Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur"

Transcription

1 Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur

2 esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur

3 Instructionl Objectives After reding this chpter the student will be ble to. Clculte dditionl stresses developed in stticlly indeterminte structures due to support settlements.. Anlyse continuous bems which re supported on yielding supports. 3. Sketch the deflected shpe of the member. 4. Drw bnding moment nd sher force digrms for indeterminte bems undergoing support settlements. 9. Introduction In the lst lesson, the force method of nlysis of stticlly indeterminte bems subjected to externl lods ws discussed. It is however, ssumed in the nlysis tht the supports re unyielding nd the temperture remins constnt. In the design of indeterminte structure, it is required to mke necessry provision for future unequl verticl settlement of supports or probble rottion of supports. It my be observed here tht, in cse of determinte structures no stresses re developed due to settlement of supports. The whole structure displces s rigid body (see Fig. 9.). Hence, construction of determinte structures is esier thn indeterminte structures. Version CE IIT, Khrgpur

4 Version CE IIT, Khrgpur

5 The stticlly determinte structure chnges their shpe due to support settlement nd this would in turn induce rections nd stresses in the system. Since, there is no externl force system cting on the structures, these forces form blnced force system by themselves nd the structure would be in equilibrium. The effect of temperture chnges, support settlement cn lso be esily included in the force method of nlysis. In this lesson few problems, concerning the effect of support settlement re solved to illustrte the procedure. 9. Support Displcements Consider two spn continuous bem, which is stticlly indeterminte to second degree, s shown in Fig. 9.. Assume the flexurl rigidity of this bem to be constnt throughout. In this exmple, the support B is ssumed to hve settled by n mount s shown in the figure. Δ b This problem ws solved in the lst lesson, when there ws no support settlement (vide section 8.). In section 8., choosing rection t B nd C s the redundnt, the totl deflection of the primry structure due to pplied externl loding nd redundnt nd R is written s, R ( Δ ) + R + Δ (9.) R ( Δ ) + R + Δ (9.b) R ( ) wherein, R nd R re the redundnts t B nd C respectively, nd Δ, nd ( Δ ) re the deflections of the primry structure t B nd C due to pplied loding. In the present cse, the support B settles by n mount Δ b in the direction of the redundnt R. This support movement cn be redily incorported in the force method of nlysis. From the physics of the problem the totl deflection t the support my be equl to the given mount of support movement. Hence, the comptibility condition my be written s, Δ Δ b (9.) Δ 0 (9.b) It must be noted tht, the support settlement Δ b must be negtive s it is displces downwrds. It is ssumed tht deflections nd rections re positive in the upwrd direction. The eqution (9.) nd (9.b) my be written in compct form s, Version CE IIT, Khrgpur

6 R R Δ Δ ( Δ ) ( Δ ) (9.3) [ A]{ R} { Δ} { ( Δ )} (9.3b) Solving the bove lgebric equtions, one could evlute redundnts due to externl loding nd support settlement. R nd R 9.3 Temperture Stresses Internl stresses re lso developed in the stticlly indeterminte structure if the free movement of the joint is prevented. For exmple, consider cntilever bem AB s shown in Fig Now, if the temperture of the member is incresed uniformly throughout its length, then the length of the member is incresed by n mount Δ T Δ α T (9.4) T In which, is the chnge in the length of the member due to temperture chnge, α is the coefficient of therml expnsion of the mteril nd T is the chnge in temperture. The elongtion (the chnge in the length of the member) nd increse in temperture re tken s positive. However if the end B is restrined to move s shown in Fig 9.4, then the bem deformtion is prevented. This would develop n internl xil force nd rections in the indeterminte structure. Next consider cntilever bem AB, subjected to different temperture, t the top nd T t the bottom s shown in Fig. 9.5() nd (b). If the top temperture T is higher thn the bottom bem surfce temperture T, then the bem will deform s shown by dotted lines. Consider smll element dx t distnce x from A. The deformtion of this smll element is shown in Fig. 9.5c. Due to rise in temperture T C on the top surfce, the top surfce elongtes by Δ T dx (9.5) T α Similrly due to rise in temperture T, the bottom fibers elongte by Δ T dx (9.5b) T α T Version CE IIT, Khrgpur

7 As the cross section of the member remins plne, the reltive ngle of rottion dθ between two cross sections t distnce dx is given by ( T T ) dx dθ α (9.6) d where, d is the depth of bem. If the end B is fixed s in Fig. 9.4, then the differentil chnge in temperture would develop support bending moment nd rections. The effect of temperture cn lso be included in the force method of nlysis quite esily. This is done s follows. Clculte the deflection corresponding to redundnt ctions seprtely due to pplied loding, due to rise in temperture (either uniform or differentil chnge in temperture) nd redundnt forces. The deflection in the primry structure due to temperture chnges is denoted by th ( Δ T ) i which denotes the deflection corresponding to i redundnt due to temperture chnge in the determinte structure. Now the comptibility eqution for stticlly indeterminte structure of order two cn be written s R R Δ Δ ( Δ ) ( Δ ) [ A]{ R} { Δ} { ( Δ )} { ( Δ )} T ( Δ ) T ( ΔT ) (9.7) wherein,{ Δ } is the vector of displcements in the primry structure corresponding to redundnt rections due to externl lods; { Δ T } is the displcements in the primry structure corresponding to redundnt rections nd Δ is the mtrix of support displcements due to temperture chnges nd { } corresponding to redundnt ctions. Eqution (9.7) cn be solved to obtin the unknown redundnts. Exmple 9. Clculte the support rections in the continuous bem ABC (see Fig. 9.6) hving constnt flexurl rigidity EI throughout, due to verticl settlement of the 4 4 support B by 5 m m s shown in the figure. E 00 GP nd I 4 0 m. Version CE IIT, Khrgpur

8 Version CE IIT, Khrgpur

9 As the given bem is stticlly indeterminte to second degree, choose rection t B ( R ) nd C ( R ) s the redundnts. In this cse the cntilever bem AC is the bsic determinte bem (primry structure). On the determinte bem only redundnt rections re cting. The first column of flexibility mtrix is evluted by first pplying unit lod long the redundnt R nd determining deflections nd respectively s shown in Fig. 9.6b EI 5 3EI () 3EI EI 6EI Simply by pplying the unit lod in the direction of redundnt evlute flexibility coefficients nd (see Fig. 9.6c). R, one could 65 6EI nd EI () The comptibility condition for the problem my be written s, 3 R + R 5 0 R + R 0 (3) The redundnt rections re, R R [ A] (4) R 3EI 000 R (5) Substituting the vlues of E nd I in the bove eqution, the redundnt rections re evluted. R kn nd R 3.7 kn R cts downwrds nd R cts in the positive direction of the rection i.e. upwrds. The remining two rections R3 nd R4 re evluted by the equtions of equilibrium. Version CE IIT, Khrgpur

10 F 0 R + R + R 0 y 3 Hence R kn M 0 R + 5 R + 0 R 0 A 4 Solving for R 4, R kn.m (counter clockwise) The sher force nd bending moment digrms re shown in Figs. 9.6d nd 9.6e respectively. Exmple 9. Compute rections nd drw bending moment digrm for the continuous bem ABCD loded s shown in Fig. 9.7, due to following support movements. Support B, m verticlly downwrds. Support C, 0.0m verticlly downwrds. 3 4 Assume, E 00GP; I.35 0 m. Version CE IIT, Khrgpur

11 The given bem is stticlly indeterminte to second degree. Select verticl rections t B ( R ) nd C ( R ) s redundnts. The primry structure in this cse is simply supported bem AD s shown in Fig. 9.7b. The deflection ( Δ ) nd ( Δ ) of the relesed structure re evluted from unit lod method. Thus, Version CE IIT, Khrgpur

12 EI ( ) 0. m Δ Δ 69 () EI ( ) 0. m The flexibility mtrix is evluted s explined in the previous exmple, i.e. by first pplying unit lod corresponding to the redundnt R nd determining deflections nd respectively s shown in Fig. 9.7c. Thus, EI () EI EI EI In this cse the comptibility equtions my be written s, R + R R + R (3) Solving for redundnt rections, R EI R (4) Substituting the vlue of E nd I in the bove eqution, R kN nd R kn Version CE IIT, Khrgpur

13 R R 3 Both nd cts in the upwrd direction. The remining two rections R nd R 4 re evluted by the equtions of sttic equilibrium. M A Hence R kn R + 0 R + 30 R F y 0 R 3+ R R R Hence R kn (5) The sher force nd bending moment digrms re now constructed nd re shown in Figs. 9.7e nd 9.7f respectively. Version CE IIT, Khrgpur

14 Summry In this lesson, the effect of support settlements on the rections nd stresses in the cse of indeterminte structures is discussed. The procedure to clculte dditionl stresses cused due to yielding of supports is explined with the help of n exmple. A formul is derived for clculting stresses due to temperture chnges in the cse of stticlly indeterminte bems. Version CE IIT, Khrgpur

COMPONENTS: COMBINED LOADING

COMPONENTS: COMBINED LOADING LECTURE COMPONENTS: COMBINED LOADING Third Edition A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering 24 Chpter 8.4 by Dr. Ibrhim A. Asskkf SPRING 2003 ENES 220 Mechnics of

More information

Cypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period:

Cypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period: Nme: SOLUTIONS Dte: Period: Directions: Solve ny 5 problems. You my ttempt dditionl problems for extr credit. 1. Two blocks re sliding to the right cross horizontl surfce, s the drwing shows. In Cse A

More information

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100 hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

More information

Basic Analysis of Autarky and Free Trade Models

Basic Analysis of Autarky and Free Trade Models Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one. 5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

More information

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

More information

Answer, Key Homework 8 David McIntyre 1

Answer, Key Homework 8 David McIntyre 1 Answer, Key Homework 8 Dvid McIntyre 1 This print-out should hve 17 questions, check tht it is complete. Multiple-choice questions my continue on the net column or pge: find ll choices before mking your

More information

Experiment 6: Friction

Experiment 6: Friction Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

Newton s Three Laws. d dt F = If the mass is constant, this relationship becomes the familiar form of Newton s Second Law: dv dt

Newton s Three Laws. d dt F = If the mass is constant, this relationship becomes the familiar form of Newton s Second Law: dv dt Newton s Three Lws For couple centuries before Einstein, Newton s Lws were the bsic principles of Physics. These lws re still vlid nd they re the bsis for much engineering nlysis tody. Forml sttements

More information

Project 6 Aircraft static stability and control

Project 6 Aircraft static stability and control Project 6 Aircrft sttic stbility nd control The min objective of the project No. 6 is to compute the chrcteristics of the ircrft sttic stbility nd control chrcteristics in the pitch nd roll chnnel. The

More information

Volumes of solids of revolution

Volumes of solids of revolution Volumes of solids of revolution We sometimes need to clculte the volume of solid which cn be obtined by rotting curve bout the x-xis. There is strightforwrd technique which enbles this to be done, using

More information

Uplift Capacity of K-Series Open Web Steel Joist Seats. Florida, Gainesville, FL 32611; email: psgreen@ce.ufl.edu

Uplift Capacity of K-Series Open Web Steel Joist Seats. Florida, Gainesville, FL 32611; email: psgreen@ce.ufl.edu Uplift Cpcity of K-Series Open Web Steel Joist Sets Perry S. Green, Ph.D, M.ASCE 1 nd Thoms Sputo, Ph.D., P.E., M.ASCE 2 1 Assistnt Professor, Deprtment of Civil nd Costl Engineering, University of Florid,

More information

Mechanics Cycle 1 Chapter 5. Chapter 5

Mechanics Cycle 1 Chapter 5. Chapter 5 Chpter 5 Contct orces: ree Body Digrms nd Idel Ropes Pushes nd Pulls in 1D, nd Newton s Second Lw Neglecting riction ree Body Digrms Tension Along Idel Ropes (i.e., Mssless Ropes) Newton s Third Lw Bodies

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

More information

Factoring Polynomials

Factoring Polynomials Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

More information

Vectors 2. 1. Recap of vectors

Vectors 2. 1. Recap of vectors Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

More information

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding 1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics. W02D3_0 Group Problem: Pulleys and Ropes Constraint Conditions

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics. W02D3_0 Group Problem: Pulleys and Ropes Constraint Conditions MSSCHUSES INSIUE OF ECHNOLOGY Deprtment of hysics 8.0 W02D3_0 Group roblem: ulleys nd Ropes Constrint Conditions Consider the rrngement of pulleys nd blocks shown in the figure. he pulleys re ssumed mssless

More information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions. Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

More information

PHY 222 Lab 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS

PHY 222 Lab 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS PHY 222 Lb 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS Nme: Prtners: INTRODUCTION Before coming to lb, plese red this pcket nd do the prelb on pge 13 of this hndout. From previous experiments,

More information

Pythagoras theorem and trigonometry (2)

Pythagoras theorem and trigonometry (2) HPTR 10 Pythgors theorem nd trigonometry (2) 31 HPTR Liner equtions In hpter 19, Pythgors theorem nd trigonometry were used to find the lengths of sides nd the sizes of ngles in right-ngled tringles. These

More information

Applications to Physics and Engineering

Applications to Physics and Engineering Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics

More information

Let us recall some facts you have learnt in previous grades under the topic Area.

Let us recall some facts you have learnt in previous grades under the topic Area. 6 Are By studying this lesson you will be ble to find the res of sectors of circles, solve problems relted to the res of compound plne figures contining sectors of circles. Ares of plne figures Let us

More information

Chapter G - Problems

Chapter G - Problems Chpter G - Problems Blinn College - Physics 2426 - Terry Honn Problem G.1 A plne flies horizonlly t speed of 280 mês in position where the erth's mgnetic field hs mgnitude 6.0µ10-5 T nd is directed t n

More information

All pay auctions with certain and uncertain prizes a comment

All pay auctions with certain and uncertain prizes a comment CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 1-2015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin

More information

Integration by Substitution

Integration by Substitution Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is

More information

Double Integrals over General Regions

Double Integrals over General Regions Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing

More information

Binary Representation of Numbers Autar Kaw

Binary Representation of Numbers Autar Kaw Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

More information

A5682: Introduction to Cosmology Course Notes. 4. Cosmic Dynamics: The Friedmann Equation. = GM s R 2 s(t).

A5682: Introduction to Cosmology Course Notes. 4. Cosmic Dynamics: The Friedmann Equation. = GM s R 2 s(t). 4. Cosmic Dynmics: The Friedmnn Eqution Reding: Chpter 4 Newtonin Derivtion of the Friedmnn Eqution Consider n isolted sphere of rdius R s nd mss M s, in uniform, isotropic expnsion (Hubble flow). The

More information

SOLUTIONS TO CONCEPTS CHAPTER 5

SOLUTIONS TO CONCEPTS CHAPTER 5 1. m k S 10m Let, ccelertion, Initil velocity u 0. S ut + 1/ t 10 ½ ( ) 10 5 m/s orce: m 5 10N (ns) 40000. u 40 km/hr 11.11 m/s. 3600 m 000 k ; v 0 ; s 4m v u ccelertion s SOLUIONS O CONCEPS CHPE 5 0 11.11

More information

Net Change and Displacement

Net Change and Displacement mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the

More information

r 2 F ds W = r 1 qe ds = q

r 2 F ds W = r 1 qe ds = q Chpter 4 The Electric Potentil 4.1 The Importnt Stuff 4.1.1 Electricl Potentil Energy A chrge q moving in constnt electric field E experiences force F = qe from tht field. Also, s we know from our study

More information

The Laws of Motion. chapter

The Laws of Motion. chapter chpter The Lws of Motion 5 5.1 The Concept of Force 5.2 Newton s First Lw nd Inertil Frmes 5.3 Mss 5.4 Newton s econd Lw 5.5 The Grvittionl Force nd Weight 5.6 Newton s Third Lw 5.7 Anlysis Models Using

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

More information

Econ 4721 Money and Banking Problem Set 2 Answer Key

Econ 4721 Money and Banking Problem Set 2 Answer Key Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in

More information

Rotating DC Motors Part II

Rotating DC Motors Part II Rotting Motors rt II II.1 Motor Equivlent Circuit The next step in our consiertion of motors is to evelop n equivlent circuit which cn be use to better unerstn motor opertion. The rmtures in rel motors

More information

Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chapter 40 Key Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

More information

Theory of Forces. Forces and Motion

Theory of Forces. Forces and Motion his eek extbook -- Red Chpter 4, 5 Competent roblem Solver - Chpter 4 re-lb Computer Quiz ht s on the next Quiz? Check out smple quiz on web by hurs. ht you missed on first quiz Kinemtics - Everything

More information

Impact Testing. Ductile to Brittle Transition in Steel. Charpy Test. Outline. Impact Testing

Impact Testing. Ductile to Brittle Transition in Steel. Charpy Test. Outline. Impact Testing Impct Testing Outline Impct Testing Chrpy Test Ductile to Brittle Trnsition Temperture Metllurgicl Fctors Affecting T T Ftigue Ftigue Testing Types of Ftigue Chrpy nd Izod tests mesure impct energy or

More information

Math 135 Circles and Completing the Square Examples

Math 135 Circles and Completing the Square Examples Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

More information

14.2. The Mean Value and the Root-Mean-Square Value. Introduction. Prerequisites. Learning Outcomes

14.2. The Mean Value and the Root-Mean-Square Value. Introduction. Prerequisites. Learning Outcomes he Men Vlue nd the Root-Men-Squre Vlue 4. Introduction Currents nd voltges often vry with time nd engineers my wish to know the men vlue of such current or voltge over some prticulr time intervl. he men

More information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers. 2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

More information

Project Recovery. . It Can Be Done

Project Recovery. . It Can Be Done Project Recovery. It Cn Be Done IPM Conference Wshington, D.C. Nov 4-7, 200 Wlt Lipke Oklhom City Air Logistics Center Tinker AFB, OK Overview Mngement Reserve Project Sttus Indictors Performnce Correction

More information

Homework #6: Answers. a. If both goods are produced, what must be their prices?

Homework #6: Answers. a. If both goods are produced, what must be their prices? Text questions, hpter 7, problems 1-2. Homework #6: Answers 1. Suppose there is only one technique tht cn be used in clothing production. To produce one unit of clothing requires four lbor-hours nd one

More information

addition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix.

addition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix. APPENDIX A: The ellipse August 15, 1997 Becuse of its importnce in both pproximting the erth s shpe nd describing stellite orbits, n informl discussion of the ellipse is presented in this ppendix. The

More information

Solutions to Section 1

Solutions to Section 1 Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this

More information

1+(dy/dx) 2 dx. We get dy dx = 3x1/2 = 3 x, = 9x. Hence 1 +

1+(dy/dx) 2 dx. We get dy dx = 3x1/2 = 3 x, = 9x. Hence 1 + Mth.9 Em Solutions NAME: #.) / #.) / #.) /5 #.) / #5.) / #6.) /5 #7.) / Totl: / Instructions: There re 5 pges nd totl of points on the em. You must show ll necessr work to get credit. You m not use our

More information

In this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists.

In this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists. Mth 52 Sec S060/S0602 Notes Mtrices IV 5 Inverse Mtrices 5 Introduction In our erlier work on mtrix multipliction, we sw the ide of the inverse of mtrix Tht is, for squre mtrix A, there my exist mtrix

More information

Lecture 3 Basic Probability and Statistics

Lecture 3 Basic Probability and Statistics Lecture 3 Bsic Probbility nd Sttistics The im of this lecture is to provide n extremely speedy introduction to the probbility nd sttistics which will be needed for the rest of this lecture course. The

More information

Arc Length. P i 1 P i (1) L = lim. i=1

Arc Length. P i 1 P i (1) L = lim. i=1 Arc Length Suppose tht curve C is defined by the eqution y = f(x), where f is continuous nd x b. We obtin polygonl pproximtion to C by dividing the intervl [, b] into n subintervls with endpoints x, x,...,x

More information

Understanding 22. 23. The frictional force acting to the left is missing. It is equal in magnitude to the applied force acting to the right.

Understanding 22. 23. The frictional force acting to the left is missing. It is equal in magnitude to the applied force acting to the right. Chpter 3 Review, pges 154 159 Knowledge 1. (c) 2. () 3. (d) 4. (d) 5. (d) 6. (c) 7. (b) 8. (c) 9. Flse. One newton is equl to 1 kg /s 2. 10. Flse. A norl force is perpendiculr force cting on n object tht

More information

Math 22B Solutions Homework 1 Spring 2008

Math 22B Solutions Homework 1 Spring 2008 Mth 22B Solutions Homework 1 Spring 2008 Section 1.1 22. A sphericl rindrop evportes t rte proportionl to its surfce re. Write differentil eqution for the volume of the rindrop s function of time. Solution

More information

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur Module Analysis of Statically Indeterminate Structures by the Matrix Force Method esson 11 The Force Method of Analysis: Frames Instructional Objectives After reading this chapter the student will be able

More information

Worksheet 24: Optimization

Worksheet 24: Optimization Worksheet 4: Optimiztion Russell Buehler b.r@berkeley.edu 1. Let P 100I I +I+4. For wht vlues of I is P mximum? P 100I I + I + 4 Tking the derivtive, www.xkcd.com P (I + I + 4)(100) 100I(I + 1) (I + I

More information

11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π.

11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π. . Fourier series Summry of the bsic ides The following is quick summry of the introductory tretment of Fourier series in MATH. We consider function f with period π, tht is, stisfying f(x + π) = f(x) for

More information

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn 33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

More information

On the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding

On the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding Dt_nlysisclm On the Mening of Regression for tegoricl nd ontinuous Vribles: I nd II; Effect oding nd Dummy oding R Grdner Deprtment of Psychology This describes the simple cse where there is one ctegoricl

More information

The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Factor of an Insulated Two-Wire Transmission Line The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the

More information

4. DC MOTORS. Understand the basic principles of operation of a DC motor. Understand the operation and basic characteristics of simple DC motors.

4. DC MOTORS. Understand the basic principles of operation of a DC motor. Understand the operation and basic characteristics of simple DC motors. 4. DC MOTORS Almost every mechnicl movement tht we see round us is ccomplished by n electric motor. Electric mchines re mens o converting energy. Motors tke electricl energy nd produce mechnicl energy.

More information

AREA OF A SURFACE OF REVOLUTION

AREA OF A SURFACE OF REVOLUTION AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.

More information

Dispersion in Coaxial Cables

Dispersion in Coaxial Cables Dispersion in Coxil Cbles Steve Ellingson June 1, 2008 Contents 1 Summry 2 2 Theory 2 3 Comprison to Welch s Result 4 4 Findings for RG58 t LWA Frequencies 5 Brdley Dept. of Electricl & Computer Engineering,

More information

1. In the Bohr model, compare the magnitudes of the electron s kinetic and potential energies in orbit. What does this imply?

1. In the Bohr model, compare the magnitudes of the electron s kinetic and potential energies in orbit. What does this imply? Assignment 3: Bohr s model nd lser fundmentls 1. In the Bohr model, compre the mgnitudes of the electron s kinetic nd potentil energies in orit. Wht does this imply? When n electron moves in n orit, the

More information

Reasoning to Solve Equations and Inequalities

Reasoning to Solve Equations and Inequalities Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

More information

An Off-Center Coaxial Cable

An Off-Center Coaxial Cable 1 Problem An Off-Center Coxil Cble Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Nov. 21, 1999 A coxil trnsmission line hs inner conductor of rdius nd outer conductor

More information

DETERMINANTS. ] of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A, where a ij. = ad bc.

DETERMINANTS. ] of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A, where a ij. = ad bc. Chpter 4 DETERMINANTS 4 Overview To every squre mtrix A = [ ij ] of order n, we cn ssocite number (rel or complex) clled determinnt of the mtrix A, written s det A, where ij is the (i, j)th element of

More information

PartialDifferentialEquations

PartialDifferentialEquations C H A P T E R 13 PrtilDifferentilEqutions 13.1 Derivtion of the Het Eqution 75 We will now consider differentil equtions tht model chnge where there is more thn one independent vrible. For exmple, the

More information

4.11 Inner Product Spaces

4.11 Inner Product Spaces 314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

More information

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1 PROBLEMS - APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.

More information

MATLAB: M-files; Numerical Integration Last revised : March, 2003

MATLAB: M-files; Numerical Integration Last revised : March, 2003 MATLAB: M-files; Numericl Integrtion Lst revised : Mrch, 00 Introduction to M-files In this tutoril we lern the bsics of working with M-files in MATLAB, so clled becuse they must use.m for their filenme

More information

AAPT UNITED STATES PHYSICS TEAM AIP 2010

AAPT UNITED STATES PHYSICS TEAM AIP 2010 2010 F = m Exm 1 AAPT UNITED STATES PHYSICS TEAM AIP 2010 Enti non multiplicnd sunt preter necessittem 2010 F = m Contest 25 QUESTIONS - 75 MINUTES INSTRUCTIONS DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD

More information

Einstein. Mechanics. In Grade 10 we investigated kinematics, or movement described in terms of velocity, acceleration, displacement, and so on.

Einstein. Mechanics. In Grade 10 we investigated kinematics, or movement described in terms of velocity, acceleration, displacement, and so on. Cmbridge University Press 978-0-521-68359-3 - Study nd Mster Physicl Sciences Grde 11 Lerner s Book Krin Kelder More informtion MODULE 1 Einstein Mechnics motion force Glileo Newton decelerte moment of

More information

HCP crystal planes are described using the Miller-Bravais indices, (hkil) Draw the hexagonal crystal planes whose Miller-Bravais indices are:

HCP crystal planes are described using the Miller-Bravais indices, (hkil) Draw the hexagonal crystal planes whose Miller-Bravais indices are: .6 How re crystllogrphic plnes indicted in HCP unit cells? In HCP unit cells, crystllogrphic plnes re indicted using four indices which correspond to four xes: three bsl xes of the unit cell,,, nd, which

More information

Algorithms Chapter 4 Recurrences

Algorithms Chapter 4 Recurrences Algorithms Chpter 4 Recurrences Outline The substitution method The recursion tree method The mster method Instructor: Ching Chi Lin 林清池助理教授 chingchilin@gmilcom Deprtment of Computer Science nd Engineering

More information

6 Energy Methods And The Energy of Waves MATH 22C

6 Energy Methods And The Energy of Waves MATH 22C 6 Energy Methods And The Energy of Wves MATH 22C. Conservtion of Energy We discuss the principle of conservtion of energy for ODE s, derive the energy ssocited with the hrmonic oscilltor, nd then use this

More information

Lecture 15 - Curve Fitting Techniques

Lecture 15 - Curve Fitting Techniques Lecture 15 - Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting - motivtion For root finding, we used given function to identify where it crossed zero where does fx

More information

Review guide for the final exam in Math 233

Review guide for the final exam in Math 233 Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered

More information

Area Between Curves: We know that a definite integral

Area Between Curves: We know that a definite integral Are Between Curves: We know tht definite integrl fx) dx cn be used to find the signed re of the region bounded by the function f nd the x xis between nd b. Often we wnt to find the bsolute re of region

More information

Week 7 - Perfect Competition and Monopoly

Week 7 - Perfect Competition and Monopoly Week 7 - Perfect Competition nd Monopoly Our im here is to compre the industry-wide response to chnges in demnd nd costs by monopolized industry nd by perfectly competitive one. We distinguish between

More information

Vector differentiation. Chapters 6, 7

Vector differentiation. Chapters 6, 7 Chpter 2 Vectors Courtesy NASA/JPL-Cltech Summry (see exmples in Hw 1, 2, 3) Circ 1900 A.D., J. Willird Gis invented useful comintion of mgnitude nd direction clled vectors nd their higher-dimensionl counterprts

More information

Warm-up for Differential Calculus

Warm-up for Differential Calculus Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:

More information

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324 A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................

More information

Scalar and Vector Quantities. A scalar is a quantity having only magnitude (and possibly phase). LECTURE 2a: VECTOR ANALYSIS Vector Algebra

Scalar and Vector Quantities. A scalar is a quantity having only magnitude (and possibly phase). LECTURE 2a: VECTOR ANALYSIS Vector Algebra Sclr nd Vector Quntities : VECTO NLYSIS Vector lgebr sclr is quntit hving onl mgnitude (nd possibl phse). Emples: voltge, current, chrge, energ, temperture vector is quntit hving direction in ddition to

More information

Section 5-4 Trigonometric Functions

Section 5-4 Trigonometric Functions 5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

More information

Two special Right-triangles 1. The

Two special Right-triangles 1. The Mth Right Tringle Trigonometry Hndout B (length of ) - c - (length of side ) (Length of side to ) Pythgoren s Theorem: for tringles with right ngle ( side + side = ) + = c Two specil Right-tringles. The

More information

SPECIAL PRODUCTS AND FACTORIZATION

SPECIAL PRODUCTS AND FACTORIZATION MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

More information

ASTR 170! 2010 S1 " Daniel Zucker! E7A 317 Mathematics as a tool for understanding physics

ASTR 170! 2010 S1  Daniel Zucker! E7A 317 Mathematics as a tool for understanding physics A New Scientific Er Grity nd Tides ASTR 170! 2010 S1 " Dniel Zucker! E7A 317 Mthemtics s tool for understnding physics zucker@science.mq.edu.u! 1 2 Velocity nd Accelertion Isc Newton (1643-1727)! Building

More information

Lectures 8 and 9 1 Rectangular waveguides

Lectures 8 and 9 1 Rectangular waveguides 1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves

More information

NCERT INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS. Trigonometric Ratios of the angle A in a triangle ABC right angled at B are defined as:

NCERT INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS. Trigonometric Ratios of the angle A in a triangle ABC right angled at B are defined as: INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS (A) Min Concepts nd Results Trigonometric Rtios of the ngle A in tringle ABC right ngled t B re defined s: side opposite to A BC sine of A = sin A = hypotenuse

More information

AP QUIZ #2 GRAPHING MOTION 1) POSITION TIME GRAPHS DISPLACEMENT Each graph below shows the position of an object as a function of time.

AP QUIZ #2 GRAPHING MOTION 1) POSITION TIME GRAPHS DISPLACEMENT Each graph below shows the position of an object as a function of time. AP QUIZ # GRAPHING MOTION ) POSITION TIME GRAPHS DISPLAEMENT Ech grph below shows the position of n object s function of time. A, B,, D, Rnk these grphs on the gretest mgnitude displcement during the time

More information

Projectile Motion CHAPTER 1

Projectile Motion CHAPTER 1 CHAPTER 1 PHYSICS ESSENTIALS STAGE 2 Projectile Motion Subject Outline In the bsence of ir resistnce nd moing under the ction of constnt grittionl force, projectile hs constnt ccelertion in the direction

More information

Chapter 6 Solving equations

Chapter 6 Solving equations Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign

More information

Sequences and Series

Sequences and Series Centre for Eduction in Mthemtics nd Computing Euclid eworkshop # 5 Sequences nd Series c 014 UNIVERSITY OF WATERLOO While the vst mjority of Euclid questions in this topic re use formule for rithmetic

More information

Matrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA

Matrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA CHAPTER 1 Mtrix Algebr PREAMBLE Tody, the importnce of mtrix lgebr is of utmost importnce in the field of physics nd engineering in more thn one wy, wheres before 1925, the mtrices were rrely used by the

More information

The Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx

The Chain Rule. rf dx. t t lim  (x) dt  (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the single-vrible chin rule extends to n inner

More information

m, where m = m 1 + m m n.

m, where m = m 1 + m m n. Lecture 7 : Moments nd Centers of Mss If we hve msses m, m 2,..., m n t points x, x 2,..., x n long the x-xis, the moment of the system round the origin is M 0 = m x + m 2 x 2 + + m n x n. The center of

More information

Polygons, Pingos, and Thermokarst! Oh my!

Polygons, Pingos, and Thermokarst! Oh my! Polygons, Pingos, nd Thermokrst! Oh my! Overview: Ares underlin with permfrost exhibit common fetures due to freezing nd thwing in such res. Ice wedge polygons, pingos, nd thermokrst re some of the most

More information