CHAPTER 9: Moments of Inertia
|
|
- Bonnie Benson
- 7 years ago
- Views:
Transcription
1 HPTER 9: Moments of nerti! Moment of nerti of res! Second Moment, or Moment of nerti, of n re! Prllel-is Theorem! Rdius of Grtion of n re! Determintion of the Moment of nerti of n re ntegrtion! Moments of nerti of omposite res! Polr Moment of nerti
2 9. Moment of nerti: Definition d(d(d ( ( d d O
3 9. Prllel-is Theorem of n re d entroidl is G d entroidl is d ( ' + d d [( ' + ( '( d + ( d ] d ( ' d + ( '( d d +, ( d d O + d ' d + d d + + d + + d JO J + d
4 9. Rdius of Grtion of n re k The rdius of grtion of n re with respect to the is is defined s the distnce k, where k. With similr definitions for the rdii of grtion of with respect to the is nd with respect to O, we hve O k k ko JO
5 9. Determintion of the Moment of nerti of n re ntegrtion The rectngulr moments of inerti nd of n re re defined s d d d These computtions re reduced to single integrtions choosing d to e thin strip prllel to one of the coordinte es. The result is d d d d 5
6 Moment of nerti of Rectngulr re. d d d (/d h d h/ d / d ' d h ( d h ( d ( h ( h/ h h 6
7 7 / h/ h d hd ( h ( d ' h d h ( / ( h h h d d d hd d (h/d
8 h/ h/ h h + d h + h h + h ( h( h 8
9 h d d Moment of nerti of Tringulr re. / d l d Using similr tringles, we hve l / h- d ntegrting d from to h, we otin h d h h d h [ h ] h h + h d d h h ( ( h h ( h h 6 d l h h l h h h d h d 9
10 Emple 9. Determine the moment of inerti of the shded re shown with respect to ech of the coordinte es. k
11 Moment of nerti. d k ( d d (-d k k k or d Sustituting nd / / ( / d / / / 7 ( 7 ( 7 / d 7 / 5/ 7 / d
12 Moment of nerti. k d d d d d ( d d 5 ( ( 5 ( ( 5 5 5
13 Emple 9. Determine the moment of inerti of the shded re shown with respect to ech of the coordinte es. (,
14 Moment of nerti. (, d ( d d d ( - d / ( 5/ ( d d ( d 7 7 / 7 7 /
15 Moment of nerti. d (, d ( d d ( - d ( d ( d ( d
16 9.5 Moment of nerti of omposite res d c similr theorem cn e used with the polr moment of inerti. The polr moment of inerti J O of n re out O nd the polr moment of inerti J of the re out its o centroid re relted to the distnce d etween points nd O the reltionship J O J + d The prllel-is theorem is used ver effectivel to compute the moment of inerti of composite re with respect to given is. 6
17 Emple 9. ompute the moment of inerti of the composite re shown. mm 5 mm 75 mm 75 mm 7
18 SOLUTON mm 5 mm 75 mm 75 mm (d ir h ( Rect ( + d ir [ ((5 ] Re [ π (5 + ( π 5 (75 ct ] ir 6 mm 8
19 Emple 9. Determine the moments of inerti of the em s cross-sectionl re shown out the nd centroidl es. 6 Dimension in mm 9
20 SOLUTON d B d D D 6 Dimension in mm ( + d + ( + d + ( + d B [ (( + ( ( ] + [ (6( + [ (( + ( ( ] + ].9 9 mm
21 Dimension in mm 6 B d d d ( ( ( B ] (5 ( (( [ ] ((6 [ ] (5 ( (( [ mm d d D D
22 Emple 9.5 (Prolem 9., Determine the moments of inerti nd the rdius of grtion of the shded re with respect to the nd es. mm mm 8 mm O 6 mm mm mm 6 mm mm mm
23 8 mm SOLUTON O mm mm B d mm mm [ (6( 6. mm d ] 6 mm mm mm 6 mm k + [ (8(8 ] B + [ (6(8 ] ( + d + ( + d + ( + d 9 mm 9 [( 6 + (8 8 + (8 6]. 9 ( + d + ( + d + ( + d B 6. k mm [( 6 + (8 8 + (8 6] [ ((6 + ( 6(7 ] + [ (8(8 + ] B + [ (8(6 + (8 6(7 B ] mm
24 Emple 9.6 (Prolem 9., Determine the moments of inerti nd the rdius of grtion of the shded re with respect to the nd es..5 m m m.5 m m O m m m m.5 m.5 m
25 .5 m m m.5 m ( + d 5 6 ( + d B ( + d B O d B d m m m [ (5(6 + ] [ (( [ (( + ( (.5 ] + ( ( ] B.5 m m m.5 m 6 m 6 k. 599 [(5 6 ( ( ] m ( + d ( + d ( + d B [ (6(5 ] [ (( ] B [ (( ] 6.5 m 6.5 k. 67 [(5 6 ( ( ] m 5
26 Emple 9.7 Determine the moments of inerti nd the rdius of grtion of the shded re with respect to the nd es nd t the centroidl es. cm cm 5 cm cm 5 cm 6
27 cm cm Moments of inerti out centroid d 5 (5(.5 5 cm cm Y.5 5 cm G.5 Y OR 5.5 cm [( ((5 + (5 ( ] + [( (5( + (5 ( ] 5.5 cm Y [(.5(5 ] + (.5( 5 (5.5 cm Moments of inerti out is [( (5( 5.5 cm + (5 ( ] + ((5 [( ((5 5 cm + (5 (.5 ] + (5( 5.5 k k. 88 cm 5 7
28 Emple 9.8 The strength of W6 57 rolled-steel em is incresed ttching 9 mm 9 mm plte to its upper flnge s shown. Determine the moment of inerti nd the rdius of grtion of the composite section with respect to n is which is prllel to the plte nd psses through the centroid of the section. 9 mm 9 mm 58 mm 7 mm 8
29 9 mm SOLUTON 9 mm Moment of nerti ' + ( ' plte ( ' wide flnge d Y 58 mm O 88.5 mm ( + d + ( + Y ' plte (9(9 + ' 6 [ 6. + (7(7.8 ] mm wide flnge + (5( mm entroid The wide-flnge shpe of W6 57 found referring to Fig mm 6. mm plte (9(9 5 mm YΣ Σ ' Rdius of Grtion k ' 6 57 mm ' k ' 9 mm (5+ 7 Y ( 5+ 7 (88.5(5 + ((7 Y 7. 8 mm 9
30 9.6 Polr Moment of nerti d The polr moment of inerti of n re with respect to the pole O is defined s r O J O r d The distnce from O to the element of re d is r. Oserving tht r +, we estlished the reltion J + O
31 Emple 9.9 ( Determine the centroidl polr moment of inerti of circulr re direct integrtion. ( Using the result of prt, determine the moment of inerti of circulr re with respect to dimeter. r O
32 SOLUTON. Polr Moment of nerti. dj O u d d πu du r O u du J O r djo u πu du ( π r u du J O π r. Moment of nerti with Respect to Dimeter. J + O π r dimeter π r
PROBLEM 4.1 SOLUTION. Knowing that the couple shown acts in a vertical plane, determine the stress at (a) point A, (b) point B.
PROBLEM.1 Knowing tht the couple shown cts in verticl plne, determine the stress t () point A, (b) point B. SOLUTON () (b) For rectngle: For cross sectionl re: 1 = bh 1 1 = 1 + + = ()(1.5) + ()(5.5) +
More informationAnswer, Key Homework 10 David McIntyre 1
Answer, Key Homework 10 Dvid McIntyre 1 This print-out should hve 22 questions, check tht it is complete. Multiple-choice questions my continue on the next column or pge: find ll choices efore mking your
More information, and the number of electrons is -19. e e 1.60 10 C. The negatively charged electrons move in the direction opposite to the conventional current flow.
Prolem 1. f current of 80.0 ma exists in metl wire, how mny electrons flow pst given cross section of the wire in 10.0 min? Sketch the directions of the current nd the electrons motion. Solution: The chrge
More informationAREA 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 informationMath 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.
Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose
More informationSection 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 informationPure C4. Revision Notes
Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd
More informationRadius of the Earth - Radii Used in Geodesy James R. Clynch February 2006
dius of the Erth - dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.
More informationPROBLEMS 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 informationExample 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 informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint
More informationv T R x m Version PREVIEW Practice 7 carroll (11108) 1
Version PEVIEW Prctice 7 crroll (08) his print-out should he 5 questions. Multiple-choice questions y continue on the next colun or pge find ll choices before nswering. Atwood Mchine 05 00 0.0 points A
More informationThe 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 informationSection 7-4 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More informationaddition, 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 informationVectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.
Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles
More informationPHY 140A: Solid State Physics. Solution to Homework #2
PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.
More informationDrawing Diagrams From Labelled Graphs
Drwing Digrms From Lbelled Grphs Jérôme Thièvre 1 INA, 4, venue de l Europe, 94366 BRY SUR MARNE FRANCE Anne Verroust-Blondet 2 INRIA Rocquencourt, B.P. 105, 78153 LE CHESNAY Cedex FRANCE Mrie-Luce Viud
More informationUse 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 informationCOMPONENTS: 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 information1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator
AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine so-clled volumes of
More informationB Conic Sections. B.1 Conic Sections. Introduction to Conic Sections. Appendix B.1 Conic Sections B1
Appendi B. Conic Sections B B Conic Sections B. Conic Sections Recognize the four bsic conics: circles, prbols, ellipses, nd hperbols. Recognize, grph, nd write equtions of prbols (verte t origin). Recognize,
More informationRIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS
RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is
More information2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration
Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 25-6 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting
More informationAPPLICATION OF INTEGRALS
APPLICATION OF INTEGRALS 59 Chpter 8 APPLICATION OF INTEGRALS One should study Mthemtics ecuse it is only through Mthemtics tht nture cn e conceived in hrmonious form. BIRKHOFF 8. Introduction In geometry,
More information10 AREA AND VOLUME 1. Before you start. Objectives
10 AREA AND VOLUME 1 The Tower of Pis is circulr bell tower. Construction begn in the 1170s, nd the tower strted lening lmost immeditely becuse of poor foundtion nd loose soil. It is 56.7 metres tll, with
More informationVolumes as integrals of cross-sections (Sect. 6.1) Volumes as integrals of cross-sections (Sect. 6.1)
Volumes s integrls of cross-sections (ect. 6.1) Te volume of simple regions in spce Volumes integrting cross-sections: Te generl cse. Certin regions wit oles. Volumes s integrls of cross-sections (ect.
More information. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2
7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6
More informationThe Acoustic Design of Soundproofing Doors and Windows
3 The Open Acoustics Journl, 1, 3, 3-37 The Acoustic Design of Soundproofing Doors nd Windows Open Access Nishimur Yuy,1, Nguyen Huy Qung, Nishimur Sohei 1, Nishimur Tsuyoshi 3 nd Yno Tkshi 1 Kummoto Ntionl
More informationApplications 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 informationThe remaining two sides of the right triangle are called the legs of the right triangle.
10 MODULE 6. RADICAL EXPRESSIONS 6 Pythgoren Theorem The Pythgoren Theorem An ngle tht mesures 90 degrees is lled right ngle. If one of the ngles of tringle is right ngle, then the tringle is lled right
More informationRadius of the Earth - Radii Used in Geodesy James R. Clynch Naval Postgraduate School, 2002
dius of the Erth - dii Used in Geodesy Jmes. Clynh vl Postgrdute Shool, 00 I. Three dii of Erth nd Their Use There re three rdii tht ome into use in geodesy. These re funtion of ltitude in the ellipsoidl
More informationScalar 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 informationGraphs 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 informationReview 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 informationReview Problems for the Final of Math 121, Fall 2014
Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since
More informationLECTURE #05. Learning Objectives. How does atomic packing factor change with different atom types? How do you calculate the density of a material?
LECTURE #05 Chpter : Pcking Densities nd Coordintion Lerning Objectives es How does tomic pcking fctor chnge with different tom types? How do you clculte the density of mteril? 2 Relevnt Reding for this
More informationGeometry 7-1 Geometric Mean and the Pythagorean Theorem
Geometry 7-1 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the
More informationP.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 informationOr more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
More information5.2. 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 informationPhysics 2102 Lecture 2. Physics 2102
Physics 10 Jonthn Dowling Physics 10 Lecture Electric Fields Chrles-Augustin de Coulomb (1736-1806) Jnury 17, 07 Version: 1/17/07 Wht re we going to lern? A rod mp Electric chrge Electric force on other
More informationCS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001
CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic
More information** Dpt. Chemical Engineering, Kasetsart University, Bangkok 10900, Thailand
Modelling nd Simultion of hemicl Processes in Multi Pulse TP Experiment P. Phnwdee* S.O. Shekhtmn +. Jrungmnorom** J.T. Gleves ++ * Dpt. hemicl Engineering, Ksetsrt University, Bngkok 10900, Thilnd + Dpt.hemicl
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationWeek 11 - Inductance
Week - Inductnce November 6, 202 Exercise.: Discussion Questions ) A trnsformer consists bsiclly of two coils in close proximity but not in electricl contct. A current in one coil mgneticlly induces n
More informationPROBLEMS 05 - ELLIPSE Page 1
PROBLEMS 0 ELLIPSE Pge 1 ( 1 ) The edpoits A d B of AB re o the X d Yis respectivel If AB > 0 > 0 d P divides AB from A i the rtio : the show tht P lies o the ellipse 1 ( ) If the feet of the perpediculrs
More informationSummary: Vectors. This theorem is used to find any points (or position vectors) on a given line (direction vector). Two ways RT can be applied:
Summ: Vectos ) Rtio Theoem (RT) This theoem is used to find n points (o position vectos) on given line (diection vecto). Two ws RT cn e pplied: Cse : If the point lies BETWEEN two known position vectos
More informationDistributions. (corresponding to the cumulative distribution function for the discrete case).
Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive
More informationwww.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)
www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input
More informationOperations 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 informationVectors and dyadics. Chapter 2. Summary. 2.1 Examples of scalars, vectors, and dyadics
Chpter 2 Vectors nd dydics Summry Circ 1900 A.D., J. Willird Gis proposed the ide of vectors nd their higher-dimensionl counterprts dydics, tridics, ndpolydics. Vectors descrie three-dimensionl spce nd
More informationIntroduction to Integration Part 2: The Definite Integral
Mthemtics Lerning Centre Introduction to Integrtion Prt : The Definite Integrl Mr Brnes c 999 Universit of Sdne Contents Introduction. Objectives...... Finding Ares 3 Ares Under Curves 4 3. Wht is the
More informationLECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.
LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 64-83.
More informationPhysics 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 informationSPH simulation of fluid-structure interaction problems
Diprtimento di ingegneri idrulic e mientle SPH simultion of fluid-structure interction prolems C. Antoci, M. Gllti, S. Siill Reserch project Prolem: deformtion of plte due to the ction of fluid (lrge displcement
More information10.6 Applications of Quadratic Equations
10.6 Applictions of Qudrtic Equtions In this section we wnt to look t the pplictions tht qudrtic equtions nd functions hve in the rel world. There re severl stndrd types: problems where the formul is given,
More information2012 Mathematics. Higher. Finalised Marking Instructions
0 Mthemts Higher Finlised Mrking Instructions Scottish Quliftions Authority 0 The informtion in this publtion my be reproduced to support SQA quliftions only on non-commercil bsis. If it is to be used
More informationVectors and dyadics. Chapter 2. Summary. 2.1 Examples of scalars, vectors, and dyadics
Chpter 2 Vectors nd dydics Summry Circ 1900 A.D., J. Willird Gis proposed the ide of vectors nd their higher-dimensionl counterprts dydics, tridics, ndpolydics. Vectors descrie three-dimensionl spce nd
More informationMECHANICAL PRINCIPLES HNC/D MOMENTS OF AREA. Define and calculate 1st. moments of areas. Define and calculate 2nd moments of areas.
MECHANICAL PRINCIPLES HNC/D MOMENTS OF AREA The concepts of first and second moments of area fundamental to several areas of engineering including solid mechanics and fluid mechanics. Students who are
More informationBayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the
More informationReasoning 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 informationSection 16: Neutral Axis and Parallel Axis Theorem 16-1
Section 16: Neutral Axis and Parallel Axis Theorem 16-1 Geometry of deformation We will consider the deformation of an ideal, isotropic prismatic beam the cross section is symmetric about y-axis All parts
More informationSECTION 7-2 Law of Cosines
516 7 Additionl Topis in Trigonometry h d sin s () tn h h d 50. Surveying. The lyout in the figure t right is used to determine n inessile height h when seline d in plne perpendiulr to h n e estlished
More informationLesson 4.1 Triangle Sum Conjecture
Lesson 4.1 ringle um onjecture Nme eriod te n ercises 1 9, determine the ngle mesures. 1. p, q 2., y 3., b 31 82 p 98 q 28 53 y 17 79 23 50 b 4. r, s, 5., y 6. y t t s r 100 85 100 y 30 4 7 y 31 7. s 8.
More informationVolumes by Cylindrical Shells: the Shell Method
olumes Clinril Shells: the Shell Metho Another metho of fin the volumes of solis of revolution is the shell metho. It n usull fin volumes tht re otherwise iffiult to evlute using the Dis / Wsher metho.
More informationDAGmaps: Space Filling Visualization of Directed Acyclic Graphs
Journl of Grph Algorithms nd Applictions http://jg.info/ vol. 13, no. 3, pp. 319 347 (2009) DAGmps: Spce Filling Visuliztion of Directed Acyclic Grphs Vssilis Tsirs 1,2 Sofi Trintfilou 1,2 Ionnis G. Tollis
More informationEffect of microscopic damage events on static and ballistic impact strength of triaxial braid composites
Astrct Sumitted to CompTest 2008 4th Interntionl Conference on Composites Testing nd Model Identifiction 20-22 Octoer, 2008 Dton, OH Effect of microscopic dmge events on sttic nd llistic impct strength
More informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More informationChapter Outline How do atoms arrange themselves to form solids? Types of Solids
Chpter Outline How do toms rrnge themselves to form solids? Fundmentl concepts nd lnguge Unit cells Crystl structures Fce-centered cubic Body-centered cubic Hexgonl close-pcked Close pcked crystl structures
More informationSensorless Force Estimation for Robots with Friction
Proc. Austrlsin Conference on Rootics nd Automtion Aucklnd, 7-9 Novemer Sensorless orce Estimtion for Roots with riction John W.L Simpson, Chris D Cook, Zheng Li School of Electricl, Computer nd Telecommunictions
More informationUnit 6: Exponents and Radicals
Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -
More informationFirm Objectives. The Theory of the Firm II. Cost Minimization Mathematical Approach. First order conditions. Cost Minimization Graphical Approach
Pro. Jy Bhttchry Spring 200 The Theory o the Firm II st lecture we covered: production unctions Tody: Cost minimiztion Firm s supply under cost minimiztion Short vs. long run cost curves Firm Ojectives
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives
More informationtrademark and symbol guidelines FOR CORPORATE STATIONARY APPLICATIONS reviewed 01.02.2007
trdemrk nd symbol guidelines trdemrk guidelines The trdemrk Cn be plced in either of the two usul configurtions but horizontl usge is preferble. Wherever possible the trdemrk should be plced on blck bckground.
More informationCalculating Principal Strains using a Rectangular Strain Gage Rosette
Clulting Prinipl Strins using Retngulr Strin Gge Rosette Strin gge rosettes re used often in engineering prtie to determine strin sttes t speifi points on struture. Figure illustrtes three ommonly used
More informationSolenoid Operated Proportional Directional Control Valve (with Pressure Compensation, Multiple Valve Series)
Solenoid Operted Proportionl Directionl Control Vlve (with Pressure Compenstion, Multiple Vlve Series) Hydrulic circuit (Exmple) v Fetures hese stcking type control vlves show pressure compensted type
More informationMA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!
MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more
More informationLectures 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 informationRotational Equilibrium: A Question of Balance
Prt of the IEEE Techer In-Service Progrm - Lesson Focus Demonstrte the concept of rottionl equilirium. Lesson Synopsis The Rottionl Equilirium ctivity encourges students to explore the sic concepts of
More informationRatio and Proportion
Rtio nd Proportion Rtio: The onept of rtio ours frequently nd in wide vriety of wys For exmple: A newspper reports tht the rtio of Repulins to Demorts on ertin Congressionl ommittee is 3 to The student/fulty
More informationLec 2: Gates and Logic
Lec 2: Gtes nd Logic Kvit Bl CS 34, Fll 28 Computer Science Cornell University Announcements Clss newsgroup creted Posted on we-pge Use it for prtner finding First ssignment is to find prtners Due this
More informationVector 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 informationThinking out of the Box... Problem It s a richer problem than we ever imagined
From the Mthemtics Techer, Vol. 95, No. 8, pges 568-574 Wlter Dodge (not pictured) nd Steve Viktor Thinking out of the Bo... Problem It s richer problem thn we ever imgined The bo problem hs been stndrd
More informationProject 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 informationEnd of term: TEST A. Year 4. Name Class Date. Complete the missing numbers in the sequences below.
End of term: TEST A You will need penil nd ruler. Yer Nme Clss Dte Complete the missing numers in the sequenes elow. 8 30 3 28 2 9 25 00 75 25 2 Put irle round ll of the following shpes whih hve 3 shded.
More informationTrowel Notch Sizes for Installation of Floor Coverings, Wood Flooring and Tiles
TKB-Tehnil Briefing Note 6 Trowel Noth s for Instlltion of Floor Coverings, Wood Flooring nd Tiles Version: My 2007 Prepred y the Tehnishe Kommission Buklestoffe (TKB) (Tehnil Commission on Constrution
More information6.5 - Areas of Surfaces of Revolution and the Theorems of Pappus
Lecture_06_05.n 1 6.5 - Ares of Surfces of Revolution n the Theorems of Pppus Introuction Suppose we rotte some curve out line to otin surfce, we cn use efinite integrl to clculte the re of the surfce.
More information9.3. 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 informationRadiographic identification of threaded endoseous dental implants
Rdiogrphic identifiction of threded endoseous dentl implnts Indir G. Shiwl, BDS, DMD, MS, Ronld D. Woody, DDS, Byron W. Benson, DDS, MS, c nd Guillermo E. Guillen, DDS d Bylor College of Dentistry, Texs
More informationCUBIC-FOOT VOLUME OF A LOG
CUBIC-FOOT VOLUME OF A LOG Wys to clculte cuic foot volume ) xylometer: tu of wter sumerge tree or log in wter nd find volume of wter displced. ) grphic: exmple: log length = 4 feet, ech section feet in
More informationMultiplication and Division - Left to Right. Addition and Subtraction - Left to Right.
Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction
More informationExercises in KS3 Mathematics Levels 7-8. R Joinson
Exercises in KS Mthemtics Levels 7-8 R Joinson Sumbooks Northwy Chester CH 8BB Exercises in KS Mthemtics - Levels 7 nd 8 First Published 00 Copyright R Joinson nd Sumbooks This pckge of worksheets is sold
More informationFAULT TREES AND RELIABILITY BLOCK DIAGRAMS. Harry G. Kwatny. Department of Mechanical Engineering & Mechanics Drexel University
SYSTEM FAULT AND Hrry G. Kwtny Deprtment of Mechnicl Engineering & Mechnics Drexel University OUTLINE SYSTEM RBD Definition RBDs nd Fult Trees System Structure Structure Functions Pths nd Cutsets Reliility
More informationAn efficient integral equation technique for the analysis of arbitrarily shaped capacitive waveguide circuits
RADIO SCIENCE, VOL. 46,, doi:10.1029/2010rs004458, 2011 An efficient integrl eqution technique for the nlysis of rbitrrily shped cpcitive wveguide circuits F. D. Quesd Pereir, 1 P. Ver Cstejón, 1 A. Álvrez
More informationPHY 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 information2 DIODE CLIPPING and CLAMPING CIRCUITS
2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of
More information