Trabalho e Energia Cinética em Física

Física

Trabalho

• Trabalho ($W$) é definido como a força ($\vec{F}$) aplicada sobre um objeto, resultando em um deslocamento ($\Delta \vec{r}$).
• Fórmula para trabalho realizado por uma força constante: $W = \vec{F} \cdot \Delta \vec{r} = F \Delta r \cos \theta$.
  • $W$ é medido em Joules.
  • $F$ é a magnitude da força em Newtons.
  • $\Delta r$ é a magnitude do deslocamento em metros.
  • $\theta$ é o ângulo entre a força e o deslocamento.
• Para forças não constantes ou caminhos não retos, usa-se a integral de linha para calcular o trabalho: $W = \int_{C} \vec{F} \cdot d\vec{r}$.

Energia Cinética

• Energia cinética ($K$) é a energia que um objeto possui devido ao seu movimento.
• A energia cinética de um objeto com massa $m$ e velocidade $v$ é $K = \frac{1}{2}mv^2$.

Teorema do Trabalho-Energia

• O trabalho total realizado sobre um objeto é igual à variação de sua energia cinética: $W_{total} = \Delta K = K_f - K_i = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$.

Potência

• Potência ($P$) é a taxa na qual o trabalho é realizado, medido em Watts.
• Fórmula geral: $P = \frac{dW}{dt}$.
• Para uma força constante, a potência pode ser expressa como $P = \vec{F} \cdot \vec{v}$.

Energia Potencial Gravitacional

• Energia potencial gravitacional ($U_g$) é dada por $U_g = mgh$, onde $m$ é a massa, $g$ é a aceleração da gravidade, e $h$ é a altura.

Energia Potencial Elástica

• Energia potencial elástica ($U_e$) armazenada em uma mola é $U_e = \frac{1}{2}kx^2$, onde $k$ é a constante da mola e $x$ é o deslocamento da mola.

Forças Conservativas

• Forças conservativas são aquelas cujo trabalho realizado independe do caminho percorrido.

Energia Mecânica

• A energia mecânica total ($E$) é a soma da energia cinética ($K$) e potencial ($U$): $E = K + U$.
  • Se apenas forças conservativas atuam, a energia mecânica total é conservada: $E_i = E_f$, ou $K_i + U_i = K_f + U_f$.

Forças Não Conservativas

• Se forças não conservativas atuam (como atrito), a energia mecânica não é conservada.
  • O trabalho realizado pelas forças não conservativas é $W_{nc} = \Delta E = E_f - E_i$.

Algorithmic Game Theory

Game Theory Definition

• Game theory studies mathematical models of strategic interactions among rational agents.
• Game theory can be applied to social science, logic, systems science and computer science.

Selfish Routing

• Selfish routing is defined by a network of $n$ agents who want to route traffic from a source to a destination.
• A graph $G = (V, E)$ represents the network.
• Each edge $e \in E$ has a cost function $c_e(x)$ related to traffice on the edge.
• Each agent tries to minimize its personal cost acting selfishly.

Braess's Paradox

• Braess's Paradox is an example where adding an edge to a network can increase the total cost due to selfish routing.
• In the example with a network of 4 nodes and 5 edges, after adding an edge from A to B with cost 0, the total cost increased from 3 to 4.

Price of Anarchy (PoA)

• Social cost is the total cost incurred by all players in a game.
• Nash Equilibrium is a set of strategies, one for each player, where no player benefit from unilaterally changing their strategy.
• Price of Anarchy measures the degradation of the system's performance due to selfishness.
• $PoA = \frac{\text{Social Cost of Worst Nash Equilibrium}}{\text{Social Optimum}}$

Static Equilibrium

Introduction

• An object is in static equilibrium when both the net force and the net torque on the object are zero.
• Static equilibrium means the object is not accelerating linearly or rotationally.
• The equations of static equilibirum are:
  • Net Force: $\sum \vec{F} = 0$
  • Net Torque: $\sum \vec{\tau} = 0$
• In two dimensions, these equations can be written as
  • $\sum F_x = 0$
  • $\sum F_y = 0$
  • $\sum \tau_z = 0$, where the z-axis is perpendicular to the plane in which the forces act.

Example 1

• A beam supported by two ropes is in static equilibirum.
• The goal is to calculate the tensions in the two ropes, $T_1$ and $T_2$.
• The equations used to solve for tension include:
  • $\sum F_y = 0$: $T_1 + T_2 - W = 0$
  • $\sum \tau_z = 0$:$-W \cdot \frac{L}{2} + T_2 \cdot \frac{3L}{4} = 0$
• For this example the tensions are:
  • $T_2 = \frac{2}{3}W$
  • $T_1 = \frac{1}{3}W$

Example 2

• A uniform ladder leaning against a smooth wall in static equilibrium.
• The goal is to calculate the minimum angle the ladder can have with the horizontal without slipping.
• Important parameters to know in this problem are:
  • $N_1$ - the normal force exerted by the wall on the ladder
  • $N_2$ - the normal force exerted by the ground on the ladder
  • $f_s$ - the force of static friction between the ladder and the ground
  • $f_s \le \mu_s N_2 = \mu_s W$
• The minimum angle occurs when $\theta \ge \tan^{-1}(\frac{1}{2\mu_s})$

Algorithmic Trading and Order Execution

Algorithmic Trading

• In Algorithmic trading pre-programmed instructions considering price, time, and volume automates trading decisions and execution.
• Algorithmic trading reduces transaction costs, improves order execution speed, increases trading capacity, reduces errors, reduces market impact and improves trading flexibility
• Common Algorithmic Trading Strategies
  • Trent Following, Mean Reversion, Arbitrage, Market Making, Statistical Arbitrage

Order Execution

• Order execution is the process or completing a buy or sell order in the market.
• Order execution involves routing orders to exchanges or market makers and filling them at the best available price.
• Order Execution Strategies: Market Order, Limit Order, Stop Order, Iceberg Order, VWAP and TWAP

Key Considerations

• Liquidity, Market Impact, Slippage, Transaction Costs and Regulatory Compliance.

Vectores

Suma de Vectores

• Hay dos métodos principales para sumar vectores:
  • Método gráfico (polígono)
  • Método analítico:
    • Componentes rectangulares
    • Vectores unitarios

Método Gráfico (Polígono)

• En este método, los vectores se colocan uno tras otro.
• El vector resultante une el origen del primero con el extremo del último.

Método Analítico

• Componentes Rectangulares:
  • Cada vector se descompone en sus componentes rectangulares ($A_x$, $A_y$).
  • $A_x = A \cos \theta$, $A_y = A \sin \theta$
  • $A = \sqrt{A_x^2 + A_y^2}$, $\theta = \arctan \frac{A_y}{A_x}$

Suma de Vectores por Componentes Rectangulares

• $\vec{R} = \vec{A} + \vec{B}$
  • $R_x = A_x + B_x$
  • $R_y = A_y + B_y$
  • $R = \sqrt{R_x^2 + R_y^2}$, $\theta = \arctan \frac{R_y}{R_x}$

Vectores Unitarios

• Un vector unitario tiene magnitud 1 y se usa para especificar una dirección.
  • $\hat{i}$: vector unitario en la dirección del eje x positivo
  • $\hat{j}$: vector unitario en la dirección del eje y positivo
  • $\vec{A} = A_x \hat{i} + A_y \hat{j}$
  • $\vec{R} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j}$

Producto de Vectores

Producto Escalar (Punto)

• El producto escalar de dos vectores es un escalar.
  • $\vec{A} \cdot \vec{B} = A B \cos \theta$
  • $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$

Propiedades del Producto Escalar

• $\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$
  • $\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}$
  • $\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1$
  • $\hat{i} \cdot \hat{j} = \hat{i} \cdot \hat{k} = \hat{j} \cdot \hat{k} = 0$

Producto Vectorial (Cruz)

• El producto vectorial de dos vectores es otro vector.
  • $\vec{A} \times \vec{B} = A B \sin \theta \hat{n}$
  • $\hat{n}$: vector unitario perpendicular al plano formado por $\vec{A}$ y $\vec{B}$

Componentes del Producto Vectorial

• $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y) \hat{i} + (A_z B_x - A_x B_z) \hat{j} + (A_x B_y - A_y B_x) \hat{k}$
  • $\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \end{vmatrix}$

Propiedades del Producto Vectorial

• $\vec{A} \times \vec{B} = - \vec{B} \times \vec{A}$
• $\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}$
• $\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0$
• $\hat{i} \times \hat{j} = \hat{k}$
• $\hat{j} \times \hat{k} = \hat{i}$
• $\hat{k} \times \hat{i} = \hat{j}$

Chemical Kinetics

Definition

• Chemical kinetics studies the rates of chemical processes.

Factors Affecting Reaction Rates

• Reactant Concentration: Higher concentration increases reaction rate due to more frequent collisions.
• Physical State and Surface Area: Reactions between different phases are limited by the contact area.
• Temperature: Higher temperature increases reaction rate by increasing collision frequency and energy.
• Presence of a Catalyst: Catalysts speed up reactions by lowering activation energy, without being consumed.

Reaction Rate

• General Reaction Formula: For the reaction $aA + bB \rightarrow cC + dD$, the rate of reaction can be expressed as: $-\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = \frac{1}{c}\frac{d[C]}{dt} = \frac{1}{d}\frac{d[D]}{dt}$

Rate Law

• Rate Law Definition: An equation that relates the rate of a reaction to the concentrations of reactants and catalysts.
• General Rate Law: $rate = k[A]^m[B]^n$
  • where:
    • $k$ is the rate constant
    • $m$ is the order of reaction with respect to reactant A
    • $n$ is the order of reaction with respect to the reactant B
  • the overall reaction order is $m+n$

Determining Reaction Order

• The reaction order must be calculated, not guessed

Integrated Rate Laws

• Integrated rate laws relate reactant concentration to time.
  • Zero-Order: $[A]_t = -kt + [A]_0$
  • First-Order: $ln[A]_t = -kt + ln[A]_0$
  • Second-Order: $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$
    • where: -$[A]_t$ is the concentration of A at time t -$[A]_0$ is the initial concentration of A -$k$ is the rate constant

Half-Life

• The half-life ($t_{1/2}$) of a reaction is the time required for the concentration of a reactant to decrease to one-half of its initial value.

Integrated Rate Law Formulas

• Zero-Order: $t_{1/2} = \frac{[A]_0}{2k}$
• First-Order: $t_{1/2} = \frac{0.693}{k}$
• Second-Order: $t_{1/2} = \frac{1}{k[A]_0}$

Reaction Mechanisms

• A reaction mechanism is the step-by-step sequence of elementary reactions by which overall chemical change occurs.
• Elementary Reactions are reactions that occur in a single step.
• Rate-Determining Step is the slowest step in a reaction mechanism, it determines the overall rate of the reaction.
• Intermediates are species that are produced in one step of a reaction mechanism and consumed in a subsequent step.

Collision Theory

• Collision theory states that chemical reactions occur when reactant molecules collide with sufficient energy and proper orientation.
• Activation Energy ($E_a$): The minimum energy required for a reaction to occur.

Arrhenius Equation

• Relates the rate constant of a reaction to the activation energy and temperature.
• Formula: $k = Ae^{-\frac{E_a}{RT}}$

Fourier Transform Properties

Linearity

• Combines signals while preserving the individual transforms.
• $F{af(t) + bg(t)} = aF(f(t)) + bF(g(t))$

Time Scaling

• Adjusts the signal's time scale, affecting its frequency spectrum.
• $F{f(at)} = \frac{1}{|a|}F(\frac{\omega}{a})$

Time Shifting

• Moves the signal in time, which introduces a linear phase shift.
• $F{f(t - t_0)} = e^{-j\omega t_0}F(\omega)$

Frequency Shifting

• Shifts the signal's frequency content, equivalent to modulation.
• $F{e^{j\omega_0 t}f(t)} = F(\omega - \omega_0)$

Time Differentiation

• Converts time differentiation to frequency multiplication.
• $F{\frac{df(t)}{dt}} = j\omega F(\omega)$

Time Integration

• Frequency division with term for DC component.
• $F{\int_{-\infty}^{t} f(\tau) d\tau} = \frac{1}{j\omega}F(\omega) + \pi F(0)\delta(\omega)$

Convolution

• Convolution in time corresponds to frequency multiplication.
• $F{f(t) * g(t)} = F(\omega)G(\omega)$

Multiplication

• Transforms time multiplication to frequency convolution.
• $F{f(t)g(t)} = \frac{1}{2\pi}[F(\omega) * G(\omega)]$

Duality

• Symmetry showing how time and frequency are interchange.
• If $f(t) \leftrightarrow F(\omega)$, then $F(t) \leftrightarrow 2\pi f(-\omega)$

Parseval's Theorem

• Signal's energy equivalence between time and frequency.
• $\int_{-\infty}^{\infty} |f(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |F(\omega)|^2 d\omega$