# Pressure. Pressure. Atmospheric pressure. Conceptual example 1: Blood pressure. Pressure is force per unit area:

Save this PDF as:

Size: px
Start display at page:

Download "Pressure. Pressure. Atmospheric pressure. Conceptual example 1: Blood pressure. Pressure is force per unit area:"

## Transcription

1 Pressure Pressure is force per unit area: F P = A Pressure Te direction of te force exerted on an object by a fluid is toward te object and perpendicular to its surface. At a microscopic level, te force is associated wit te atoms and molecules in te fluid bouncing elastically from te surfaces of te object. Te SI unit for pressure is te pascal. Pa = N/m Te arrows below illustrate te directions of te force of pressure acting on a sark at various point of its body. Tey sow tat te force is always pointing towards and perpendicular to te surface of te sark. Atmosperic pressure atm =.0 x 0 5 Pa = 4.7 psi = 760 torr (psi = pounds / square inc) At atmosperic pressure, every square meter as a force of 00,000 N exerting on it, coming from air molecules bouncing off it! Wy don t we, and oter tings, collapse because of tis pressure? We, uman, ave an internal pressure of atmospere. General solid objects do not collapse because tey possess an elastic modulus (i.e., rigidity) enabling tem to deform only 4 little wen compressed by te atm. pressure. Conceptual example : Blood pressure Conceptual example. Rank by pressure Question A typical reading for blood pressure is 0 over 80. Wat do te two numbers represent? Wat units are tey in? Ans. 0 mm Hg (millimeters of mercury) is a typical systolic pressure, te pressure wen te eart contracts. 80 mm Hg is a typical diastolic pressure, te blood pressure wen te eart relaxes after a contraction. 760 mm Hg is te typical atmosperic pressure. Te blood pressure readings are not absolute tey tell us ow muc above atmosperic pressure te blood pressure is. 5 A container, closed on te rigt side but open to te atmospere on te left, is almost completely filled wit water, as sown. Tree points are marked in te container. Rank tese according to te pressure at te points, from igest pressure to lowest.. A = B > C. B > A > C. B > A = C 4. C > B > A 5. C > A = B 6. some oter order See explanation on te next page. P B P C P B 6

2 Pressure in a static fluid Pressure in a Static Fluid A static fluid is a fluid at rest. In a static fluid: Pressure increases wit dept. Two points at te same vertical position experience te same pressure, no matter wat te sape of te container. If point is a vertical distance below point, and te pressure at point is P, te pressure at point is: P = P+ ρg Point does not ave to be directly below point - wat matters is te vertical distance. To understand te equation, P = P + ρg, consider a column of fluid wit area A and eigt as sown. Te weigt of te fluid column is ρga. Tis weigt will cause an extra pressure of ρga/a = ρg at te bottom of te fluid column and explains te equation. Area, A P P 7 8 Measuring pressure Conceptual example b. Rank by pressure Te relationsip between pressure and dept is exploited in manometers (or barometers) tat measure pressure. A standard barometer is a tube wit one end sealed. Te sealed end is close to zero pressure, wile te oter end is open to te atmospere. Te pressure difference between te two ends of te tube can maintain a column of fluid in te tube, wit te eigt of te column being proportional to te pressure difference. P = P+ρ g P (usually ~ 0) P 9 Wat is te pressure at tis level?. Cannot be determined. P A ρ fluid g(0.m). P A + ρ fluid g(0.m) 4. Atmosperic pressure 5. and 4 6. and 4 0 Example. Water pressure At te surface of a body of water, te pressure you experience is atmosperic pressure. Estimate ow deep you ave to dive to experience a pressure of atmosperes. Given tat density of water is 000 kg/m and g = 0 m/s (or N/kg). Solution: P = P+ ρg Pa = Pa + (000 kg/m ) (0 N/kg) works out to 0 m. Every 0 m down in water increases te pressure by atmospere. Te origin of te buoyant force P = P+ ρg Te net upward buoyant force is te vector sum of te various forces from te fluid pressure. Because te fluid pressure increases wit dept, te upward force on te bottom surface is larger tan te downward force on te upper surface of te immersed object. F = Δ P A= ρ g A= ρ gv net fluid fluid Tis is for a fully immersed object. For a floating object, is te eigt below te water level, so we get: F = ρ gv net fluid disp

3 Conceptual example 4. Wen te object goes deeper Question: If we displace an object deeper into a fluid, wat appens to te buoyant force acting on te object? You may assume tat te fluid density is te same at all depts. Te buoyant force Conceptual example 4. Wen te object goes deeper (cont d) Explanation: P = P+ ρg. increases. decreases. stays te same If te fluid density does not cange wit dept, all te forces increase by te same amount, leaving te buoyant force uncanged! 4 Fluid Dynamics Flowing fluids fluid dynamics We ll start wit an idealized fluid tat:. Has streamline flow (no turbulence). Is incompressible (constant density). Has no viscosity (flows witout resistance) 4. Is irrotational (no swirling eddies) 5 6 Te continuity equation We generally apply two equations to flowing fluids. One comes from te idea tat te rate at wic mass flowing past a point is constant, oterwise fluid builds up in regions of low flow rate. Δm ρδv ρaδx mass flow rate = = = = ρav Δt Δt Δt Te mass flow rate of a fluid must be continuous. Oterwise fluid accumulates at some points. So, we ave: ρ A v = ρ A v Density of fluid Av= Av (continuity equation) doesn t cange 8 An application of continuity Te concept of continuity (a constant mass flow rate) is applied in our own circulatory systems. Wen blood flows from a large artery to small capillaries, te rate at wic blood leaves te artery equals te sum of te rates at wic blood flows troug te various capillaries attaced to te artery. Te fluid flows faster in narrow sections of te tube. 7

4 Applying energy conservation to fluids Our second equation for flowing fluids comes from energy conservation. Bernoulli s equation U + K + W nc = U + K P A P A mgy + mv + W = mgy + mv nc ρgy + ρv + P = ρgy + ρv + P W = FΔx F Δ x = PAΔx P A Δx nc mgy + mv + P A Δ x = mgy + mv + P A Δx Dividing by volume: ρgy + ρv + P = ρgy + ρv + P (Bernoulli s equation) 9 0 Simple Interpretation of Bernoulli s equation W nc by te forces of pressure in moving a unit te fluid from point to point. ρ gy + ρv + P + ( P ) = ρgy + ρv question lower tan tat of points A and B. If te fluid is at rest, rank te points based on teir pressure. PE per unit point KE per unit point PE per unit point KE per unit point. Equal for all four. A>B=C>D v A > v B P A < P B question lower tan tat of points A and B. If te fluid is flowing from left to rigt, rank te points based on te fluid speed. question By similar argument, P C < P D lower tan tat of points A and B. If te fluid is flowing from left to rigt, rank te points based on te pressure.. Equal for all four. A>B=C>D. Equal for all four. A>B=C>D Use te concept of continuity to figure out tis problem. 4 4

5 In going from B to A, we still ave v > v P B > P A question 4 lower tan tat of points A and B. If te fluid is flowing from rigt to left, rank te points based on te pressure.. Equal for all four. A>B=C>D P B -P A By similar argument, PD > P C P C -P D Te answer ere being te same as tat to question is because Bernoulli s 5 equation, depending on v and v, does not depend on te sign of te velocities. Tree oles in a cylinder A cylinder, open to te atmospere at te top, is filled wit water. It stands uprigt on a table. Tere are tree oles on te side of te cylinder, but tey are covered to start wit. One ole is /4 of te way down from te top, wile te oter two are / and /4 of te way down. Wen te oles are uncovered, water soots out. Wic ole soots te water fartest orizontally on te table?. Te ole closest to te top.. Te ole alfway down.. Te ole closest to te bottom. 4. It's a tree-way tie. 6 Tree oles in a cylinder Te pressure at te top of te cylinder is te atmosperic pressure (P atm ) since te cylinder is open to te atmospere and normally te fluid speed tere is zero. Similarly, te pressure at point is also P atm. P atm At point : P = P atm + ρg v = 0 At point : P = P atm ρgy+ ρv + P = ρgy + ρv + P x x (y = y ) P atm Hence, v = (g) / It sows tat te deeper te ole is, te iger te speed of te fluid will be wen it emerges from te ole. 7 A flexible tube can be used as a simple sipon to transfer fluid from one container to a lower container. Te fluid as a density of 000. kg/m. See te dimensions given in te figure, and take atmosperic pressure to be 00 kpa and g = 0 m/s. If te tube as a cross-sectional area tat is muc smaller tan te cross sectional area of te iger container, wat is te absolute pressure at: (a) Point P? (b) Point Z 00 kpa 00 kpa Wat is te speed of : P (c) Point Z? (d) Point Y? (e) Point? All tree are te same equal.46 m/s Wat is te absolute pressure at: (e) Point Y? (f) Point? 9 kpa 94 kpa 8 (c) Solution for te speed of Z (v): First, note tat te pressures at P and Z are te same and equal atm. Apply te Bernoulli s equation to points P and Z and take =0 at point P: atm = atm + ρg(-0.6m) + (/)ρv (/)ρv = ρg(0.6m) v = (g(0.6)) / =.46 m/s P (e) Solution for pressure at Y (P Y ): Apply te Bernoulli s equation to points P and Y, and take =0 at point P: atm = P Y + ρg(0.m) + (/)ρv 00 kpa = P Y + (000)(0)(0.) + (000)(0)(0.6) P Y = (00 6) kpa= 9 kpa P 9 0 5

### Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 12.

Capter 6. Fluid Mecanics Notes: Most of te material in tis capter is taken from Young and Freedman, Cap. 12. 6.1 Fluid Statics Fluids, i.e., substances tat can flow, are te subjects of tis capter. But

### Chapter 14 - Fluids. -Archimedes, On Floating Bodies. David J. Starling Penn State Hazleton PHYS 213. Chapter 14 - Fluids. Objectives (Ch 14)

Any solid lighter than a fluid will, if placed in the fluid, be so far immersed that the weight of the solid will be equal to the weight of the fluid displaced. -Archimedes, On Floating Bodies David J.

### Fluid Statics. [Ans.(c)] (iv) How is the metacentric height, MG expressed? I I

Fluid Statics Q1. Coose te crect answer (i) Te nmal stress is te same in all directions at a point in a fluid (a) only wen te fluid is frictionless (b) only wen te fluid is frictionless and incompressible

### Math 113 HW #5 Solutions

Mat 3 HW #5 Solutions. Exercise.5.6. Suppose f is continuous on [, 5] and te only solutions of te equation f(x) = 6 are x = and x =. If f() = 8, explain wy f(3) > 6. Answer: Suppose we ad tat f(3) 6. Ten

### Surface Areas of Prisms and Cylinders

12.2 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.10.B G.11.C Surface Areas of Prisms and Cylinders Essential Question How can you find te surface area of a prism or a cylinder? Recall tat te surface area of

### ACT Math Facts & Formulas

Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as

### Chapter 28 Fluid Dynamics

Chapter 28 Fluid Dynamics 28.1 Ideal Fluids... 1 28.2 Velocity Vector Field... 1 28.3 Mass Continuity Equation... 3 28.4 Bernoulli s Principle... 4 28.5 Worked Examples: Bernoulli s Equation... 7 Example

### AP2 Fluids. Kinetic Energy (A) stays the same stays the same (B) increases increases (C) stays the same increases (D) increases stays the same

A cart full of water travels horizontally on a frictionless track with initial velocity v. As shown in the diagram, in the back wall of the cart there is a small opening near the bottom of the wall that

### ACTIVITY: Deriving the Area Formula of a Trapezoid

4.3 Areas of Trapezoids a trapezoid? How can you derive a formula for te area of ACTIVITY: Deriving te Area Formula of a Trapezoid Work wit a partner. Use a piece of centimeter grid paper. a. Draw any

### Physics 1114: Unit 6 Homework: Answers

Physics 1114: Unit 6 Homework: Answers Problem set 1 1. A rod 4.2 m long and 0.50 cm 2 in cross-sectional area is stretched 0.20 cm under a tension of 12,000 N. a) The stress is the Force (1.2 10 4 N)

### Chapter 13 - Solutions

= Chapter 13 - Solutions Description: Find the weight of a cylindrical iron rod given its area and length and the density of iron. Part A On a part-time job you are asked to bring a cylindrical iron rod

### Derivatives Math 120 Calculus I D Joyce, Fall 2013

Derivatives Mat 20 Calculus I D Joyce, Fall 203 Since we ave a good understanding of its, we can develop derivatives very quickly. Recall tat we defined te derivative f x of a function f at x to be te

### Gauge Pressure, Absolute Pressure, and Pressure Measurement

Gauge Pressure, Absolute Pressure, and Pressure Measurement Bởi: OpenStaxCollege If you limp into a gas station with a nearly flat tire, you will notice the tire gauge on the airline reads nearly zero

### Chapter 13 Fluids. Copyright 2009 Pearson Education, Inc.

Chapter 13 Fluids 13-1 Phases of Matter The three common phases of matter are solid, liquid, and gas. A solid has a definite shape and size. A liquid has a fixed volume but can be any shape. A gas can

### Fluids flow conform to shape of container. Mass: mass density, Forces: Pressure Statics: Human body 50-75% water, live in a fluid (air)

Chapter 11 - Fluids Fluids flow conform to shape of container liquids OR gas Mass: mass density, Forces: Pressure Statics: pressure, buoyant force Dynamics: motion speed, energy friction: viscosity Human

### Areas and Centroids. Nothing. Straight Horizontal line. Straight Sloping Line. Parabola. Cubic

Constructing Sear and Moment Diagrams Areas and Centroids Curve Equation Sape Centroid (From Fat End of Figure) Area Noting Noting a x 0 Straigt Horizontal line /2 Straigt Sloping Line /3 /2 Paraola /4

### SAT Subject Math Level 1 Facts & Formulas

Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses

### A Low-Temperature Creep Experiment Using Common Solder

A Low-Temperature Creep Experiment Using Common Solder L. Roy Bunnell, Materials Science Teacer Soutridge Hig Scool Kennewick, WA 99338 roy.bunnell@ksd.org Copyrigt: Edmonds Community College 2009 Abstract:

### Solution Derivations for Capa #7

Solution Derivations for Capa #7 1) Consider te beavior of te circuit, wen various values increase or decrease. (Select I-increases, D-decreases, If te first is I and te rest D, enter IDDDD). A) If R1

### F mg (10.1 kg)(9.80 m/s ) m

Week 9 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution

### Lecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function

Lecture 10: Wat is a Function, definition, piecewise defined functions, difference quotient, domain of a function A function arises wen one quantity depends on anoter. Many everyday relationsips between

### Tangent Lines and Rates of Change

Tangent Lines and Rates of Cange 9-2-2005 Given a function y = f(x), ow do you find te slope of te tangent line to te grap at te point P(a, f(a))? (I m tinking of te tangent line as a line tat just skims

### Temperature Measure of KE At the same temperature, heavier molecules have less speed Absolute Zero -273 o C 0 K

Temperature Measure of KE At the same temperature, heavier molecules have less speed Absolute Zero -273 o C 0 K Kinetic Molecular Theory of Gases 1. Large number of atoms/molecules in random motion 2.

### ME422 Mechanical Control Systems Modeling Fluid Systems

Cal Poly San Luis Obispo Mecanical Engineering ME422 Mecanical Control Systems Modeling Fluid Systems Owen/Ridgely, last update Mar 2003 Te dynamic euations for fluid flow are very similar to te dynamic

### CHAPTER 11: ANSWERS TO ASSIGNED PROBLEMS Hauser- General Chemistry I revised 12/02/08

CAPTER 11: ANSWERS TO ASSIGNED PROBLEMS auser- General Chemistry I revised 12/02/08 11.15 Describe the intermolecular forces that must be overcome to convert each of the following from a liquid or solid

### Physics Principles of Physics

Physics 1408-002 Principles of Physics Lecture 21 Chapter 13 April 2, 2009 Sung-Won Lee Sungwon.Lee@ttu.edu Announcement I Lecture note is on the web Handout (6 slides/page) http://highenergy.phys.ttu.edu/~slee/1408/

### Reinforced Concrete Beam

Mecanics of Materials Reinforced Concrete Beam Concrete Beam Concrete Beam We will examine a concrete eam in ending P P A concrete eam is wat we call a composite eam It is made of two materials: concrete

### Forces. Definition Friction Falling Objects Projectiles Newton s Laws of Motion Momentum Universal Forces Fluid Pressure Hydraulics Buoyancy

Forces Definition Friction Falling Objects Projectiles Newton s Laws of Motion Momentum Universal Forces Fluid Pressure Hydraulics Buoyancy Definition of Force Force = a push or pull that causes a change

### Module 2. The Science of Surface and Ground Water. Version 2 CE IIT, Kharagpur

Module Te Science of Surface and Ground Water Version CE IIT, Karagpur Lesson 6 Principles of Ground Water Flow Version CE IIT, Karagpur Instructional Objectives On completion of te lesson, te student

### 14-1. Fluids in Motion There are two types of fluid motion called laminar flow and turbulent flow.

Fluid Dynamics Sections Covered in the Text: Chapter 15, except 15.6 To complete our study of fluids we now examine fluids in motion. For the most part the study of fluids in motion was put into an organized

### A. 1 cm 3 /s, in B. 1 cm 3 /s, out C. 10 cm 3 /s, in D. 10 cm 3 /s, out E. It depends on the relative size of the tubes.

The figure shows volume flow rates (in cm 3 /s) for all but one tube. What is the volume flow rate through the unmarked tube? Is the flow direction in or out? A. 1 cm 3 /s, in B. 1 cm 3 /s, out C. 10 cm

### Fluid Mechanics. Fluid Statics [3-1] Dr. Mohammad N. Almasri. [3] Fall 2010 Fluid Mechanics Dr. Mohammad N. Almasri [3-1] Fluid Statics

1 Fluid Mechanics Fluid Statics [3-1] Dr. Mohammad N. Almasri Fluid Pressure Fluid pressure is the normal force exerted by the fluid per unit area at some location within the fluid Fluid pressure has the

### When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid.

Fluid Statics When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid. Consider a small wedge of fluid at rest of size Δx, Δz, Δs

### Understanding the Derivative Backward and Forward by Dave Slomer

Understanding te Derivative Backward and Forward by Dave Slomer Slopes of lines are important, giving average rates of cange. Slopes of curves are even more important, giving instantaneous rates of cange.

### EC201 Intermediate Macroeconomics. EC201 Intermediate Macroeconomics Problem set 8 Solution

EC201 Intermediate Macroeconomics EC201 Intermediate Macroeconomics Prolem set 8 Solution 1) Suppose tat te stock of mone in a given econom is given te sum of currenc and demand for current accounts tat

### CHAPTER 3: FORCES AND PRESSURE

CHAPTER 3: FORCES AND PRESSURE 3.1 UNDERSTANDING PRESSURE 1. The pressure acting on a surface is defined as.. force per unit. area on the surface. 2. Pressure, P = F A 3. Unit for pressure is. Nm -2 or

### 7.6 Complex Fractions

Section 7.6 Comple Fractions 695 7.6 Comple Fractions In tis section we learn ow to simplify wat are called comple fractions, an eample of wic follows. 2 + 3 Note tat bot te numerator and denominator are

### Determine the perimeter of a triangle using algebra Find the area of a triangle using the formula

Student Name: Date: Contact Person Name: Pone Number: Lesson 0 Perimeter, Area, and Similarity of Triangles Objectives Determine te perimeter of a triangle using algebra Find te area of a triangle using

### New Vocabulary volume

-. Plan Objectives To find te volume of a prism To find te volume of a cylinder Examples Finding Volume of a Rectangular Prism Finding Volume of a Triangular Prism 3 Finding Volume of a Cylinder Finding

### Problems DISCUSSION QUESTIONS. How Long to Drain? BRIDGING PROBLEM

Discussion Questions 393 BRIDGING PROBLEM How Long to Drain? A large cylindrical tank wit diameter D is open to te air at te top. Te tank contains water to a eigt H. A small circular ole wit diameter d,

### Fluids and Solids: Fundamentals

Fluids and Solids: Fundamentals We normally recognize three states of matter: solid; liquid and gas. However, liquid and gas are both fluids: in contrast to solids they lack the ability to resist deformation.

### 1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution

1.6 Analyse Optimum Volume and Surface Area Estimation and oter informal metods of optimizing measures suc as surface area and volume often lead to reasonable solutions suc as te design of te tent in tis

### Proof of the Power Rule for Positive Integer Powers

Te Power Rule A function of te form f (x) = x r, were r is any real number, is a power function. From our previous work we know tat x x 2 x x x x 3 3 x x In te first two cases, te power r is a positive

### Mercury is poured into a U-tube as in Figure (14.18a). The left arm of the tube has crosssectional

Chapter 14 Fluid Mechanics. Solutions of Selected Problems 14.1 Problem 14.18 (In the text book) Mercury is poured into a U-tube as in Figure (14.18a). The left arm of the tube has crosssectional area

### Fluid Mechanics Definitions

Definitions 9-1a1 Fluids Substances in either the liquid or gas phase Cannot support shear Density Mass per unit volume Specific Volume Specific Weight % " = lim g#m ( ' * = +g #V \$0& #V ) Specific Gravity

### Finite Difference Approximations

Capter Finite Difference Approximations Our goal is to approximate solutions to differential equations, i.e., to find a function (or some discrete approximation to tis function) tat satisfies a given relationsip

### General Physics (PHY 2130)

General Physics (PHY 30) Lecture 3 Solids and fluids buoyant force Archimedes principle Fluids in motion http://www.physics.wayne.edu/~apetrov/phy30/ Lightning Review Last lecture:. Solids and fluids different

### Physics 6B. Philip Lubin

Physics 6B Philip Lubin prof@deepspace.ucsb.edu http://www.deepspace.ucsb.edu/classes/physics-6b-spring-2015 Course Outline Text College Physics Freedman 2014 Cover Chap 11-13, 16-21 Chap 11- Fluid Chap

### This supplement is meant to be read after Venema s Section 9.2. Throughout this section, we assume all nine axioms of Euclidean geometry.

Mat 444/445 Geometry for Teacers Summer 2008 Supplement : Similar Triangles Tis supplement is meant to be read after Venema s Section 9.2. Trougout tis section, we assume all nine axioms of uclidean geometry.

### Warm medium, T H T T H T L. s Cold medium, T L

Refrigeration Cycle Heat flows in direction of decreasing temperature, i.e., from ig-temperature to low temperature regions. Te transfer of eat from a low-temperature to ig-temperature requires a refrigerator

### Chapter 27 Static Fluids

Chapter 27 Static Fluids 27.1 Introduction... 1 27.2 Density... 1 27.3 Pressure in a Fluid... 2 27.4 Pascal s Law: Pressure as a Function of Depth in a Fluid of Uniform Density in a Uniform Gravitational

### Chapter 10: Refrigeration Cycles

Capter 10: efrigeration Cycles Te vapor compression refrigeration cycle is a common metod for transferring eat from a low temperature to a ig temperature. Te above figure sows te objectives of refrigerators

### Chapter 9: The Behavior of Fluids

Chapter 9: The Behavior of Fluids 1. Archimedes Principle states that A. the pressure in a fluid is directly related to the depth below the surface of the fluid. B. an object immersed in a fluid is buoyed

### Math Test Sections. The College Board: Expanding College Opportunity

Taking te SAT I: Reasoning Test Mat Test Sections Te materials in tese files are intended for individual use by students getting ready to take an SAT Program test; permission for any oter use must be sougt

### Sections 3.1/3.2: Introducing the Derivative/Rules of Differentiation

Sections 3.1/3.2: Introucing te Derivative/Rules of Differentiation 1 Tangent Line Before looking at te erivative, refer back to Section 2.1, looking at average velocity an instantaneous velocity. Here

### M(0) = 1 M(1) = 2 M(h) = M(h 1) + M(h 2) + 1 (h > 1)

Insertion and Deletion in VL Trees Submitted in Partial Fulfillment of te Requirements for Dr. Eric Kaltofen s 66621: nalysis of lgoritms by Robert McCloskey December 14, 1984 1 ackground ccording to Knut

### Why Study Fluids? Solids and How They Respond to Forces. Solids and How They Respond to Forces. Crystal lattice structure:

States of Matter Gas In a gas, the molecules are far apart and the forces between them are very small Solid In a solid, the molecules are very close together, and the form of the solid depends on the details

### = 800 kg/m 3 (note that old units cancel out) 4.184 J 1000 g = 4184 J/kg o C

Units and Dimensions Basic properties such as length, mass, time and temperature that can be measured are called dimensions. Any quantity that can be measured has a value and a unit associated with it.

### Density (r) Chapter 10 Fluids. Pressure 1/13/2015

1/13/015 Density (r) Chapter 10 Fluids r = mass/volume Rho ( r) Greek letter for density Units - kg/m 3 Specific Gravity = Density of substance Density of water (4 o C) Unitless ratio Ex: Lead has a sp.

### Chapter 8 Fluid Flow

Chapter 8 Fluid Flow GOALS When you have mastered the contents of this chapter, you will be able to achieve the following goals: Definitions Define each of the following terms, and use it in an operational

Name: Class: Date: Exam 4--PHYS 101--F14 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A wheel, initially at rest, rotates with a constant acceleration

### Trapezoid Rule. y 2. y L

Trapezoid Rule and Simpson s Rule c 2002, 2008, 200 Donald Kreider and Dwigt Lar Trapezoid Rule Many applications of calculus involve definite integrals. If we can find an antiderivative for te integrand,

### The modelling of business rules for dashboard reporting using mutual information

8 t World IMACS / MODSIM Congress, Cairns, Australia 3-7 July 2009 ttp://mssanz.org.au/modsim09 Te modelling of business rules for dasboard reporting using mutual information Gregory Calbert Command, Control,

### Gases. Macroscopic Properties. Petrucci, Harwood and Herring: Chapter 6

Gases Petrucci, Harwood and Herring: Chapter 6 CHEM 1000A 3.0 Gases 1 We will be looking at Macroscopic and Microscopic properties: Macroscopic Properties of bulk gases Observable Pressure, volume, mass,

### oil liquid water water liquid Answer, Key Homework 2 David McIntyre 1

Answer, Key Homework 2 David McIntyre 1 This print-out should have 14 questions, check that it is complete. Multiple-choice questions may continue on the next column or page: find all choices before making

### The differential amplifier

DiffAmp.doc 1 Te differential amplifier Te emitter coupled differential amplifier output is V o = A d V d + A c V C Were V d = V 1 V 2 and V C = (V 1 + V 2 ) / 2 In te ideal differential amplifier A c

### Research on the Anti-perspective Correction Algorithm of QR Barcode

Researc on te Anti-perspective Correction Algoritm of QR Barcode Jianua Li, Yi-Wen Wang, YiJun Wang,Yi Cen, Guoceng Wang Key Laboratory of Electronic Tin Films and Integrated Devices University of Electronic

### Physics 181- Summer 2011 - Experiment #8 1 Experiment #8, Measurement of Density and Archimedes' Principle

Physics 181- Summer 2011 - Experiment #8 1 Experiment #8, Measurement of Density and Archimedes' Principle 1 Purpose 1. To determine the density of a fluid, such as water, by measurement of its mass when

### Buoyant Force and Archimedes Principle

Buoyant Force and Archimedes Principle Predict the behavior of fluids as a result of properties including viscosity and density Demonstrate why objects sink or float Apply Archimedes Principle by measuring

### Pipe Flow Analysis. Pipes in Series. Pipes in Parallel

Pipe Flow Analysis Pipeline system used in water distribution, industrial application and in many engineering systems may range from simple arrangement to extremely complex one. Problems regarding pipelines

### Projective Geometry. Projective Geometry

Euclidean versus Euclidean geometry describes sapes as tey are Properties of objects tat are uncanged by rigid motions» Lengts» Angles» Parallelism Projective geometry describes objects as tey appear Lengts,

### Kinetic Theory. Bellringer. Kinetic Theory, continued. Visual Concept: Kinetic Molecular Theory. States of Matter, continued.

Bellringer You are already familiar with the most common states of matter: solid, liquid, and gas. For example you can see solid ice and liquid water. You cannot see water vapor, but you can feel it in

### Area of a Parallelogram

Area of a Parallelogram Focus on After tis lesson, you will be able to... φ develop te φ formula for te area of a parallelogram calculate te area of a parallelogram One of te sapes a marcing band can make

### TAKE-HOME EXP. # 2 A Calculation of the Circumference and Radius of the Earth

TAKE-HOME EXP. # 2 A Calculation of te Circumference and Radius of te Eart On two dates during te year, te geometric relationsip of Eart to te Sun produces "equinox", a word literally meaning, "equal nigt"

### So if ω 0 increases 3-fold, the stopping angle increases 3 2 = 9-fold.

Name: MULTIPLE CHOICE: Questions 1-11 are 5 points each. 1. A safety device brings the blade of a power mower from an angular speed of ω 1 to rest in 1.00 revolution. At the same constant angular acceleration,

### Applied Fluid Mechanics

Applied Fluid Mechanics 1. The Nature of Fluid and the Study of Fluid Mechanics 2. Viscosity of Fluid 3. Pressure Measurement 4. Forces Due to Static Fluid 5. Buoyancy and Stability 6. Flow of Fluid and

### 01 The Nature of Fluids

01 The Nature of Fluids WRI 1/17 01 The Nature of Fluids (Water Resources I) Dave Morgan Prepared using Lyx, and the Beamer class in L A TEX 2ε, on September 12, 2007 Recommended Text 01 The Nature of

### Name Partner Date Class

Name Partner Date Class FLUIDS Part 1: Archimedes' Principle Equipment: Dial-O-Gram balance, small beaker (150-250ml), metal specimen, string, calipers. Object: To find the density of an object using Archimedes'

### Lecture 5 Hemodynamics. Description of fluid flow. The equation of continuity

1 Lecture 5 Hemodynamics Description of fluid flow Hydrodynamics is the part of physics, which studies the motion of fluids. It is based on the laws of mechanics. Hemodynamics studies the motion of blood

### Faraday s Law of Induction Motional emf

Faraday s Law of Induction Motional emf Motors and Generators Lenz s Law Eddy Currents Te needle deflects momentarily wen te switc is closed A current flows troug te loop wen a magnet is moved near it,

### f(a + h) f(a) f (a) = lim

Lecture 7 : Derivative AS a Function In te previous section we defined te derivative of a function f at a number a (wen te function f is defined in an open interval containing a) to be f (a) 0 f(a + )

### Min-218 Fundamentals of Fluid Flow

Excerpt from "Chap 3: Principles of Airflow," Practical Mine Ventilation Engineerg to be Pubished by Intertec Micromedia Publishing Company, Chicago, IL in March 1999. 1. Definition of A Fluid A fluid

### Perimeter, Area and Volume of Regular Shapes

Perimeter, Area and Volume of Regular Sapes Perimeter of Regular Polygons Perimeter means te total lengt of all sides, or distance around te edge of a polygon. For a polygon wit straigt sides tis is te

### Math Warm-Up for Exam 1 Name: Solutions

Disclaimer: Tese review problems do not represent te exact questions tat will appear te exam. Tis is just a warm-up to elp you begin studying. It is your responsibility to review te omework problems, webwork

### 1. Fluids Mechanics and Fluid Properties. 1.1 Objectives of this section. 1.2 Fluids

1. Fluids Mechanics and Fluid Properties What is fluid mechanics? As its name suggests it is the branch of applied mechanics concerned with the statics and dynamics of fluids - both liquids and gases.

### Exam 2 Review. . You need to be able to interpret what you get to answer various questions.

Exam Review Exam covers 1.6,.1-.3, 1.5, 4.1-4., and 5.1-5.3. You sould know ow to do all te omework problems from tese sections and you sould practice your understanding on several old exams in te exam

### 22.1 Finding the area of plane figures

. Finding te area of plane figures a cm a cm rea of a square = Lengt of a side Lengt of a side = (Lengt of a side) b cm a cm rea of a rectangle = Lengt readt b cm a cm rea of a triangle = a cm b cm = ab

### 1206 Concepts in Physics. Monday, October 19th

1206 Concepts in Physics Monday, October 19th Notes Problems with webpage over the weekend If this is not fixed by tonight, I will find another way to display the material Either webct or printouts The

### Section 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations

Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of

### δy θ Pressure is used to indicate the normal force per unit area at a given point acting on a given plane.

2 FLUID PRESSURES By definition, a fluid must deform continuously when a shear stress of any magnitude is applied. Therefore when a fluid is either at rest or moving in such a manner that there is no relative

### CE 204 FLUID MECHANICS

CE 204 FLUID MECHANICS Onur AKAY Assistant Professor Okan University Department of Civil Engineering Akfırat Campus 34959 Tuzla-Istanbul/TURKEY Phone: +90-216-677-1630 ext.1974 Fax: +90-216-677-1486 E-mail:

### Theoretical calculation of the heat capacity

eoretical calculation of te eat capacity Principle of equipartition of energy Heat capacity of ideal and real gases Heat capacity of solids: Dulong-Petit, Einstein, Debye models Heat capacity of metals

### Basic Fluid Mechanics. Prof. Young I Cho

Basic Fluid Mechanics MEM 220 Prof. Young I Cho Summer 2009 Chapter 1 Introduction What is fluid? Give some examples of fluids. Examples of gases: Examples of liquids: What is fluid mechanics? Mechanics

### An Interest Rate Model

An Interest Rate Model Concepts and Buzzwords Building Price Tree from Rate Tree Lognormal Interest Rate Model Nonnegativity Volatility and te Level Effect Readings Tuckman, capters 11 and 12. Lognormal

### Typhoon Haiyan 1. Force of wind blowing against vertical structure. 2. Destructive Pressure exerted on Buildings

Typhoon Haiyan 1. Force of wind blowing against vertical structure 2. Destructive Pressure exerted on Buildings 3. Atmospheric Pressure variation driving the Typhoon s winds 4. Energy of Typhoon 5. Height

### Gases. Gas: fluid, occupies all available volume Liquid: fluid, fixed volume Solid: fixed volume, fixed shape Others?

CHAPTER 5: Gases Chemistry of Gases Pressure and Boyle s Law Temperature and Charles Law The Ideal Gas Law Chemical Calculations of Gases Mixtures of Gases Kinetic Theory of Gases Real Gases Gases The

### Gas Properties and Balloons & Buoyancy SI M Homework Answer K ey

Gas Properties and Balloons & Buoyancy SI M Homework Answer K ey 1) In class, we have been discussing how gases behave and how we observe this behavior in our daily lives. In this homework assignment,

### Gas Laws. The kinetic theory of matter states that particles which make up all types of matter are in constant motion.

Name Period Gas Laws Kinetic energy is the energy of motion of molecules. Gas state of matter made up of tiny particles (atoms or molecules). Each atom or molecule is very far from other atoms or molecules.