Supply Chain Management (SCM)

• Supply Chain (SC): A comprehensive system that includes organizations, people, activities, information, and resources involved in taking products from suppliers to customers.
• Supply Chain Management (SCM): Overseeing the flow of goods and services to maximize efficiency and customer satisfaction.

Importance of SCM

• Improves operational efficiency
• Reduces costs
• Enhances customer satisfaction
• Provides a competitive advantage

Key SCM Components

• Planning: Includes demand forecasting and capacity planning
• Sourcing: Involves selecting suppliers and contract negotiation
• Making: Encompasses production scheduling and quality control
• Delivering: Deals with logistics and distribution
• Returning: Manages returns and customer support

Main Flows of a Supply Chain

• Material Flow: The movement of raw materials, components, and finished goods
• Information Flow: The exchange of order information, demand forecasts, and inventory levels
• Financial Flow: Credit terms, payment schedules, and ownership arrangements

SCM Strategies

Inventory Management

• Just-in-Time (JIT): Aims to minimize inventory levels, reduce waste, and increase efficiency
• Vendor-Managed Inventory (VMI): The supplier manages inventory levels at the customer's location to improve turnover and reduce stockouts

Lean Manufacturing

• Focused on eliminating waste, improving efficiency, and prioritizing customer value

Agile Supply Chain

• Designed to quickly respond to demand changes with flexibility and adaptability

Bullwhip Effect

• Small demand fluctuations at retail level causing progressively larger fluctuations up the supply chain

Causes

• Lack of Information Sharing
• Order Batching
• Price Fluctuations
• Rationing and Shortage Gaming

Solutions

• Information Sharing
• Channel Alignment
• Operational Efficiency

SCM Technologies

Enterprise Resource Planning (ERP)

• Integrates all business aspects for improved data visibility and process automation

Customer Relationship Management (CRM)

• Manages customer interactions, improves satisfaction, and increases sales

SCM Software

• Manages the supply chain to improve efficiency and reduce costs

SCM Challenges

• Complexity
• Uncertainty
• Sustainability

Future of SCM

• Digitalization for greater visibility and responsiveness
• Automation to streamline operations
• Sustainability to minimize environmental impact

Chemical Reaction Engineering

• Chapter 18 focuses on the resistance to mass transfer in chemical reactions

Film Theory

Foundation and Core Idea

• Based on Whitman's work (1923)
• Suggests a stagnant film present at the interface through which mass must be transferred

Film Theory Assumptions

• Mass transfer happens through molecular diffusion in a stagnant film
• Absence of convection within the film
• Linear concentration gradient inside the film
• Equilibrium maintenance at the interface

Mass Transfer Rate

• $N_A = k_g(p_A - p_{Ai}) = k_l(c_{Ai} - c_A)$
• $N_A$ = molar flux of component A
• $k_g$, $k_l$ = gas and liquid phase mass transfer coefficients
• $p_A$, $c_A$ = bulk phase concentrations
• $p_{Ai}$, $c_{Ai}$ = interfacial concentrations

Overall Mass Transfer Coefficient

• $N_A = K_g(p_A - p_A^) = K_l(c_A^ - c_A)$
• $K_g$, $K_l$ = overall gas and liquid mass transfer coefficients
• $p_A^$, $c_A^$ = hypothetical concentrations in equilibrium with $c_A$ and $p_A$

Coefficient Relationship

• $\frac{1}{K_g} = \frac{1}{k_g} + \frac{H}{k_l}$
• $\frac{1}{K_l} = \frac{1}{Hk_g} + \frac{1}{k_l}$
• $H$ = Henry's Law constant

Penetration Theory

Origin and Primary Concept

• Introduced by Higbie (1935)
• Involves unsteady-state diffusion into a liquid element

Penetration Theory Assumptions

• Diffusion under unsteady-state conditions in a liquid element at the interface
• Consistent exposure time for each liquid element at the surface
• Preservation of equilibrium at interface
• Surface element replaced by fresh liquid from bulk

Mass Transfer Rate

• $N_A = \sqrt{\frac{D_A}{\pi t_e}}(c_{Ai} - c_A)$
• $t_e$ = exposure time

Surface Renewal Theory

Background and Basic Principle

• Proposed by Danckwerts (1951)
• Centers around unsteady-state diffusion into a liquid element at the interface

Theory Assumptions

• Unsteady-state diffusion into a liquid element for brief exposure at an interface
• $\phi(t) = se^{-st}$ age distribution function for residence times
• Maintenance of equilibrium at the interface

Mass Transfer Rate

• $N_A = \sqrt{D_A s}(c_{Ai} - c_A)$
• $s$ = fractional rate of surface renewal

October 29, 2012 - Lecture 21 Review

• Differentiability definition for $f: A \subseteq \mathbb{R}^n \rightarrow \mathbb{R}^m$ at a uses a linear transformation T

Differentiability Criteria

• $\lim_{\mathbf{h} \rightarrow \mathbf{0}} \frac{\mathbf{f(a + h)} - \mathbf{f(a)} - T(\mathbf{h})}{||\mathbf{h}||} = \mathbf{0}$
• Jacobian Matrix: $\mathbf{f'(a)} = \begin{bmatrix} \frac{\partial f_1}{\partial x_1}(\mathbf{a}) & \cdots & \frac{\partial f_1}{\partial x_n}(\mathbf{a})\ \vdots & \ddots & \vdots\ \frac{\partial f_m}{\partial x_1}(\mathbf{a}) & \cdots & \frac{\partial f_m}{\partial x_n}(\mathbf{a}) \end{bmatrix}$

Chain Rule

• If $\mathbf{f}: A \subseteq \mathbb{R}^n \rightarrow \mathbb{R}^m$ is differentiable at $\mathbf{a}$, and $\mathbf{g}: B \subseteq \mathbb{R}^m \rightarrow \mathbb{R}^k$ is differentiable at $\mathbf{f(a)}$, and $\mathbf{f(A)} \subseteq B$, then $\mathbf{g \circ f}: A \subseteq \mathbb{R}^n \rightarrow \mathbb{R}^k$ differentiable at $\mathbf{a}$
• $(\mathbf{g \circ f})'(\mathbf{a}) = \mathbf{g'(f(a))} \mathbf{f'(a)}$

Alternate Notation

• If $\mathbf{y} = \mathbf{f(x)}$ and $\mathbf{z} = \mathbf{g(y)}$, then $\frac{d\mathbf{z}}{d\mathbf{x}} = \frac{d\mathbf{z}}{d\mathbf{y}} \frac{d\mathbf{y}}{d\mathbf{x}}$

Example Task

• Finding partial derivatives with variable substitutions e.g., converting $z=f(x, y)$ to polar by using $x = r\cos\theta$, $y = r\sin\theta$

Solution Examples

• $\frac{\partial z}{\partial r} = \frac{\partial z}{\partial x} \cos\theta + \frac{\partial z}{\partial y} \sin\theta$
• $\frac{\partial z}{\partial \theta} = \frac{\partial z}{\partial x} (-r\sin\theta) + \frac{\partial z}{\partial y} (r\cos\theta)$

Higher Order Derivatives

• For $f: A \subseteq \mathbb{R}^n \rightarrow \mathbb{R}$, if $\frac{\partial f}{\partial x_i}$ is differentiable

Notation

• The notation for second-order partial derivative: $\frac{\partial}{\partial x_j} \left( \frac{\partial f}{\partial x_i} \right) = \frac{\partial^2 f}{\partial x_j \partial x_i}$

Clairaut's Theorem

• If $\frac{\partial^2 f}{\partial x_j \partial x_i}$ and $\frac{\partial^2 f}{\partial x_i \partial x_j}$ are continuous, $\frac{\partial^2 f}{\partial x_j \partial x_i}(\mathbf{a}) = \frac{\partial^2 f}{\partial x_i \partial x_j}(\mathbf{a})$

Example

• Deriving mixed partial derivatives, showing derivatives are equal

Derived Equations

• $\frac{\partial^2 f}{\partial y \partial x} = \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) = 2x + e^{xy} + xy e^{xy} - 2x\sin(x^2 + y)$
• $\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial y} \right) = 2x + e^{xy} + xe^{xy} - 2x\sin(x^2 + y)$

Taylor's Theorem

• Describes function approximation using derivatives at a single point

Univariable case

• $f(x) = f(a) + f'(a)(x - a) + \varphi_a (x) (x - a)$ with $\lim_{x \rightarrow a} \varphi_a(x) = 0$

Further function approximation

• $f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2 + \psi_a(x) (x - a)^2$ with $\lim_{x \rightarrow a} \psi_a(x) = 0$

Multivariable Version

• Expands around a point with partial derivatives and a remainder term

Multivariable Theorem

${f(\mathbf{x}) = f(\mathbf{a}) + \sum_{i=1}^{n} \frac{\partial f}{\partial x_i}(\mathbf{a})(x_i - a_i) + \frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n} \frac{\partial^2 f}{\partial x_i \partial x_j}(\mathbf{a})(x_i - a_i)(x_j - a_j) + \Psi_{\mathbf{a}}(\mathbf{x}) ||\mathbf{x - a}||^2}$

$\lim_{\mathbf{x} \rightarrow \mathbf{a}} \Psi_{\mathbf{a}}(\mathbf{x}) = 0$

Quadratic Form

• Expression involving second-order terms.

Quadratic Form Equation

• $Q(\mathbf{x}) = \mathbf{x}^T A \mathbf{x}$ where $A$ is a symmetric matrix

Rewritten Taylor’s Theorem

• Taylor’s theorem can be rewritten as $f(\mathbf{x}) = f(\mathbf{a}) + \sum_{i=1}^{n} \frac{\partial f}{\partial x_i}(\mathbf{a})(x_i - a_i) + \frac{1}{2} (\mathbf{x - a})^T H_{\mathbf{a}} (\mathbf{x - a}) + \Psi_{\mathbf{a}}(\mathbf{x}) ||\mathbf{x - a}||^2$

Física - Tarea 1

Problemas a Resolver

1. Peso en la Luna:

   • Gravedad en la Luna ($g_L$) = $1.62 \ m/s^2$
   • Peso en la Tierra ($P_T$) = 900 N
   • Determinar el peso en la Luna ($P_L$)

   Ecuación: $P_L = m \cdot g_L$, donde $m = \frac{P_T}{g_T}$ y $g_T = 9.8 \ m/s^2$ (gravedad en la Tierra)

2. Análisis Dimensional:

   • Demostrar dimensionalmente que $x = v_0t + \frac{1}{2}at^2$ es correcta.

   Análisis:

   • $[x] = L$ (Longitud)
   • $[v_0t] = LT^{-1} \cdot T = L$
   • $[\frac{1}{2}at^2] = LT^{-2} \cdot T^2 = L$

   Conclusión: La suma de términos con dimensión de longitud resulta en una longitud; por lo tanto, es dimensionalmente correcta.

3. Movimiento en Dos Dimensiones:

   • Parte 1: Velocidad hacia el este ($v_E$) = $50 \ km/h$ durante 1 hora.
   • Parte 2: Velocidad hacia el norte ($v_N$) = $80 \ km/h$ durante 30 minutos (0.5 horas).

   a) Distancias recorridas:

   • Distancia al este ($d_E$) = $v_E \cdot t_E = 50 \ km$
   • Distancia al norte ($d_N$) = $v_N \cdot t_N = 40 \ km$

   b) Velocidad media en todo el recorrido:

   • Desplazamiento total ($\vec{d}$) = $\sqrt{d_E^2 + d_N^2} = \sqrt{50^2 + 40^2} \ km$
   • Tiempo total ($t_{total}$) = 1 hora + 0.5 horas = 1.5 horas
   • Velocidad media ($\vec{v}{avg}$) = $\frac{\vec{d}}{t{total}}$

   c) Rapidez media en todo el recorrido:

   • Distancia total ($d_{total}$) = $d_E + d_N = 50 + 40 \ km$
   • Rapidez media ($v_{avg}$) = $\frac{d_{total}}{t_{total}}$

4. Vectores en Componentes:

   • $\vec{A}$ componentes: $A_x = -8.7 \ cm$, $A_y = 15 \ cm$
   • $\vec{B}$ componentes: $B_x = 13.2 \ cm$, $B_y = -6.6 \ cm$
   • $\vec{A} - \vec{B} + 3\vec{C} = 0$, encontrar las componentes de $\vec{C}$ $\vec{C} = \frac{\vec{B} - \vec{A}}{3}$

5. Movimiento Acelerado en Dos Dimensiones:

   • Velocidad inicial ($\vec{v}_0$) = $2.0 \times 10^6 \ m/s$ en la dirección x
   • Aceleración constante ($\vec{a}$) = $3.0 \times 10^{12} \ m/s^2$ en la dirección y
   • Coordenada x del electrón = 4.0 cm

   Fórmulas:

   • $x = v_{0x} \cdot t$, encontrar $t$ usando $x = 0.04 \ m$
   • $y = v_{0y} \cdot t + \frac{1}{2} a_y t^2$, calcular $y$ usando el valor de $t$ obtenido.

Heat Transfer

• Heat transfer is the thermal energy in transit due to a temperature difference.

Heat Transfer Modes

Conduction

• Energy transfer between particles due to interactions
• Fourier's Law, $\dot{q}_x'' = -k\frac{dT}{dx}$
  • $\dot{q}_x''$ is heat flux in $\left[\frac{W}{m^2}\right]$
  • $k$ is thermal conductivity in $\left[\frac{W}{mK}\right]$
  • $\frac{dT}{dx}$ is temperature gradient in $\left[\frac{K}{m}\right]$

Convection

• Energy transfer between surface and moving fluid at different temperatures
• Newton's Law of Cooling, $\dot{q}'' = h(T_s - T_{\infty})$
  • $\dot{q}''$ is heat flux in $\left[\frac{W}{m^2}\right]$
  • $h$ is convection heat transfer coefficient in $\left[\frac{W}{m^2K}\right]$
  • $T_s$ is surface temperature in $[K]$
  • $T_{\infty}$ is fluid temperature in $[K]$

Radiation

• Energy emitted due matter due to change in electron configurations
• Stefan-Boltzmann Law, $\dot{q}'' = \epsilon \sigma T_s^4$
  • $\dot{q}''$ is heat flux in $\left[\frac{W}{m^2}\right]$
  • $\epsilon$ is emissivity of the surface
  • $\sigma$ is Stefan-Boltzmann constant, $\sigma = 5.67 \times 10^{-8} \frac{W}{m^2K^4}$
  • $T_s$ is surface temperature in $[K]$

Other Terms

Thermal Resistance

• Calculated with formula $R = \frac{\Delta T}{\dot{Q}}$
• $R$ is the thermal resistance in $\left[\frac{K}{W}\right]$
• $\Delta T$ is the temperature difference in $[K]$
• $\dot{Q}$ is the heat transfer rate in $[W]$

Contact Resistance

• It exists when two materials are in contact due to imperfect surface contact
• It can be minimized by using using thermal grease

Overall Heat Transfer Coefficient

• Calculated with formula $U = (\frac{1}{h_i} + \sum \frac{L}{k} + \frac{1}{h_o})^{-1}$ $U$: overall heat transfer coefficient $\left[\frac{W}{m^2K}\right]$
• $h_i$: inner heat transfer coefficient $\left[\frac{W}{m^2K}\right]$
• $L$: thickness of the material $[m]$
• $k$: thermal conductivity $\left[\frac{W}{mK}\right]$
• $h_o$: outer heat transfer coefficient $\left[\frac{W}{m^2K}\right]$

What is Economics?

• The study of how societies use scarce resources to make valuable goods and services
• How they distribute them among different individuals

Key Ideas

• Resources or goods are limited or scarce
• Society must make efficient use of limited resources

Branches of Economics

• Microeconomics: Focuses on behavior of aspects of an economy, such as markets, firms, and households.
• Macroeconomics: Concentrates on overall performance of the economy.

Central Questions of Economics

• What goods and services are produced, and in what quantities?
• How are these goods produced?
• For whom are the goods produced; how are they distributed?

Economic Types

• Market Economy: Resources allocated via markets where individuals and firms voluntarily agree to exchange goods and usually through monetary payment.
• Authoritarian Economy: The government makes most economic decisions.
• Mixed Economies: Combines market and authoritarian economics

Technological Possibilities of Society

• Every economy is limited by resources like labor, technical knowledge, and capital.

Inputs and Products

• Inputs: Services that are used to create goods and services
• Products: Resulting goods or services from the production process, consumed or used for further production

Production Possibility Frontier (PPF)

• Represents maximum quantities an economy can produce, given its technological knowledge and quantity of inputs

Example

• A point system represents examples of what resources can be dedicated to and what they can output (listed in document)

Key Terms

• Opportunity Cost: The value of the next best alternative when making a decision.
• Efficiency: When the most output is being created with the least amount of resources

Efficiency

• Productive Efficiency:* Achieved when economy cannot produce more of one good without reducing production of another; economy located on its PPF.

Unidad 1: Vectores

• Introducción a las magnitudes físicas y operaciones vectoriales.

1.1 Magnitudes Escalares y Vectoriales

• Distinción entre magnitudes que se describen con un valor y aquellas que requieren dirección y sentido.

Magnitud Escalar

• Definición: Queda completamente determinada con un número y sus unidades. Ejemplos: Masa, temperatura, energía, tiempo, volumen, densidad.

Magnitud Vectorial

• Definición: Requiere módulo, dirección y sentido. Ejemplos: Velocidad, aceleración, fuerza, peso, desplazamiento, impulso.

1.2 Representación de Vectores

• Formas de representar vectores geométrica y analíticamente.

Geométrica

• Un vector es una flecha con: Módulo, dirección, sentido y punto de aplicación. Descripción: $\vec{A}$ = Vector A. $|\vec{A}| = A =$ Módulo del vector A

Analítica

• Se representa con sus componentes en un sistema de coordenadas. Tipos:
• Cartesianas: $\vec{A} = (A_x, A_y, A_z)$.
• Polares:* $\vec{A} = (A, θ)$.

Relación entre coordenadas cartesianas y polares

$A_x = A \cdot cos θ$, $A_y = A \cdot sen θ$, $A = \sqrt{A_x^2 + A_y^2}$, $θ = arctan (\frac{A_y}{A_x})$.

Vectores unitarios cartesianos

Vectores de módulo 1 en la dirección de los ejes coordenados ($\hat{i}, \hat{j}, \hat{k}$).

Expresión de un Vector

• Mediante vectores unitarios cartesianos: $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$.

1.3 Operaciones con Vectores

• Suma, resta, producto escalar y vectorial.

Suma de vectores

• Gráfica: Métodos del paralelogramo y poligonal.
• Analítica:* Sumar las componentes correspondientes.
• $\vec{A} + \vec{B} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j}$.

Resta de Vectores

• Restar las componentes correspondientes: $\vec{A} - \vec{B} = (A_x - B_x) \hat{i} + (A_y - B_y) \hat{j}$.

Producto de un Escalar por un Vector

• El producto de un escalar por un vector da otro vector con la misma dirección pero diferente módulo.
• $k \vec{A} = k A_x \hat{i} + k A_y \hat{j}$

Producto Escalar (Producto Punto)

• Definición: Da como resultado un escalar. Fórmulas: $\vec{A} \cdot \vec{B} = A B cos θ = A_x B_x + A_y B_y$.

Propiedades del Producto Escalar

Conmutativa y distributiva; vectores perpendiculares si el producto es cero; vectores unitarios ($\hat{i} \cdot \hat{i} = 1$, $\hat{i} \cdot \hat{j} = 0$).

Producto Vectorial (Producto Cruz)

• Definición: Resulta en un vector perpendicular a ambos vectores originales. Fórmulas:
• $\vec{A} \times \vec{B} = A B sen θ \hat{n}$. $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y) \hat{i} + (A_z B_x - A_x B_z) \hat{j} + (A_x B_y - A_y B_x) \hat{k}$.

Propiedades del Producto Vectorial

• No conmutativo, distributivo; vectores paralelos si el producto es cero; vectores unitarios ($\hat{i} \times \hat{i} = 0$, $\hat{i} \times \hat{j} = \hat{k}$).

Fonction Exponentielle : Study notes

I. Définition

• The exponential function (exp) is the only differentiable function on $\mathbb{R}$.
• It is defined with $\exp' = \exp$ and $\exp(0) = 1$.
• For real number x, $\exp(x) = e^x$ with $e = \exp(1) \approx 2.718$.

II. Propriétés

1. Propriétés Algébriques

• For real numbers $x$ and $y$:
• $e^{x+y} = e^x \times e^y$
• $e^{x-y} = \frac{e^x}{e^y}$
• $e^{-x} = \frac{1}{e^x}$
• $(e^x)^y = e^{xy}$

2. Propriétés Analytiques

• The exponential function is positive on $\mathbb{R}$. For all real numbers $x, e^x > 0$
• It is strictly increasing on $\mathbb{R}$.
• $\lim_{x \to -\infty} e^x = 0$
• $\lim_{x \to +\infty} e^x = +\infty$

3. Dérivée

• The exponential function is differentiable on $\mathbb{R}$, and its derivative is itself: $(e^x)' = e^x$
• More generally, if $u$ is a differentiable function on an interval $I \Rightarrow (e^u)' = u'e^u$

4. Représentation Graphique

• The graph of the exponential function is above the x-axis and increasing.
• It passes through the point $(0; 1)$.

5. Équations et Inéquations

• For real numbers $x$ and $y$:
• $e^x = e^y \Leftrightarrow x = y$
• $e^x < e^y \Leftrightarrow x < y$
• $e^x > e^y \Leftrightarrow x > y$

III. Exercices Types

1. Résolution d'équations

• Example: Solve equation $e^{x^2 - x} = 1$.
• $0$ and $1$ are the solutions.

2. Dérivation de fonctions

• Example: Let $f(x) = e^{x^2 + 1}$
• Calculate $f'(x)$
• Sol: $f'(x) = (x^2 + 1)' e^{x^2 + 1} = 2x e^{x^2 + 1}$

3. Limites de fonctions

• Example: Limit calculation $\lim_{x \to +\infty} e^{-x}$.
• Sol: $\lim_{x \to +\infty} e^{-x} = \lim_{x \to +\infty} \frac{1}{e^x} = 0$ since $\lim_{x \to +\infty} e^x = +\infty$.

Lecture 19: Mutual Information

Motivation

• Measuring how much one random variable tells about another
• Measuring uncertainty reduction about one random variable (Y) given another (X)

Quantified by

• Mutual Information

Definition 1

• $I(X; Y) = H(Y) - H(Y|X)$
  • $H(Y)$ is the entropy of Y
  • $H(Y|X)$ conditional entropy of Y given X

Definition 2

• $I(X;Y) = H(X) - H(X|Y)$
  • $H(X)$ is the entropy of X
  • $H(X|Y)$ is the conditional entropy of X given Y

Definition 3

• $I(X; Y) = \sum_{x \in \mathcal{X}} \sum_{y \in \mathcal{Y}} p(x,y) log\frac{p(x,y)}{p(x)p(y)}$
  • $\mathcal{X}$ support of X, $\mathcal{Y}$ support of Y
  • $p(x,y)$ joint pmf of X and Y
  • $p(x)$ marginal pmf of X, $p(y)$ marginal pmf of Y

Example 1 Binary symmetric channel (BSC) with crossover probability ϵ

• $X \in {0,1}$ Input
• $Y \in {0,1}$ Output
• $p(Y=1|X=0) = \epsilon \text{ Noise}$

Example Question

• Assume $X \sim \text{Bernoulli}(\frac{1}{2})$, i.e., $p(X=0) = p(X=1) = \frac{1}{2}$. What is $I(X;Y)$?

Answer

• The answer is $I(X;Y) = 1 - H(\epsilon)$

Properties of Mutual Information

• $I(X;Y) = I(Y;X)$ (Symmetry)
• $I(X;Y) \geq 0$
• $I(X;Y) = H(X) + H(Y) - H(X,Y)$
• $I(X;X) = H(X)$
• $I(X;Y|Z) = H(X|Z) - H(X|Y,Z)$

Chain rule

• $I(X; Y_1, Y_2,..., Y_n) = I(X; Y_1) + I(X; Y_2 | Y_1) +... + I(X; Y_n | Y_1,..., Y_{n-1})$

Relationship to KL Divergence

• $I(X;Y) = D(p(x,y) || p(x)p(y))$
• Mutual information is the KL divergence between joint distribution and product of marginal distributions

Lab 2: Measurements

Objectives

• Familiarization w/ lab measuring devices.
• Estimating measurement uncertainty and propagating it

Measuring

• Accurate measurement skills are useful in science.
• Every measurement has an uncertainty.

Equipment

• Ruler, DMM, stopwatch, triple beam balance, cylindrical metal object
• Vernier caliper or micrometer

Procedures

Measuring Length via equipment

• Ruler to measure length, width, thickness of object
• Caliper to measure length, width, thickness of object
  • Record w/ estimation.

Measuring Mass via equipment

• Triple Beam Balance to measure object.
• Record w/ estimation.

Measuring Time via equipment

• Stopwatch
  • Record values. Calculate the average and SD to est. the uncertainty.

Measuring Resistance via equipment

• DMM to measure resistor resistance.

Equations for measurement

Volume Calc

• Volume = Length * Width * Thickness, $V = l * w * t$

Uncertainty Volume

• $\sigma_V = V * \sqrt{(\frac{\sigma_l}{l})^2 + (\frac{\sigma_w}{w})^2 + (\frac{\sigma_t}{t})^2}$

Mass Calc

• Use above measurements calculate with Density = Mass/Volume, $ρ = \frac{m}{V}$

Uncertainty Mass

• $\sigma_ρ = ρ * \sqrt{(\frac{\sigma_m}{m})^2 + (\frac{\sigma_V}{V})^2}$

Sample Questions

List of Questions

• Which measuring device (ruler or caliper) gave you a more precise measurement of the dimensions of the metal object? Explain your answer
• How does the uncertainty in your measurements affect the uncertainty in your calculated values?
• What are some possible sources of error in this experiment?
• How could you reduce the uncertainty in your measurements?