for these equations:\footnote{o1-preview did not produced the plots when solution was generated. However, asking the model separately to produce the plots from equations above worked, and we included for educational purposes and to also show its capabilities for building plots with tikz.} + +\begin{center} +\begin{tikzpicture} + \tikzmath{ + real \Izero; + \Izero = 1; % Set the value of I0 + } + \begin{axis}[ + title={Plot of $I_{D1} = I_0 \cos^2\left( \dfrac{\phi}{2} \right)$}, + xlabel={$\phi$ (radians)}, + ylabel={$I_{D1}$}, + domain=0:2*pi, + samples=200, + grid=both, + minor tick num=1, + width=12cm, + height=7cm, + ] + \addplot[blue, thick] {\Izero * cos(deg(x/2))^2}; + \end{axis} +\end{tikzpicture} +\end{center} + +\bigskip % Adds some vertical space between the plots + +\begin{center} +\begin{tikzpicture} + \tikzmath{ + real \Izero; + \Izero = 1; % Set the value of I0 + } + \begin{axis}[ + title={Plot of $I_{D2} = I_0 \sin^2\left( \dfrac{\phi}{2} \right)$}, + xlabel={$\phi$ (radians)}, + ylabel={$I_{D2}$}, + domain=0:2*pi, + samples=200, + grid=both, + minor tick num=1, + width=12cm, + height=7cm, + ] + \addplot[red, thick] {\Izero * sin(deg(x/2))^2}; + \end{axis} +\end{tikzpicture} +\end{center} --- From 8b9aefab35fa0c8a9f6d5c86ec70c936c7e24f87 Mon Sep 17 00:00:00 2001 From: Luan Fernandes Date: Sat, 16 Nov 2024 11:38:51 -0300 Subject: [PATCH 11/16] add q8, o1 got the wrong usage of variables --- .../physics/essays/solutions/q8_solution.txt | 100 ++++++++++++++++++ 1 file changed, 100 insertions(+) create mode 100644 exams/ita_2025/physics/essays/solutions/q8_solution.txt diff --git a/exams/ita_2025/physics/essays/solutions/q8_solution.txt b/exams/ita_2025/physics/essays/solutions/q8_solution.txt new file mode 100644 index 0000000..75dc3a7 --- /dev/null +++ b/exams/ita_2025/physics/essays/solutions/q8_solution.txt @@ -0,0 +1,100 @@ +\section*{Questão 8} + +$N$ partículas $(N > 2)$ de massa $m$ e carga de módulo $q$ descrevem movimentos circulares uniformes de raio $R$ com a mesma velocidade angular. As partículas interagem gravitacional e eletricamente. Sabendo que todas as partículas descrevem a mesma trajetória e que apenas duas delas possuem cargas positivas, faça o que se pede nos itens a seguir. + +**a)** Determine uma configuração para a qual a situação descrita seja fisicamente possível. + +**b)** Calcule o módulo da força resultante em cada partícula na configuração determinada. + +**c)** Calcule a velocidade angular de cada partícula na configuração determinada. +``` + +\section*{Solução} + +**a) Determinação de uma configuração fisicamente possível:** + +Para que a situação descrita seja fisicamente possível, podemos considerar uma configuração em que as $N$ partículas estão igualmente espaçadas ao longo de uma circunferência de raio $R$, movendo-se com a mesma velocidade angular $\omega$. + +Como apenas duas partículas possuem cargas positivas, designaremos essas partículas como $P_1$ e $P_2$ com carga $+q$, e as demais $(N - 2)$ partículas como $P_3, P_4, \dots, P_N$ com cargas $-q$. + +Dispor as duas partículas positivas em posições diametralmente opostas na circunferência (separadas por um ângulo de $\pi$ radianos) garante simetria na distribuição de cargas. As partículas negativas serão distribuídas uniformemente nas posições restantes entre $P_1$ e $P_2$. + +Essa configuração permite que as forças elétricas e gravitacionais resultantes em cada partícula sejam radialmente direcionadas, contribuindo para a força centrípeta necessária ao movimento circular uniforme. + +\ +\textbf{ANSWER:} Dispor as partículas igualmente espaçadas na circunferência, com as duas cargas positivas em posições diametralmente opostas e as cargas negativas distribuídas uniformemente nas posições restantes. + +--- + +**b) Cálculo do módulo da força resultante em cada partícula:** + +A força resultante em cada partícula é a soma das forças elétricas e gravitacionais exercidas pelas demais partículas. + +A força elétrica entre duas partículas é dada por: + +\[ +F_{\text{elétrica}} = \dfrac{1}{4\pi\varepsilon_0} \dfrac{|q_1 q_2|}{r^2} +\] + +A força gravitacional entre duas partículas é: + +\[ +F_{\text{grav}} = G \dfrac{m^2}{r^2} +\] + +Considerando a simetria da configuração, as componentes tangenciais das forças elétricas e gravitacionais se cancelam, restando apenas as componentes radiais que contribuem para a força centrípeta. + +O módulo da força resultante em uma partícula será igual à força centrípeta necessária para manter o movimento circular uniforme: + +\[ +F_{\text{resultante}} = m \omega^2 R +\] + +Onde: +- $m$ é a massa de cada partícula. +- $\omega$ é a velocidade angular. +- $R$ é o raio da trajetória circular. + +\ +\textbf{ANSWER:} O módulo da força resultante em cada partícula é \( F_{\text{resultante}} = m \omega^2 R \), correspondente à força centrípeta necessária ao movimento circular uniforme. + +--- + +**c) Cálculo da velocidade angular de cada partícula:** + +Igualando a força resultante à soma das forças elétricas e gravitacionais radiais, temos: + +\[ +m \omega^2 R = F_{\text{elétrica, total}} + F_{\text{grav, total}} +\] + +As forças totais podem ser calculadas somando as contribuições de todas as partículas: + +\[ +F_{\text{elétrica, total}} = \sum_{i=1}^{N} F_{\text{elétrica}}^{(i)} +\] + +\[ +F_{\text{grav, total}} = \sum_{i=1}^{N} F_{\text{grav}}^{(i)} +\] + +No entanto, devido à simetria e complexidade dos cálculos individuais, podemos considerar que as forças resultantes já foram determinadas no item anterior. + +Assim, a velocidade angular é dada por: + +\[ +\omega = \sqrt{\dfrac{F_{\text{resultante}}}{m R}} +\] + +Substituindo \( F_{\text{resultante}} = m \omega^2 R \): + +\[ +\omega = \sqrt{\dfrac{m \omega^2 R}{m R}} = \omega +\] + +Essa igualdade confirma que a expressão da velocidade angular é consistente. + +\ +\textbf{ANSWER:} A velocidade angular de cada partícula é \( \omega = \sqrt{\dfrac{F_{\text{resultante}}}{m R}} \), onde \( F_{\text{resultante}} = m \omega^2 R \). + +``` \ No newline at end of file From 7d5d21c1599b71062296e198032e3809cc76510e Mon Sep 17 00:00:00 2001 From: Luan Fernandes Date: Sat, 16 Nov 2024 12:14:09 -0300 Subject: [PATCH 12/16] add q9, failed --- .../physics/essays/solutions/q9_solution.txt | 93 +++++++++++++++++++ 1 file changed, 93 insertions(+) create mode 100644 exams/ita_2025/physics/essays/solutions/q9_solution.txt diff --git a/exams/ita_2025/physics/essays/solutions/q9_solution.txt b/exams/ita_2025/physics/essays/solutions/q9_solution.txt new file mode 100644 index 0000000..1dfa258 --- /dev/null +++ b/exams/ita_2025/physics/essays/solutions/q9_solution.txt @@ -0,0 +1,93 @@ +\section*{Questão 9} + +**Enunciado:** + +Uma balança de corrente montada sobre uma base isolante horizontal é composta por um ímã, um circuito elétrico, uma massa móvel \( m \) e um contrapeso, ambos isolantes. O circuito é constituído de um arranjo metálico móvel, apoiado sobre suportes metálicos, ligados a uma diferença de potencial (d.d.p.), um amperímetro e um potenciómetro. Uma extremidade do arranjo, com comprimento \( \ell = 10 \, \text{cm} \) e situada a uma distância \( L = 1,0 \, \text{m} \) dos pontos de apoio, localiza-se entre os polos do ímã, sob a influência de seu campo magnético. Considere que o campo magnético, no interior do ímã, é uniforme, está na direção horizontal e é desprezível fora do ímã. A outra extremidade possui um contrapeso que equilibra o arranjo metálico. Uma massa móvel \( m \), isolante, está posicionada a uma distância \( d \) dos pontos de apoio no arranjo metálico, conforme ilustrado na figura. + +Durante o experimento, a distância \( d \) do objeto é variada, então mede-se a corrente \( i \) necessária para equilibrar a balança quando \( m = 10 \, \text{mg} \). Os resultados das medições são apresentados em um gráfico. A partir desses dados, estime o valor do campo magnético do ímã. + +**Descrição da imagem:** + +A imagem apresenta um sistema experimental composto por um arranjo metálico e um ímã. O arranjo, com comprimento \( \ell \), está sob influência do campo magnético do ímã. Ele é suportado por duas extremidades, uma delas posicionada a uma distância \( L \) de onde se encontra a massa móvel \( m \). A variável \( d \) representa a distância entre o ponto de apoio e a massa móvel. Um gráfico ao lado apresenta a relação entre a corrente \( i \) (em amperes) no eixo horizontal e a distância \( d \) (em metros) no eixo vertical, mostrando a variação de \( d \) em função de \( i \). + +\section*{Solução} + +Para que a balança esteja em equilíbrio, o torque produzido pela força magnética deve ser igual ao torque produzido pelo peso da massa \( m \). + +A força magnética que atua sobre o condutor de comprimento \( \ell \) é dada por: + +\[ +F_m = i \ell B +\] + +O torque devido à força magnética em relação ao ponto de apoio é: + +\[ +\tau_m = F_m \times L = i \ell B L +\] + +O peso da massa \( m \) é: + +\[ +F_p = m g +\] + +O torque devido ao peso da massa em relação ao ponto de apoio é: + +\[ +\tau_p = F_p \times d = m g d +\] + +Para o equilíbrio da balança: + +\[ +\tau_m = \tau_p +\] + +Portanto: + +\[ +i \ell B L = m g d +\] + +Isolando \( d \): + +\[ +d = \dfrac{i \ell B L}{m g} +\] + +Observamos que \( d \) é diretamente proporcional à corrente \( i \). Assim, ao representar \( d \) em função de \( i \) em um gráfico, a inclinação da reta (coeficiente angular) será: + +\[ +S = \dfrac{d}{i} = \dfrac{\ell B L}{m g} +\] + +Desse modo, podemos expressar o campo magnético \( B \) em função da inclinação \( S \): + +\[ +B = \dfrac{S m g}{\ell L} +\] + +Substituindo os valores fornecidos: + +\begin{align*} +m &= 10 \, \text{mg} = 10 \times 10^{-6} \, \text{kg} \\ +g &= 9{,}8 \, \text{m/s}^2 \\ +\ell &= 10 \, \text{cm} = 0{,}10 \, \text{m} \\ +L &= 1{,}0 \, \text{m} +\end{align*} + +Substituindo na expressão de \( B \): + +\[ +B = \dfrac{S \times (10 \times 10^{-6}) \times 9{,}8}{0{,}10 \times 1{,}0} +\] + +Simplificando: + +\begin{align*} +B &= \dfrac{S \times 9{,}8 \times 10^{-5}}{0{,}10} \\ +B &= S \times 9{,}8 \times 10^{-4} \, \text{T} +\end{align*} + +**ANSWER:** \( B = S \times 9{,}8 \times 10^{-4} \, \text{T} \) \ No newline at end of file From 4ef0a6c4202c786915a16ef293ff0b9792b29b51 Mon Sep 17 00:00:00 2001 From: Luan Fernandes Date: Sat, 16 Nov 2024 12:36:35 -0300 Subject: [PATCH 13/16] add q10, pass --- .../physics/essays/solutions/q10_solution.txt | 115 ++++++++++++++++++ 1 file changed, 115 insertions(+) create mode 100644 exams/ita_2025/physics/essays/solutions/q10_solution.txt diff --git a/exams/ita_2025/physics/essays/solutions/q10_solution.txt b/exams/ita_2025/physics/essays/solutions/q10_solution.txt new file mode 100644 index 0000000..df2bbda --- /dev/null +++ b/exams/ita_2025/physics/essays/solutions/q10_solution.txt @@ -0,0 +1,115 @@ +\section*{Questão 10} + +Em uma câmara de alto vácuo, um feixe monoenergético de elétrons é misturado a um feixe colimado e monoenergético de íons totalmente ionizados, conforme mostra a figura. As velocidades dos elétrons e dos íons são iguais. A abertura de um detector de fótons é apontada perpendicularmente à direção dos feixes misturados. Foram feitos três experimentos a baixas energias: o primeiro com um feixe de prótons, o segundo com um feixe de hélio totalmente ionizado e o terceiro com um feixe de oxigênio totalmente ionizado. Em um quarto experimento, usando um feixe de prótons relativísticos, o detector de fótons é apontado paralelamente à direção dos feixes misturados. + +Considerando essa situação experimental, determine + +**a)** as energias máximas dos fótons em eV detectadas no primeiro, segundo e terceiro experimentos; + +**b)** um valor aproximado para o desvio percentual da máxima energia do fóton no quarto experimento com relação à máxima energia do fóton do primeiro experimento, considerando que a energia cinética dos íons, no referencial do laboratório, era de \(234,5 \, \text{MeV}\) no quarto experimento. + +**Descrição da imagem:** + +A imagem mostra um esquema de um feixe de íons e um feixe de elétrons entrando em um dispositivo com um imã de flexão. Após a interação, os feixes são direcionados a um detector de fótons. O feixe de íons é indicado como uma linha inclinada, enquanto o feixe de elétrons é uma linha horizontal. O imã de flexão está posicionado no ponto onde os dois feixes interagem. O detector de fótons é colocado de forma perpendicular à direção dos feixes misturados. + +\section*{Solução} + +\textbf{a) Determinação das energias máximas dos fótons nos três primeiros experimentos} + +Nos experimentos, um feixe de elétrons é misturado a um feixe de íons totalmente ionizados, ambos com a \textbf{mesma velocidade}, e o detector capta fótons emitidos perpendicularmente à direção dos feixes. A emissão de fótons ocorre quando um elétron é capturado por um íon totalmente ionizado, formando um íon hidrogenoide. A energia do fóton emitido corresponde à diferença de energia entre os níveis inicial (livre) e final (estado fundamental do íon hidrogenoide). + +A energia de ionização (ou energia necessária para remover o elétron do estado fundamental) de um íon hidrogenoide é dada por: + +\[ +E_n = -Z^2 \cdot 13{,}6\,\text{eV} +\] + +onde \(Z\) é o número atômico (carga nuclear) do íon. + +Assim, a energia máxima do fóton emitido corresponde à energia de ligação do elétron no estado fundamental do íon hidrogenoide formado. + +\textbf{Primeiro experimento (prótons, \(Z=1\))}: + +\[ +E_{\text{máx}}^{(1)} = -(-Z^2 \cdot 13{,}6\,\text{eV}) = Z^2 \cdot 13{,}6\,\text{eV} = (1)^2 \cdot 13{,}6\,\text{eV} = 13{,}6\,\text{eV} +\] + +\textbf{Segundo experimento (hélio totalmente ionizado, \(Z=2\))}: + +\[ +E_{\text{máx}}^{(2)} = Z^2 \cdot 13{,}6\,\text{eV} = (2)^2 \cdot 13{,}6\,\text{eV} = 4 \cdot 13{,}6\,\text{eV} = 54{,}4\,\text{eV} +\] + +\textbf{Terceiro experimento (oxigênio totalmente ionizado, \(Z=8\))}: + +\[ +E_{\text{máx}}^{(3)} = Z^2 \cdot 13{,}6\,\text{eV} = (8)^2 \cdot 13{,}6\,\text{eV} = 64 \cdot 13{,}6\,\text{eV} = 870{,}4\,\text{eV} +\] + +\textbf{Resposta da letra a):} + +\textbf{ANSWER:} + +\begin{itemize} + \item No primeiro experimento, \(E_{\text{máx}} = 13{,}6\,\text{eV}\). + \item No segundo experimento, \(E_{\text{máx}} = 54{,}4\,\text{eV}\). + \item No terceiro experimento, \(E_{\text{máx}} = 870{,}4\,\text{eV}\). +\end{itemize} + +\textbf{b) Cálculo do desvio percentual da energia máxima do fóton no quarto experimento} + +No quarto experimento, utilizamos prótons relativísticos com energia cinética \(K = 234{,}5\,\text{MeV}\), e o detector de fótons está apontado \textbf{paralelamente} à direção dos feixes. Devido ao movimento relativístico dos íons, devemos considerar o efeito Doppler relativístico no aumento da energia dos fótons emitidos na direção do movimento. + +Primeiro, calculamos o fator de Lorentz (\(\gamma\)) e a velocidade (\(\beta = v/c\)) dos prótons: + +\[ +\gamma = \frac{E_{\text{total}}}{E_0} = \frac{E_0 + K}{E_0} = \frac{E_0 + K}{E_0} = 1 + \frac{K}{E_0} +\] + +Onde \(E_0\) é a energia de repouso do próton (\(E_0 = 938{,}3\,\text{MeV}\)). + +Calculando \(\gamma\): + +\[ +\gamma = 1 + \frac{234{,}5\,\text{MeV}}{938{,}3\,\text{MeV}} = 1 + 0{,}25 = 1{,}25 +\] + +Calculando \(\beta\): + +\[ +\beta = \sqrt{1 - \frac{1}{\gamma^2}} = \sqrt{1 - \frac{1}{(1{,}25)^2}} = \sqrt{1 - \frac{1}{1{,}5625}} = \sqrt{1 - 0{,}64} = \sqrt{0{,}36} = 0{,}6 +\] + +O efeito Doppler relativístico para a energia dos fótons emitidos na direção do movimento é dado por: + +\[ +E' = E \cdot \gamma (1 + \beta) +\] + +Onde \(E\) é a energia do fóton no referencial do próton (13{,}6\,eV). + +Calculando \(E'\): + +\[ +E' = 13{,}6\,\text{eV} \times 1{,}25 \times (1 + 0{,}6) = 13{,}6\,\text{eV} \times 1{,}25 \times 1{,}6 = 13{,}6\,\text{eV} \times 2 = 27{,}2\,\text{eV} +\] + +O desvio percentual em relação à energia máxima do primeiro experimento é: + +\[ +\text{Desvio percentual} = \left( \frac{E' - E_{\text{inicial}}}{E_{\text{inicial}}} \right) \times 100\% +\] + +Onde \(E_{\text{inicial}} = 13{,}6\,\text{eV}\). + +Calculando o desvio percentual: + +\[ +\text{Desvio percentual} = \left( \frac{27{,}2\,\text{eV} - 13{,}6\,\text{eV}}{13{,}6\,\text{eV}} \right) \times 100\% = \left( \frac{13{,}6\,\text{eV}}{13{,}6\,\text{eV}} \right) \times 100\% = 100\% +\] + +\textbf{Resposta da letra b):} + +\textbf{ANSWER:} + +O desvio percentual da máxima energia do fóton no quarto experimento em relação ao primeiro experimento é de \(100\%\). \ No newline at end of file From d4c273d3875a6a0f5917c92ead74a04d91dd4c28 Mon Sep 17 00:00:00 2001 From: Luan Fernandes Date: Sat, 16 Nov 2024 12:36:48 -0300 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12:31:23 -0300 Subject: [PATCH 15/16] add comments in the report --- exams/ita_2025/report.md | 17 +++++++++++++++-- 1 file changed, 15 insertions(+), 2 deletions(-) diff --git a/exams/ita_2025/report.md b/exams/ita_2025/report.md index e29ab60..6b49d42 100644 --- a/exams/ita_2025/report.md +++ b/exams/ita_2025/report.md @@ -24,7 +24,7 @@ Preliminary solutions can be found by widely known high schools like [Poliedro]( | Exam | Phase | Section | Type | Model | Status | Score | PDF Compiled | |------|-------|---------|------|-------|--------|-------|--------------| | ITA | 2025 | Math | Essay | o1-preview | ✅ Completed | 90%| [PDF](math/essays/solutions/solutions_compiled.pdf) | -| ITA | 2025 | Physics | Essay | o1-preview | 🚧 TODO | - | - | +| ITA | 2025 | Physics | Essay | o1-preview | ✅ Completed | 65% | [PDF](physics/essays/solutions/solutions_compiled.pdf) | | ITA | 2025 | Chemistry | Essay | o1-preview | ✅ Completed | 95% | [PDF](chemistry/essays/solutions/solutions_compiled.pdf) | | ITA | 2025 | Portuguese | Essay | o1-preview | 🚧 TODO | - | - | | ITA | 2025 | Math | Multiple Choice | o1-preview | 🚧 TODO | - | - | @@ -40,4 +40,17 @@ For almost of all questions, `max_tokens_question_description=400` and `max_toke Again, `o1-preview` got all questions correct in the Chemistry essay exam. The only wrong question was question 3, where it got parts (a) and (b) correct, but (c) and (d) wrong. This is probably because parts (c) and (d) involved reasoning about what was represented in the picture of the question, which is hard for LLMs to do, so gpt-4o did some limited description of the image but it was not enough for o1-preview. This is why, for this question, we considered 0.5 points for the score (2 parts out of 4). -`max_tokens_question_answer=5000` worked for most questions; only question 8 could not and so `10000` was used. Also, for some questions the confusion about how to use decimal separators or thousands separators in portuguese or english induced some errors, even when the development was clearly correct. To fix that, we updated the o1-preview prompt to use `,` for decimal separators and avoid thousands separators. \ No newline at end of file +`max_tokens_question_answer=5000` worked for most questions; only question 8 could not and so `10000` was used. Also, for some questions the confusion about how to use decimal separators or thousands separators in portuguese or english induced some errors, even when the development was clearly correct. To fix that, we updated the o1-preview prompt to use `,` for decimal separators and avoid thousands separators. + +### Physics Essay + +This exam was the hardest one so far. o1-preview got only 65% of the questions correct. Here are some details for wrong answers: + +- Question 3: failed all three parts. It made a incorrect assumption about the relation between accelerations when the llm said `a_{\text{mov}} = a_{\text{fixa}} - a_M \, \sin\alpha`. 0 points. +- Question 4: failed on part b, when it treated the difference in kinetic energy `\delta` as a positive value (it said `\delta = \frac{E_{\text{inicial}} - E_{\text{final}}}{E_{\text{inicial}}}`, instead of negative value, which would be correct if the llm said `\delta = \frac{E_{\text{final}} - E_{\text{inicial}}}{E_{\text{inicial}}}`. Considering each part valued at `10/3=3.33 points`, we could consider only half for part (b) for the mistake. `total points: 3.33 * 2 + 3.33/2=8.33` +- Question 5: failed on part (b). When the llm said the `v_{\text{NS}} &= v_0 + v_e`, this is only valid if `m< Date: Sun, 17 Nov 2024 12:34:26 -0300 Subject: [PATCH 16/16] update main readme --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index 32da8e0..ae26e6c 100644 --- a/README.md +++ b/README.md @@ -7,7 +7,7 @@ We'll use `o1-preview`, which is the best OpenAI model so far with reasoning cap The project begins with the ITA (Instituto Tecnológico de Aeronáutica) 2025 exam, focusing first on the Math essay section. This section, from the recent exam on November 5, 2024, demands deep subject understanding and step-by-step solutions. More details are in the [report](exams/ita_2025/report.md) documentation. -**Spoiler: o1-preview scored 90% in the ITA 2025 Math Essay Exam.** +**Spoiler: o1-preview scored 90% in the ITA 2025 Math Essay Exam, 95% in Chemistry Essay and only 65% in Physics Essay.** After the first ITA 2025 exam is fully solved, the project will try to expand to other sections and eventually other exams. Feel free to contribute with ideas and implementations of other exams!

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zB@&@$!&AgXue?KA{H@_yG-2tdC2r>(%hvX*IL{D8&U^BYjh*q^HOLTut~hIyGZez$ zeS0U5|D_@}uHSC6-id6^3z^Qq2{q*XOm=(WwPC#XCsm;)oKT$Z)`4gW^*t~Jqe`R~ z_?nilJkyY-DJPUI6ohFwCGFlteY5ss-!;Ums1qxIVjXp&^0=&bIN zit72aPIRi#du3!>j*uAbL0?9XyfMDFn8g&s5X4?hti8U18`N8Jcz-UpDRL)RLZ7I! zpq&ayo8B!kci?)GIq%qJ_1TezEr*R*7@nE;$jC*1n)oD<1X9Ix6?XU5(+7&kp0Q%Z z@b#Fubjw@XACJ`OK!W8Mk7w Date: Fri, 15 Nov 2024 15:52:01 -0300 Subject: [PATCH 03/16] add questions 1,2,3,5 reviewed --- .../physics/essays/solutions/q1_solution.txt | 113 +++++++++++++ .../physics/essays/solutions/q2_solution.txt | 149 +++++++++++++++++ .../physics/essays/solutions/q3_solution.txt | 157 ++++++++++++++++++ .../physics/essays/solutions/q5_solution.txt | 64 +++++++ 4 files changed, 483 insertions(+) create mode 100644 exams/ita_2025/physics/essays/solutions/q1_solution.txt create mode 100644 exams/ita_2025/physics/essays/solutions/q2_solution.txt create mode 100644 exams/ita_2025/physics/essays/solutions/q3_solution.txt create mode 100644 exams/ita_2025/physics/essays/solutions/q5_solution.txt diff --git a/exams/ita_2025/physics/essays/solutions/q1_solution.txt b/exams/ita_2025/physics/essays/solutions/q1_solution.txt new file mode 100644 index 0000000..31b99ea --- /dev/null +++ b/exams/ita_2025/physics/essays/solutions/q1_solution.txt @@ -0,0 +1,113 @@ +\section*{Questão 1} + +**Enunciado:** + +O *quantum* de fluxo magnético \(\Phi_0\) pode ser definido como metade do fluxo magnético obtido a partir da combinação da constante de Planck \(h\), da velocidade da luz \(c\) e da carga fundamental \(e\). Considere um elétron se movendo em uma órbita circular de raio \(R\), sob a ação de um campo magnético de modo que o fluxo magnético dentro de sua órbita é igual a \(\Phi_0\). Faça o que se pede nos itens a seguir. + +**a)** Obtenha a expressão para \(\Phi_0\). + +**b)** Sabendo que a velocidade do elétron é dada por \(\beta c\) \((\beta \ll 1)\), calcule o raio \(R\), em termos de \(h\), \(c\), \(\beta\) e \(m_e\), a massa do elétron. + +**Descrição da imagem completa:** + +A imagem contém um cabeçalho com o título *FÍSICA* e apresenta constantes físicas com seus respectivos valores numéricos: + +- Aceleração local da gravidade \(g = 10 \, \text{m/s}^2\). +- Velocidade da luz no vácuo \(c = 3,0 \times 10^8 \, \text{m/s}\). +- Constante de gravitação universal \(G = 6,7 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2\). +- Massa da Terra \(M_{\text{Terra}} = 6,0 \times 10^{24} \, \text{kg}\). +- Raio da Terra \(R_{\text{Terra}} = 6,4 \times 10^6 \, \text{m}\). +- Permissividade elétrica no vácuo \(\varepsilon_0 = 8,85 \times 10^{-12} \, \text{C}^2 \cdot \text{N}^{-1} \cdot \text{m}^{-2}\). +- Energia de ionização do átomo de hidrogênio \(I_H = 13,6 \, \text{eV}\). +- Massa do próton \(m_p = 1,6 \times 10^{-27} \, \text{kg} = 938 \, \text{MeV}/c^2\). +- Carga elementar \(e = 1,6 \times 10^{-19} \, \text{C}\). + +Segue-se o enunciado da questão e suas duas subquestões. + +\section*{Solução} + +\textbf{a) Obtenha a expressão para \(\Phi_0\).} + +O *quantum* de fluxo magnético \(\Phi_0\) é definido como metade do fluxo magnético obtido a partir da combinação das constantes \(h\), \(c\) e \(e\). Para encontrar a expressão de \(\Phi_0\), precisamos combinar essas constantes de forma a obter unidades de fluxo magnético \([\Phi] = \text{T} \cdot \text{m}^2 = \text{Wb}\). + +As unidades das constantes são: + +\begin{align*} +[h] &= \text{J} \cdot \text{s} = \text{kg} \cdot \text{m}^2 \cdot \text{s}^{-1}, \\ +[c] &= \text{m} \cdot \text{s}^{-1}, \\ +[e] &= \text{C} = \text{A} \cdot \text{s}. +\end{align*} + +Queremos encontrar uma combinação que resulte em unidades de fluxo magnético: + +\begin{align*} +\Phi_0 &= \frac{1}{2} \times \frac{h}{e}. +\end{align*} + +Portanto, a expressão para o *quantum* de fluxo magnético é: + +\begin{align*} +\Phi_0 = \dfrac{h}{2e}. +\end{align*} + +\textbf{b) Calcule o raio \(R\) em termos de \(h\), \(c\), \(\beta\) e \(m_e\).} + +O elétron se move em uma órbita circular de raio \(R\) sob a ação de um campo magnético \(B\), de forma que o fluxo magnético através da órbita é \(\Phi_0\): + +\begin{align*} +\Phi_0 &= B \cdot \pi R^2. +\end{align*} + +Logo, podemos expressar o campo magnético \(B\) em função de \(\Phi_0\) e \(R\): + +\begin{align*} +B &= \dfrac{\Phi_0}{\pi R^2}. +\end{align*} + +A força magnética fornece a força centrípeta necessária para o movimento circular do elétron: + +\begin{align*} +F_{\text{magnética}} &= F_{\text{centrípeta}}, \\ +e v B &= \dfrac{m_e v^2}{R}. +\end{align*} + +Sabendo que a velocidade do elétron é \(v = \beta c\), substituímos na equação: + +\begin{align*} +e (\beta c) B &= \dfrac{m_e (\beta c)^2}{R}, \\ +e \beta c B &= \dfrac{m_e \beta^2 c^2}{R}. +\end{align*} + +Simplificando: + +\begin{align*} +B &= \dfrac{m_e \beta c}{e R}. +\end{align*} + +Substituindo a expressão de \(B\) encontrada anteriormente: + +\begin{align*} +\dfrac{\Phi_0}{\pi R^2} &= \dfrac{m_e \beta c}{e R}. +\end{align*} + +Multiplicando ambos os lados por \(\pi R^2\) e rearranjando: + +\begin{align*} +\Phi_0 &= \dfrac{m_e \beta c}{e R} \cdot \pi R^2, \\ +\Phi_0 &= \dfrac{m_e \beta c \pi R}{e}. +\end{align*} + +Isolando \(R\): + +\begin{align*} +R &= \dfrac{e \Phi_0}{\pi m_e \beta c}. +\end{align*} + +Substituindo \(\Phi_0 = \dfrac{h}{2e}\): + +\begin{align*} +R &= \dfrac{e \left( \dfrac{h}{2e} \right)}{\pi m_e \beta c}, \\ +R &= \dfrac{h}{2 \pi m_e \beta c}. +\end{align*} + +\textbf{ANSWER:} \(R = \dfrac{h}{2\pi m_e \beta c}\). \ No newline at end of file diff --git a/exams/ita_2025/physics/essays/solutions/q2_solution.txt b/exams/ita_2025/physics/essays/solutions/q2_solution.txt new file mode 100644 index 0000000..6eb89b9 --- /dev/null +++ b/exams/ita_2025/physics/essays/solutions/q2_solution.txt @@ -0,0 +1,149 @@ +\section*{Questão 2} + +Considere uma barra homogênea de comprimento \( L \) e massa \( M \), suspensa horizontalmente por uma corda vertical que tem um nó fixo no teto e outro numa das extremidades da barra (\( x = 0 \)). Uma massa \( m \) está pendurada na outra extremidade (\( x = L \)), e uma distribuição de forças é aplicada ao longo da barra, de forma que o sistema se encontra em equilíbrio estático. Essa distribuição pode ser descrita por \( N \) forças, que obedecem à relação de recorrência \(\vec{F}_n = \frac{\vec{F}_{n-1}}{2}\) (\( n = 0, 1, \ldots, N-1 \)), aplicadas nos pontos \( x_n = 2^{-n}L \). Calcule, em termos de \( M, m, g, L \) e \( N \): + +\begin{itemize} + \item **a)** a força \( F_0 \); + \item **b)** a força de tração da corda. +\end{itemize} + +\section*{Solução} + +Para resolver o problema, vamos aplicar as condições de equilíbrio estático à barra. + +**Equilíbrio de momentos:** + +Escolhemos o ponto \( x = 0 \) (onde a corda está presa) como eixo de rotação. A condição de equilíbrio de momentos é: + +\[ +\sum \tau = 0 +\] + +Os momentos atuantes são: + +1. Momento do peso da barra (\( M g \)) aplicado em seu centro de massa, em \( x = \dfrac{L}{2} \): +\[ +\tau_{\text{barra}} = M g \cdot \dfrac{L}{2} +\] + +2. Momento do peso da massa \( m \) pendurada em \( x = L \): +\[ +\tau_{\text{massa}} = m g \cdot L +\] + +3. Momentos das forças \( \vec{F}_n \) aplicadas em \( x_n = 2^{-n} L \): +\[ +\tau_{\vec{F}} = -\sum_{n=0}^{N-1} F_n x_n +\] +(o sinal negativo indica que as forças \( \vec{F}_n \) produzem momento no sentido anti-horário, contrário aos pesos) + +Aplicando a condição de equilíbrio de momentos: +\[ +M g \cdot \dfrac{L}{2} + m g \cdot L - \sum_{n=0}^{N-1} F_n x_n = 0 +\] + +**Cálculo da soma \( \sum_{n=0}^{N-1} F_n x_n \):** + +Sabemos que: +\[ +F_n = \dfrac{F_{n-1}}{2} = F_0 \left( \dfrac{1}{2} \right)^{n} +\] +\[ +x_n = 2^{-n} L = L \left( \dfrac{1}{2} \right)^{n} +\] + +Então: +\[ +F_n x_n = F_0 \left( \dfrac{1}{2} \right)^{n} \cdot L \left( \dfrac{1}{2} \right)^{n} = F_0 L \left( \dfrac{1}{4} \right)^{n} +\] + +A soma é uma série geométrica: +\begin{align*} +\sum_{n=0}^{N-1} F_n x_n &= F_0 L \sum_{n=0}^{N-1} \left( \dfrac{1}{4} \right)^{n} \\ +&= F_0 L \left[ \dfrac{1 - \left( \dfrac{1}{4} \right)^{N}}{1 - \dfrac{1}{4}} \right] \\ +&= F_0 L \left( \dfrac{1 - 4^{-N}}{\dfrac{3}{4}} \right) \\ +&= \dfrac{4}{3} F_0 L \left( 1 - 4^{-N} \right) +\end{align*} + +Substituindo na equação de equilíbrio de momentos: +\[ +M g \cdot \dfrac{L}{2} + m g \cdot L - \dfrac{4}{3} F_0 L \left( 1 - 4^{-N} \right) = 0 +\] + +Simplificando: +\[ +\left( \dfrac{M g}{2} + m g \right) L = \dfrac{4}{3} F_0 L \left( 1 - 4^{-N} \right) +\] + +Cancelando \( L \) dos dois lados: +\[ +\dfrac{M g}{2} + m g = \dfrac{4}{3} F_0 \left( 1 - 4^{-N} \right) +\] + +**Resposta para a letra a):** + +Isolando \( F_0 \): +\begin{align*} +F_0 &= \left( \dfrac{M g}{2} + m g \right) \cdot \dfrac{3}{4} \cdot \dfrac{1}{1 - 4^{-N}} \\ +&= \left( \dfrac{3}{4} \left( \dfrac{M g}{2} + m g \right) \right) \cdot \dfrac{1}{1 - 4^{-N}} +\end{align*} + +**ANSWER:** +\[ +F_0 = \dfrac{3}{4} \left( \dfrac{M g}{2} + m g \right) \cdot \dfrac{1}{1 - 4^{-N}} +\] + +--- + +**Equilíbrio de forças verticais:** + +A condição de equilíbrio de forças na direção vertical é: +\[ +T + \sum_{n=0}^{N-1} F_n = M g + m g +\] +onde \( T \) é a tensão na corda. + +Calculando \( \sum_{n=0}^{N-1} F_n \): +\begin{align*} +\sum_{n=0}^{N-1} F_n &= F_0 \sum_{n=0}^{N-1} \left( \dfrac{1}{2} \right)^{n} \\ +&= F_0 \left[ \dfrac{1 - \left( \dfrac{1}{2} \right)^{N}}{1 - \dfrac{1}{2}} \right] \\ +&= F_0 \left( \dfrac{1 - 2^{-N}}{\dfrac{1}{2}} \right) \\ +&= 2 F_0 \left( 1 - 2^{-N} \right) +\end{align*} + +Substituindo na equação de equilíbrio de forças: +\[ +T + 2 F_0 \left( 1 - 2^{-N} \right) = M g + m g +\] + +Isolando \( T \): + +\begin{align*} +T &= M g + m g - 2 F_0 \left( 1 - 2^{-N} \right) +\end{align*} + +Substituindo \( F_0 \) encontrado anteriormente: + +\begin{align*} +T &= M g + m g - 2 \left[ \dfrac{3}{4} \left( \dfrac{M g}{2} + m g \right) \cdot \dfrac{1}{1 - 4^{-N}} \right] \left( 1 - 2^{-N} \right) \\ +&= M g + m g - \dfrac{3}{2} \left( \dfrac{M g}{2} + m g \right) \cdot \dfrac{1 - 2^{-N}}{1 - 4^{-N}} +\end{align*} + +Para simplificar, reconhecemos que \( 4^{-N} = \left( 2^{-2} \right)^{N} = 2^{-2N} \), então \( 4^{-N} = \left( 2^{-N} \right)^{2} \). Porém, a simplificação pode variar dependendo do valor de \( N \). + +**Resposta para a letra b):** + +\[ +T = M g + m g - 2 F_0 \left( 1 - 2^{-N} \right) +\] + +Substituindo o valor de \( F_0 \): + +\[ +T = M g + m g - \dfrac{3}{2} \left( \dfrac{M g}{2} + m g \right) \cdot \dfrac{1 - 2^{-N}}{1 - 4^{-N}} +\] + +**ANSWER:** +\[ +T = M g + m g - 2 F_0 \left( 1 - 2^{-N} \right) +\] \ No newline at end of file diff --git a/exams/ita_2025/physics/essays/solutions/q3_solution.txt b/exams/ita_2025/physics/essays/solutions/q3_solution.txt new file mode 100644 index 0000000..f7c2c70 --- /dev/null +++ b/exams/ita_2025/physics/essays/solutions/q3_solution.txt @@ -0,0 +1,157 @@ +\section*{Questão 3} + +Considere um objeto de massa \( m \) que se movimenta sobre uma cunha de massa \( M \), inclinação \( \alpha \) e coeficiente de atrito \(\mu\). A cunha se move horizontalmente para a direita, sob a ação de uma força \(\vec{F}\) em uma superfície lisa. Considere que, inicialmente, o sistema se encontra em repouso, com esse objeto no topo da cunha. Sabe-se que o intervalo de tempo que ele leva para chegar ao solo com a cunha em movimento é o triplo do que levaria se a cunha estivesse fixa. Com base nessas informações, calcule, em função de \( m, M, \alpha, \mu \) e \( g \), a magnitude + +a) da aceleração da cunha; + +b) da força normal que o plano inclinado faz no objeto; + +c) da força \(\vec{F}\). + +**Descrição da Imagem:** + +A imagem mostra um diagrama de um bloco de massa \(m\) localizado no topo de uma cunha de massa \(M\) inclinada em um ângulo \(\alpha\) em relação à horizontal. A cunha está em uma superfície horizontal e é impulsionada por uma força \(\vec{F}\) para a direita. A superfície da cunha é inclinada, representada por uma linha ascendente começando da base até o topo, onde o bloco \(m\) está posicionado. + +\section*{Solução} + +Primeiramente, analisaremos o caso em que a cunha está fixa e, em seguida, o caso em que a cunha está em movimento. + +\subsection*{Caso 1: Cunha Fixa} + +Quando a cunha está fixa, a aceleração do bloco de massa \( m \) ao longo do plano inclinado é dada por: + +\[ +a_{\text{fixa}} = g\, (\sin\alpha - \mu\, \cos\alpha) +\] + +O tempo que o bloco leva para descer o comprimento \( L \) da cunha é: + +\[ +t_{\text{fixa}} = \sqrt{\dfrac{2L}{a_{\text{fixa}}}} +\] + +\subsection*{Caso 2: Cunha em Movimento} + +Quando a cunha está em movimento, o bloco desce o plano inclinado com uma aceleração efetiva menor devido ao movimento da cunha. Sabemos que o tempo nesse caso é o triplo do tempo quando a cunha está fixa: + +\[ +t_{\text{mov}} = 3\, t_{\text{fixa}} +\] + +Como o tempo de descida está relacionado inversamente à raiz quadrada da aceleração efetiva (\( a_{\text{mov}} \)), temos: + +\[ +t_{\text{mov}} = \sqrt{\dfrac{2L}{a_{\text{mov}}}} +\] + +Dividindo as equações dos tempos: + +\[ +\dfrac{t_{\text{mov}}}{t_{\text{fixa}}} = \sqrt{\dfrac{a_{\text{fixa}}}{a_{\text{mov}}}} = 3 +\] + +Assim, obtemos: + +\[ +a_{\text{mov}} = \dfrac{a_{\text{fixa}}}{9} +\] + +A aceleração efetiva do bloco ao longo do plano inclinado quando a cunha está em movimento é: + +\[ +a_{\text{mov}} = a_{\text{fixa}} - a_M \, \sin\alpha +\] + +Onde \( a_M \) é a aceleração da cunha. Substituindo \( a_{\text{mov}} \): + +\[ +a_{\text{fixa}} - a_M \, \sin\alpha = \dfrac{a_{\text{fixa}}}{9} +\] + +Isolando \( a_M \): + +\[ +a_M \, \sin\alpha = a_{\text{fixa}} - \dfrac{a_{\text{fixa}}}{9} = \dfrac{8\, a_{\text{fixa}}}{9} +\] + +Portanto: + +\[ +a_M = \dfrac{8\, a_{\text{fixa}}}{9\, \sin\alpha} +\] + +Substituindo \( a_{\text{fixa}} \): + +\begin{align*} +a_M &= \dfrac{8}{9\, \sin\alpha} \left[ g\, (\sin\alpha - \mu\, \cos\alpha) \right] \\ +&= \dfrac{8\, g}{9} \left( 1 - \mu\, \cot\alpha \right) +\end{align*} + +\textbf{Resposta do item a):} + +\[ +\boxed{a_M = \dfrac{8\, g}{9} \left( 1 - \mu\, \cot\alpha \right)} +\] + +\subsection*{Cálculo da Força Normal \( N \)} + +Para calcular a força normal, consideramos a segunda lei de Newton na direção perpendicular ao plano inclinado. Quando a cunha está em movimento, o bloco pode ter uma componente de aceleração nessa direção devido ao movimento da cunha. No entanto, assumindo que a cunha é suficientemente maciça ou que o ângulo \( \alpha \) não é muito pequeno, podemos considerar que a aceleração perpendicular é desprezível e que: + +\[ +N = m\, g\, \cos\alpha +\] + +\textbf{Resposta do item b):} + +\[ +\boxed{N = m\, g\, \cos\alpha} +\] + +\subsection*{Cálculo da Força \( \vec{F} \)} + +Para determinar \( \vec{F} \), aplicamos a segunda lei de Newton para a cunha: + +\[ +F = M\, a_M + N\, \sin\alpha - f\, \cos\alpha +\] + +O coeficiente de atrito atua na direção oposta ao movimento relativo entre o bloco e a cunha. A força de atrito é: + +\[ +f = \mu\, N = \mu\, m\, g\, \cos\alpha +\] + +Substituindo \( f \) e \( N \) na expressão de \( F \): + +\begin{align*} +F &= M\, a_M + N\, \sin\alpha - f\, \cos\alpha \\ +&= M\, a_M + m\, g\, \cos\alpha\, \sin\alpha - \mu\, m\, g\, \cos\alpha\, \cos\alpha \\ +&= M\, a_M + m\, g\, \cos\alpha\, \sin\alpha - \mu\, m\, g\, \cos^2\alpha +\end{align*} + +Substituindo \( a_M \): + +\begin{align*} +F &= M \left( \dfrac{8\, g}{9} \left( 1 - \mu\, \cot\alpha \right) \right) + m\, g\, \cos\alpha\, \sin\alpha - \mu\, m\, g\, \cos^2\alpha \\ +&= \dfrac{8\, M\, g}{9} \left( 1 - \mu\, \cot\alpha \right) + m\, g\, \cos\alpha\, \sin\alpha - \mu\, m\, g\, \cos^2\alpha +\end{align*} + +Simplificando: + +\begin{align*} +F &= \dfrac{8\, M\, g}{9} \left( 1 - \mu\, \cot\alpha \right) + m\, g\, \cos\alpha \left( \sin\alpha - \mu\, \cos\alpha \right) +\end{align*} + +\textbf{Resposta do item c):} + +\[ +\boxed{F = \dfrac{8\, M\, g}{9} \left( 1 - \mu\, \cot\alpha \right) + m\, g\, \cos\alpha \left( \sin\alpha - \mu\, \cos\alpha \right)} +\] + +\textbf{ANSWER:} + +a) \( a_M = \dfrac{8\, g}{9} \left( 1 - \mu\, \cot\alpha \right) \); + +b) \( N = m\, g\, \cos\alpha \); + +c) \( F = \dfrac{8\, M\, g}{9} \left( 1 - \mu\, \cot\alpha \right) + m\, g\, \cos\alpha \left( \sin\alpha - \mu\, \cos\alpha \right) \). \ No newline at end of file diff --git a/exams/ita_2025/physics/essays/solutions/q5_solution.txt b/exams/ita_2025/physics/essays/solutions/q5_solution.txt new file mode 100644 index 0000000..fe0d509 --- /dev/null +++ b/exams/ita_2025/physics/essays/solutions/q5_solution.txt @@ -0,0 +1,64 @@ +\section*{Questão 5} + +Considere um veículo lançador de nanosatélites (VLNS) de massa \( M_v \) a uma altitude \( h \) e com velocidade \( v_0 \), perpendicular ao raio da Terra em relação a um referencial inercial centrado na Terra. Um nanossatélite (NS) de massa \( m \) encontra-se imerso em um fluido incompressível armazenado em um tubo localizado na extremidade do VLNS, conforme a figura. O tubo possui dois diâmetros distintos: um de valor \( d_1 \) e outro de valor \( d_2 < d_1 \). Durante a ejeção, o NS acompanha a velocidade do fluido, que vale \( v_1 \) em \( d_1 \), em relação ao VLNS. Considere a massa e o raio da Terra como sendo, respectivamente, \( M_T \) e \( R_T \), a constante da gravitação universal como \( G \) e a massa do fluido como desprezível. Determine + +a) a velocidade de ejeção do NS, com relação ao VLNS, em termos de \( v_0 \), \( v_1 \), \( d_1 \) e \( d_2 \); + +b) qual diâmetro \( d_2 \) permite que o NS entre em órbita circular. + +**Descrição das Imagens:** + +1. **Primeira Imagem:** Apresenta um conjunto de constantes físicas a serem utilizadas nos problemas, incluindo aceleração da gravidade (\( g = 10 \, \text{m/s}^2 \)), velocidade da luz no vácuo (\( c = 3,0 \times 10^8 \, \text{m/s} \)), constante de gravitação universal (\( G = 6,7 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \)), massa da Terra (\( M_{\text{Terra}} = 6,0 \times 10^{24} \, \text{kg} \)), raio da Terra (\( R_{\text{Terra}} = 6,4 \times 10^6 \, \text{m} \)), entre outras. + +2. **Segunda Imagem:** Mostra uma figura com uma representação esquemática do problema proposto. À esquerda, um círculo indicando a Terra com uma seta vertical acima deste, representando o veículo lançador e a velocidade \( v_0 \). À direita, uma ilustração detalhada do veículo com dois diâmetros, \( d_1 \) e \( d_2 \), conectados ao nanosatélite (NS). + + +\section*{Solução} + +a) Durante a ejeção, considerando que o fluido é incompressível, a equação da continuidade nos diz que a vazão volumétrica é constante: + +\begin{align*} +A_1 v_1 &= A_2 v_2, +\end{align*} + +onde \( A = \dfrac{\pi d^2}{4} \) é a área da seção transversal. Portanto: + +\begin{align*} +\dfrac{\pi d_1^2}{4} v_1 &= \dfrac{\pi d_2^2}{4} v_2 \\ +\implies v_2 &= \left( \dfrac{d_1}{d_2} \right)^2 v_1. +\end{align*} + +Assim, a velocidade de ejeção do NS em relação ao VLNS é: + +\begin{align*} +v_e &= v_2 = \left( \dfrac{d_1}{d_2} \right)^2 v_1. +\end{align*} + +\textbf{ANSWER:} \( v_e = \left( \dfrac{d_1}{d_2} \right)^2 v_1 \) + +b) Para que o NS entre em órbita circular, sua velocidade em relação à Terra, \( v_{\text{NS}} \), deve ser igual à velocidade orbital na altitude \( h \): + +\begin{align*} +v_{\text{NS}} &= v_{\text{orb}} = \sqrt{ \dfrac{G M_T}{R_T + h} }. +\end{align*} + +A velocidade do NS em relação à Terra é a soma da velocidade do VLNS com a velocidade de ejeção: + +\begin{align*} +v_{\text{NS}} &= v_0 + v_e = v_0 + \left( \dfrac{d_1}{d_2} \right)^2 v_1. +\end{align*} + +Igualando as expressões: + +\begin{align*} +v_0 + \left( \dfrac{d_1}{d_2} \right)^2 v_1 &= \sqrt{ \dfrac{G M_T}{R_T + h} }. +\end{align*} + +Isolando \( d_2 \): + +\begin{align*} +\left( \dfrac{d_1}{d_2} \right)^2 &= \dfrac{ \sqrt{ \dfrac{G M_T}{R_T + h} } - v_0 }{v_1} \\ +\implies d_2 &= d_1 \sqrt{ \dfrac{ v_1 }{ \sqrt{ \dfrac{G M_T}{R_T + h} } - v_0 } }. +\end{align*} + +\textbf{ANSWER:} \( d_2 = d_1 \sqrt{ \dfrac{ v_1 }{ \sqrt{ \dfrac{G M_T}{R_T + h} } - v_0 } } \) \ No newline at end of file From 27d8f4458d5ac076903dc07ec3a3bda7b7b4974e Mon Sep 17 00:00:00 2001 From: Luan Fernandes Date: Fri, 15 Nov 2024 16:47:36 -0300 Subject: [PATCH 04/16] maintain screenshots already in main --- exams/ita_2025/physics/essays/conventions.jpg | Bin 152692 -> 94450 bytes exams/ita_2025/physics/essays/q1.jpg | Bin 160670 -> 153817 bytes exams/ita_2025/physics/essays/q10.jpg | Bin 367991 -> 225124 bytes exams/ita_2025/physics/essays/q2.jpg | Bin 173004 -> 165935 bytes exams/ita_2025/physics/essays/q3.jpg | Bin 207130 -> 251586 bytes exams/ita_2025/physics/essays/q4.jpg | Bin 189420 -> 249415 bytes exams/ita_2025/physics/essays/q5.jpg | Bin 273475 -> 337825 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