Simple Harmonic Motion Lab

Lab 3: Simple Harmonic Motion

• Simple harmonic motion (SHM) is periodic motion where the restoring force is proportional to displacement, resulting in oscillations around an equilibrium.
• A mass attached to a spring exemplifies SHM; Hooke's Law describes the restoring force: F = -kx.
• F is the restoring force.
• k is the spring constant (stiffness).
• x is the displacement from equilibrium.
• The negative sign indicates the force opposes the displacement.

Objectives

• Investigate displacement, velocity, and acceleration relationships in SHM.
• Determine a spring's spring constant using static and dynamic methods.
• Examine the effect of mass on the period of oscillation in a spring-mass system.

Part 1: Static Determination of Spring Constant

• Hang a spring vertically and attach a mass hanger.
• Record the initial spring position without added mass.
• Add known masses, recording the new equilibrium position for each mass (at least 5 different masses).
• Plot force (weight of added mass) versus displacement from the initial position.
• Find the slope of the graph to determine the spring constant k.

Part 2: Dynamic Determination of Spring Constant

• Attach a known mass to the spring.
• Use a motion sensor to record the mass's position as it oscillates vertically.
• Displace the mass slightly and release it, recording position changes over time.
• Determine the oscillation period T from the position vs. time graph.
• Calculate angular frequency ω using ω = (2π)/T.
• Calculate the spring constant k using k = mω^2, where m is the mass.

Part 3: Effect of Mass on Period

• Repeat the Part 2 procedure with at least 3 different masses, recording the oscillation period for each.
• Plot the square of the period T^2 versus the mass m.
• The slope of the graph should be equal to 4π^2/k.
• Compare the k value obtained from this graph with prior k values.

Data Analysis

• Static Spring Constant: Table of added mass (kg) vs. displacement (m); graph of force (N) vs. displacement (m); calculate k from the graph's slope.
• Dynamic Spring Constant: Table of mass (kg) vs. period (s); angular frequency ω for each mass; calculate k for each mass; find the average k.
• Effect of Mass on Period: Table of mass (kg) vs. period squared (s*^2*); graph of T^2 vs. m; determine the slope and calculate k.

Questions

• Compare k values from static and dynamic methods; discuss differences and error sources.
• How does oscillation period change with increased mass? Does this match theoretical predictions?
• What are real-world applications of SHM?

Lab Report

• Include Title, Abstract, Introduction, Procedure, Data and Analysis, Results, Discussion, and Conclusion sections.
• Provide background on SHM and lab objectives, detailed experimental setup and procedure descriptions.
• Report tables of raw data, graphs, calculations, and error analysis.
• Give a summary of the main findings, including the spring constant values obtained from different methods.
• Interpret the results, discuss possible error sources, and answer the questions posed.
• Summarize what was learned and the extent to which the objectives were achieved.

¿Qué es la economía?

• Economics studies how societies use scarce resources to produce valuable goods/services and distribute them among individuals.

Dos ideas clave

• Goods are scarce.
• Society must use resources efficiently.

Microeconomía

• Focuses on individual components' behavior (industries, firms, households).

Macroeconomía

• Focuses on the overall performance of the economy.

Errores comunes en el razonamiento económico

• The post hoc fallacy.
• Failure to hold other things constant.
• The fallacy of composition.

Economía positiva

• Deals with the facts and behavior of an economy.

Economía normativa

• Involves ethical value judgments and norms of fairness.

Las tres preguntas fundamentales de la organización económica

• What commodities are produced and in what quantities?
• How are things produced?
• For whom are they produced?

Economías de mercado

• Decisions are made in markets where individuals/firms voluntarily exchange goods/services, usually with money.

Economía planificada

• The government makes most economic decisions.

Economía mixta

• Most economies today contain elements of both market and planned economies.

Frontera de posibilidades de producción (FPP)

• Production-possibility frontier: Represents the maximum amounts of goods/services an economy can produce when fully and efficiently using its resources.

La FPP ilustra muchos principios económicos básicos

• Scarcity.
• Efficiency.
• Tradeoffs.
• Opportunity cost.

Costo de oportunidad

• The value of the forgone good or service.

Reaction Rates

• For a reaction aA + bB → cC + dD, the rate is given by
• Rate = -(1/a)(d[A]/dt) = -(1/b)(d[B]/dt) = (1/c)(d[C]/dt) = (1/d)(d[D]/dt).

Factors Affecting Reaction Rate

• Concentration of Reactants: Higher concentration increases rate.
• Temperature: Higher temperature increases rate.
• Surface Area of Solid Reactant: Larger area increases rate.
• Pressure of Gaseous Reactants: Higher pressure increases rate.
• Presence of Catalyst: Catalysts increase rate.
• Intensity of Radiation: Light accelerates specific reactions.

Rate Law Expression

• For a reaction aA + bB → cC + dD, Rate = k[A]^x[B]^y.
• k is the rate constant.
• x is the order with respect to A.
• y is the order with respect to B.
• x + y is overall reaction order.

Determining Rate Laws

• Initial Rates Method: Vary initial concentrations and measure initial reaction rate.
• Integrated Rate Laws: Use calculus to relate reactant concentrations to time.

Integrated Rate Law: Zero-Order Reactions

• Rate = k
• [A]_t = -kt + [A]_0
• t_{1/2} = [A]_0 / (2k)

Integrated Rate Law: First-Order Reactions

• Rate = k[A]
• ln[A]_t = -kt + ln[A]_0
• t_{1/2} = 0.693 / k

Integrated Rate Law: Second-Order Reactions

• Rate = k[A]^2
• 1 / [A]_t = kt + 1 / [A]_0
• t_{1/2} = 1 / (k[A]_0)

Collision Theory basic principles

• Molecules must collide
• Sufficient energy
• Correct orientation

Activation Energy

• Activation Energy (Ea) is the minimum required energy.
• Reactions with lower Ea progress faster.
• Arrhenius Equation: k = Ae^(-Ea/RT).

Elementary Steps

• The sequence of steps is the reaction mechanism
• The slowest step determines the overall rate

Catalysis

• A catalyst speeds up a reaction without being consumed.
• Catalysts offer an alternative reaction pathway with a lower activation energy.

Types of Catalysis

• Homogeneous Catalysis: Catalyst matches phase of reactants.
• Heterogeneous Catalysis: Catalyst differs in phase from reactants.

Reaction Orders Summary

General organization of circulatory systems principles

• Carry fluid (blood or hemolymph).
• Use tubes (vessels).
• Use a pump (heart).

Types of circulatory systems

• Gastrovascular cavity:
  • Central cavity for digestion and nutrient distribution, e.g., cnidarians, flatworms.
• Open circulatory system:
  • Circulatory fluid bathes organs directly, e.g., arthropods, mollusks.
  • Involves hemolymph.
  • Features lower pressure and less energy expenditure.
• Closed circulatory system:
  • Blood is confined to vessels and distinct from interstitial fluid, e.g., annelids, cephalopods, vertebrates.
  • Involves blood.
  • Features higher pressure and greater energy expenditure.

Overview of Vertebrate Cardiovascular Systems

• Heart has 2 or more chambers (atria receive blood; ventricles pump blood).
• Vessels include arteries (carry blood away), veins (carry blood to heart), and capillaries (allow exchange with interstitial fluid).

Circulatory systems of Fish

• 2-chambered heart (1 atrium, 1 ventricle).
• Blood passes through two capillary beds (gills, rest of body).

Circulatory systems of Amphibians

• 3-chambered heart (2 atria, 1 ventricle).
• Double circulation: Pulmocutaneous circuit (picks up O₂ through lungs and skin) and systemic circuit (delivers O₂ to body).
• Higher pressure to organs but mixing of oxygenated/deoxygenated blood in ventricle.

Circulatory systems of Reptiles

• 3-chambered heart (2 atria, 1 ventricle with partially divided septum).
• Reduced mixing of oxygenated/deoxygenated blood.

Circulatory systems of Mammals and birds

• 4-chambered heart (2 atria, 2 ventricles).
• Double circulation: pulmonary/systemic circuits.
• No mixing of oxygenated/deoxygenated blood; can supply more O₂ enabling endothermy.

Mammalian cardiovascular system - detailed look Cardiac cycle

• Systole: Heart muscle contracts.
• Diastole: Heart muscle relaxes.
• Cardiac output: blood pumped per minute; dependent on heart rate and stroke volume.
• Atrioventricular node (AV node) delays signal to apex, then Purkinje fibers trigger ventricle contraction.

Blood vessels:

• Arteries: Endothelium, smooth muscle, connective tissue.
• Veins: Same as arteries, plus valves to prevent backflow.
• Capillaries: Endothelium and basal lamina.

Blood flow:

• Velocity is slowest in capillary beds to allow material exchange.

Blood pressure:

• Systolic pressure (ventricular systole).
• Diastolic pressure (ventricular diastole).
• Influenced by cardiac output and peripheral resistance.

Blood composition:

• Plasma: Liquid matrix (water, ions, proteins, etc.).
• Cells: Erythrocytes (red blood cells for O₂ transport), leukocytes (white blood cells for defense), and platelets (blood clotting).