Work and Energy

Work Done by a Constant Force

• Work equals force dotted with displacement: $W = F \cdot d = |F||d|cos\theta$
• The value of work done ($W$) can be postive, negative or zero based on the angle ($\theta$)

Kinetic Energy and the Work-Energy Theorem

• Kinetic Energy is the is equal to $K = \frac{1}{2}mv^2$
• The network done on an object is equal to the change in kinetic energy: $W_{net} = \Delta K = K_f - K_i$
• Kinetic energy increases when $W_{net} > 0$
• Kinetic energy decreases when $W_{net} < 0$

Work Done by a Varying Force

• In one dimension, Work is equal to $W = \int_{x_i}^{x_f} F_x dx$
• Work done is the area under the force vs position curve

Potential Energy

• Potential energy relates to the system's configuration
• Potential energy converts into kinetic energy
• Potential energy is defined for conservative forces

Gravitational Potential Energy

• Gravitational potential energy is equal to $U_g = mgy$

Elastic Potential Energy

• Elastic potential energy is equal to $U_s = \frac{1}{2}kx^2$

Conservative Forces

• Work is independent of path
• Work done is equal to the negative change in potential energy, $W_c = -\Delta U$
• Gravity, spring force, and electromagnetic forces are examples of conservative forces

Non-Conservative Forces

• Work depends on path
• Work is not equal to the negative change in potential energy
• Friction, air resistance, and tension are examples of non-conservative forces

Conservation of Energy

• Only conservative forces present: $E_i = E_f$, which is: $K_i + U_i = K_f + U_f$
• Non-conservative forces present: $\Delta E = W_{nc}$, which is: $\Delta K + \Delta U = W_{nc}$

Power

• Rate at which work is done
• Average power: $P_{avg} = \frac{W}{\Delta t}$
• Instantaneous power: $P = \frac{dW}{dt} = F \cdot v$
• Units: Watts (W)

Lab 1: Introduction to Vectors

Learning Objectives

• Perform vector math
• Calculate the magnitude and direction of a vector
• Resolve a vector into components in 2D and 3D
• Perform dot and cross products of vectors

Definition of a Vector

• A vector is a physical quantity with both magnitude and direction.
• Vectors are typically represented by arrows.

Vector Notation

• Vectors are commonly denoted in several ways.
• Using boldface letters: v
• Using an arrow above the letter: $\overrightarrow{v}$
• Specifying components in a coordinate system: $\overrightarrow{v} = (v_x, v_y, v_z)$

Vector Addition

• Add corresponding components: $\overrightarrow{c} = (a_x + b_x, a_y + b_y, a_z + b_z)$

Vector Subtraction

• Subtract corresponding components: $\overrightarrow{c} = (a_x - b_x, a_y - b_y, a_z - b_z)$

Scalar Multiplication

• Multiply each component by the scalar: $\overrightarrow{b} = (ka_x, ka_y, ka_z)$

Magnitude of a Vector

• Calculated by: $|\overrightarrow{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$
• In 2D: $|\overrightarrow{v}| = \sqrt{v_x^2 + v_y^2}$

Direction of a Vector

• In 2D, direction is given by the angle $\theta$ with the positive x-axis: $\theta = \arctan(\frac{v_y}{v_x})$

Resolving a Vector into Components

• In 2D:
  • $v_x = |\overrightarrow{v}|\cos(\theta)$
  • $v_y = |\overrightarrow{v}|\sin(\theta)$

Definition of Dot Product

• $\overrightarrow{a} \cdot \overrightarrow{b} = |\overrightarrow{a}||\overrightarrow{b}|\cos(\theta)$
• In component form: $\overrightarrow{a} \cdot \overrightarrow{b} = a_xb_x + a_yb_y + a_zb_z$

Properties of Dot Product

• Commutative: $\overrightarrow{a} \cdot \overrightarrow{b} = \overrightarrow{b} \cdot \overrightarrow{a}$
• Distributive: $\overrightarrow{a} \cdot (\overrightarrow{b} + \overrightarrow{c}) = \overrightarrow{a} \cdot \overrightarrow{b} + \overrightarrow{a} \cdot \overrightarrow{c}$
• Orthogonal if $\overrightarrow{a} \cdot \overrightarrow{b} = 0$

Definition of Cross Product

• $|\overrightarrow{c}| = |\overrightarrow{a}||\overrightarrow{b}|\sin(\theta)$
• In component form: $\overrightarrow{a} \times \overrightarrow{b} = (a_yb_z - a_zb_y, a_zb_x - a_xb_z, a_xb_y - a_yb_x)$

Properties of Cross Product

• Anti-commutative: $\overrightarrow{a} \times \overrightarrow{b} = -\overrightarrow{b} \times \overrightarrow{a}$
• Distributive: $\overrightarrow{a} \times (\overrightarrow{b} + \overrightarrow{c}) = \overrightarrow{a} \times \overrightarrow{b} + \overrightarrow{a} \times \overrightarrow{c}$
• Parallel if $\overrightarrow{a} \times \overrightarrow{b} = 0$

Cardiovascular System

Blood Vessels

Arteries

• Carry blood away from the heart

Artery Walls

• Walls have 3 layers:
  • Tunica intima: endothelium
  • Tunica media: smooth muscle (vasoconstriction/vasodilation)
  • Tunica externa: connective tissue

Key Artery Types

Anastomoses

• Vessels unite

Capillaries

• Smallest vessels
• Walls: endothelium + basement membrane
• Exchange of gases, nutrients, wastes, hormones

Capillary Types

• Continuous: most tissues; tight junctions, intercellular clefts
• Fenestrated: more permeable; pores
• Sinusoid: most permeable; large fenestrations; liver, bone marrow

Veins

• Carry blood toward the heart
• Walls have 3 layers (thinner than arteries)
• Low pressure
• Valves prevent backflow
• Venules: smallest veins
• Venous sinuses: flattened veins

Blood Flow

Definition

• Volume of blood flowing through a vessel, organ, or entire circulation in a given period
• Relatively constant at rest
• Varies widely with activity

Factors Affecting Blood Flow

• Blood pressure (BP): Force per unit area exerted on a vessel wall by the contained blood, measured in mm Hg.
• Resistance: Opposition to flow

Sources of Resistance

• Blood viscosity
• Vessel length
• Vessel diameter (greatest influence)
• Blood flow is directly proportional to blood pressure gradient
• Blood flow is inversely proportional to resistance

Blood Pressure

Systolic Pressure

• Pressure exerted during ventricular contraction

Diastolic Pressure

• Pressure exerted during ventricular relaxation

Pulse Pressure

• systolic - diastolic

Mean Arterial Pressure

• MAP = diastolic pressure + 1/3 pulse pressure

Regulation of Blood Pressure

Short-term Regulation

• Nervous system (neural controls)
  • Baroreceptors
  • Chemoreceptors
  • Higher brain centers
• Bloodborne chemicals (hormonal controls)
  • Epinephrine, norepinephrine
  • Atrial natriuretic peptide (ANP)
  • Antidiuretic hormone (ADH)
  • Angiotensin II
  • Endothelium-derived factors

Long-term Regulation

• Renal regulation (blood volume)
  • Direct: alters blood volume independently of hormones
  • Indirect: renin-angiotensin-aldosterone mechanism

Velocity of Blood Flow

• Fastest in aorta
• Slowest in capillaries
• Increases again in veins

Capillary Exchange

• Gases, nutrients, wastes
• Diffusion
• Vesicular transport
• Bulk flow:
  • Fluid leaves capillaries at arterial end (BP higher)
  • Fluid enters capillaries at venous end (osmotic pressure higher)

Circulatory Pathways

• Pulmonary circuit: heart -> lungs -> heart
• Systemic circuit: heart -> tissues -> heart

Special Circulations

• Hepatic portal system
• Fetal circulation

Reglas de Inferencia (Inference Rules)

Definition of Inference Rule

• A scheme for constructing valid arguments to derive conclusions from premises.

Modus Ponens (MP)

• If $P \rightarrow Q$ is true, and $P$ is true, then $Q$ is true. $$\frac{P \rightarrow Q, P}{\therefore Q}$$

Modus Tollens (MT)

• If $P \rightarrow Q$ is true, and $Q$ is false, then $P$ is false. $$\frac{P \rightarrow Q, \neg Q}{\therefore \neg P}$$

Silogismo Hipotético (Hypothetical Syllogism - HS)

• If $P \rightarrow Q$ and $Q \rightarrow R$ are true, then $P \rightarrow R$ is true. $$\frac{P \rightarrow Q, Q \rightarrow R}{\therefore P \rightarrow R}$$

Silogismo Disyuntivo (Disjunctive Syllogism - DS)

• If $P \lor Q$ is true, and $P$ is false, then $Q$ is true. $$\frac{P \lor Q, \neg P}{\therefore Q}$$

Adición (Addition - ADD)

• If $P$ is true, then $P \lor Q$ is true. $$\frac{P}{\therefore P \lor Q}$$

Simplificación (Simplification - SIMP)

• If $P \land Q$ is true, then $P$ is true. $$\frac{P \land Q}{\therefore P}$$

Conjunción (Conjunction - CONJ)

• If $P$ is true and $Q$ is true, then $P \land Q$ is true. $$\frac{P, Q}{\therefore P \land Q}$$

Resolución (Resolution - RES)

• If $P \lor Q$ and $\neg P \lor R$ are true, then $Q \lor R$ is true. $$\frac{P \lor Q, \neg P \lor R}{\therefore Q \lor R}$$