ME 101: Engineering Mechanics
|
|
- Rosemary Martin
- 6 years ago
- Views:
Transcription
1 ME 0: Engineering Mechanics Rajib Kumar Bhattacharja Department of Civil Engineering ndian nstitute of Technolog Guwahati M Block : Room No 005 : Tel: 8
2 Area Moments of nertia Parallel Ais Theorem Consider moment of inertia of an area A with respect to the ais AA da The ais BB passes through the area centroid and is called a centroidal ais. Parallel Ais theorem: an ais centroidal ais Ad The two aes should be parallel to each other. da da d ( d ) da da d Second term 0 since centroid lies on BB ( da c A, and c 0 da Ad Parallel Ais theorem
3 Area Moments of nertia Parallel Ais Theorem Moment of inertia T of a circular area with respect to a tangent to the circle, T 5 π r Ad π r ( π r ) Moment of inertia of a triangle with respect to a centroidal ais, AA BB BB AA 36 bh Ad Ad 3 bh 3 r bh ( h) The moment of inertia of a composite area A about a given ais is obtained b adding the moments of inertia of the component areas A, A, A 3,..., with respect to the same ais. 3
4 Area Moments of nertia: Standard Ms Moment of inertia about -ais Moment of inertia about -ais Answer 3 h 3 h Moment of inertia about -ais Moment of inertia about -ais h h Moment of inertia about -ais passing through C h h
5 Moment of inertia about -ais h Moment of inertia about -ais 36 h
6 Moment of inertia about -ais Moment of inertia about -ais passing through O
7 Moment of inertia about -ais 8 Moment of inertia about -ais passing through O
8 Moment of inertia about -ais 6 Moment of inertia about -ais passing through O 8
9 Moment of inertia about -ais Moment of inertia about -ais Moment of inertia about -ais passing through O
10 Area Moments of nertia Eample: Determine the moment of inertia of the shaded area with respect to the ais. SOLUTON: Compute the moments of inertia of the bounding rectangle and half-circle with respect to the ais. The moment of inertia of the shaded area is obtained b subtracting the moment of inertia of the half-circle from the moment of inertia of the rectangle.
11 Area Moments of nertia Eample: Solution ( )( 90) r a 38. mm 3π 3π b 0 - a 8.8 mm A πr π ( 90) mm SOLUTON: Compute the moments of inertia of the bounding rectangle and half-circle with respect to the ais. Rectangle: bh mm 3 3 ( )( ) Half-circle: moment of inertia with respect to AA, 6 πr mm AA π 8 8 ( ) Moment of inertia with respect to, AA Aa mm 6 3 ( ) (.7 0 )( 38.) moment of inertia with respect to, Ab mm 6 3 (.7 0 )( 8.8)
12 Area Moments of nertia Eample: Solution The moment of inertia of the shaded area is obtained b subtracting the moment of inertia of the half-circle from the moment of inertia of the rectangle mm mm mm
13 Consider area () 3 h Consider area () Consider area (3) h
14 Determine the moment of inertia and the radius of gration of the area shown in the fig
15 Area Moments of nertia Products of nertia: for problems involving unsmmetrical cross-sections and in calculation of M about rotated aes. t ma be ve, -ve, or zero Product of nertia of area A w.r.t. - aes: da and are the coordinates of the element of area da When the ais, the ais, or both are an ais of smmetr, the product of inertia is zero. Parallel ais theorem for products of inertia: A - Quadrants -
16 Area Moments of nertia Rotation of Aes Product of inertia is useful in calculating inclined aes. Determination of aes about which the M is a maimum and a minimum da Moments and product of inertia w.r.t. new aes and? Note: cosθ sinθ da cosθ sinθ da ' ' ' ' ' ' da da ' ' da ( cosθ sinθ ) da ( cosθ sinθ ) da ( cosθ sinθ )( cosθ sinθ )da cos θ cos θ sin θ cos θ sinθ cosθ / sin θ cos θ sin θ cos θ ' cos θ cos θ sin θ cos θ sin θ sin θ
17 Area Moments of nertia Rotation of Aes cosθ cosθ ' sinθ cosθ sinθ sinθ Adding first two eqns: z The Polar O Angle which makes and either ma or min can be found b setting the derivative of either or w.r.t. θ equal to zero: ( ) 0 d sinθ cosθ dθ Denoting this critical angle b α ' tanα two values of α which differ b π since tanα tan(απ) two solutions for α will differ b π/ one value of α will define the ais of maimum M and the other defines the ais of minimum M These two rectangular aes are called the principal aes of inertia
18 Area Moments of nertia Rotation of Aes θ θ θ θ θ θ cos sin sin cos sin cos ' cos sin tan α α α Substituting in the third eqn for critical value of θ: 0 Product of nertia is zero for the Principal Aes of inertia Substituting sinα and cosα in first two eqns for Principal Moments of nertia: ( ) ( ) min ma α
19 Squaring both the equation and adding
20 Defining And Which is a equation of circle with center,0 and radius
21 Area Moments of nertia Mohr s Circle of nertia: Following relations can be represented graphicall b a diagram called Mohr s Circle For given values of,, &, corresponding values of,, & ma be determined from the diagram for an desired angle θ. θ θ θ θ θ θ cos sin sin cos sin cos ' tan α ( ) ( ) min ma α ( ) ave ave R R At the points A and B, 0 and is a maimum and minimum, respectivel. R ave ± ma,min
22 Area Moments of nertia Mohr s Circle of nertia: Construction tanα cosθ cosθ ' sinθ cosθ ma α 0 ( ) ( ) sinθ sinθ Choose horz ais M Choose vert ais P Point A known {, } Point B known {, - } Circle with dia AB Angle α for Area Angle α to horz (same sense) ma, min Angle to θ Angle OA to OC θ Same sense Point C, Point D
23 Area Moments of nertia Eample: Product of nertia SOLUTON: Determine the product of inertia using direct integration with the parallel ais theorem on vertical differential area strips Appl the parallel ais theorem to evaluate the product of inertia with respect to the centroidal aes. Determine the product of inertia of the right triangle (a) with respect to the and aes and (b) with respect to centroidal aes parallel to the and aes.
24 Area Moments of nertia Eamples SOLUTON: Determine the product of inertia using direct integration with the parallel ais theorem on vertical differential area strips b h d b h d da b h el el ntegrating d from 0 to b, ( ) b b b el el b b h d b b h d b h da d h b
25 Area Moments of nertia Eamples SOLUTON Appl the parallel ais theorem to evaluate the product of inertia with respect to the centroidal aes. b 3 3 With the results from part a, b h h A ( b)( h)( bh) 3 3 b h 7
26 Area Moments of nertia Eample: Mohr s Circle of nertia SOLUTON: Plot the points (, ) and (,- ). Construct Mohr s circle based on the circle diameter between the points. The moments and product of inertia with respect to the and aes are 7.06 mm,.606 mm, and mm. Using Mohr s circle, determine (a) the principal aes about O, (b) the values of the principal moments about O, and (c) the values of the moments and product of inertia about the and aes Based on the circle, determine the orientation of the principal aes and the principal moments of inertia. Based on the circle, evaluate the moments and product of inertia with respect to the aes.
27 Area Moments of nertia Eample: Mohr s Circle of nertia SOLUTON: Plot the points (, ) and (,- ). Construct Mohr s circle based on the circle diameter between the points. OC CD R ( ) ( ) ave ( CD) ( DX ) mm 6 mm 6 mm mm mm 6 mm Based on the circle, determine the orientation of the principal aes and the principal moments of inertia. DX tan θ m.097 θ m 7. 6 θ m 3. 8 CD
28 Area Moments of nertia Eample: Mohr s Circle of nertia OC ave R mm 6 mm Based on the circle, evaluate the moments and product of inertia with respect to the aes. The points X and Y corresponding to the and aes are obtained b rotating CX and CY counterclockwise through an angle θ (60 o ) 0 o. The angle that CX forms with the horz is φ 0 o o 7. o. ϕ Rcos7. ' OF OC CX cos ave OG OC CY cosϕ Rcos7. ' ave 6 mm o o mm R sin 7. ' FX CY sinϕ o mm
29 Determine the product of the inertia of the shaded area shown below about the - aes.
30 Solution: Parallel ais theorem: Ī d d A Both areas () and () are their centroidal Ais Ī 0 for both area. Therefore, for Area (): d d A mm Similarl, for Area (): d d A mm Total mm
31 <, () >, () >, (-) <, (-)
32 Area Moments of nertia Mass Moments of nertia (): mportant in Rigid Bod Dnamics - is a measure of distribution of mass of a rigid bod w.r.t. the ais in question (constant propert for that ais). - Units are (mass)(length) kg.m Consider a three dimensional bod of mass m Mass moment of inertia of this bod about ais O-O: r dm ntegration is over the entire bod. r perpendicular distance of the mass element dm from the ais O-O
33 Area Moments of nertia Moments of nertia of Thin Plates For a thin plate of uniform thickness t and homogeneous material of densit ρ, the mass moment of inertia with respect to ais AA contained in the plate is AA r dm ρt r ρt AA, area da Similarl, for perpendicular ais BB which is also contained in the plate, BB ρt BB, area For the ais CC which is perpendicular to the plate, CC AA BB ( ) ρt JC, area ρt AA, area BB, area
34 Area Moments of nertia Moments of nertia of Thin Plates For the principal centroidal aes on a rectangular plate, 3 ( ) a b 3 ( ) ab m( a ) A A ρt AA, area ρt ma B B ρt BB, area ρt mb C C AA, mass BB, mass b For centroidal aes on a circular plate, ( r ) A A BB ρt AA area ρt π, mr
35 Area Moments of nertia Moments of nertia of a 3D Bod b ntegration Moment of inertia of a homogeneous bod is obtained from double or triple integrations of the form ρ r dv For bodies with two planes of smmetr, the moment of inertia ma be obtained from a single integration b choosing thin slabs perpendicular to the planes of smmetr for dm. The moment of inertia with respect to a particular ais for a composite bod ma be obtained b adding the moments of inertia with respect to the same ais of the components.
36 Q. No. Determine the moment of inertia of a slender rod of length L and mass m with respect to an ais which is perpendicular to the rod and passes through one end of the rod. Solution
37 Q. No. For the homogeneous rectangular prism shown, determine the moment of inertia with respect to z ais Solution
38 Area Moments of nertia M of some common geometric shapes
39 Q. Determine the angleαwhich locates the principal aes of inertia through point O for the rectangular area (Figure 5). Construct the Mohr s circle of inertia and specif the corresponding values of ma and min. b b O α
40 () () ( )() With this data, plot the Mohr s circle, and using trigonometr calculate the angle α α tan - (b /b ) 5 o Therefore, α.5 o (clockwise w.r.t. ) From Mohr s Circle: ma 3.08 min 0.5
41 Q. No. Determine the moments and product of inertia of the area of the square with respect to the '- ' aes. ; 0 ( ) Withθ30 o, using the equation of moment of inertia about an inclined aes we get, ' 0 60 ( ) 0.68 ' 0 60 ( ) ' 0( ) ( )0.50 Alternativel, Mohr s Circle ma be used to determine the three quantities.
Distributed Forces: Moments of Inertia
Distributed Forces: Moments of nertia Contents ntroduction Moments of nertia of an Area Moments of nertia of an Area b ntegration Polar Moments of nertia Radius of Gration of an Area Sample Problems Parallel
More informationSTATICS. Moments of Inertia VECTOR MECHANICS FOR ENGINEERS: Ninth Edition CHAPTER. Ferdinand P. Beer E. Russell Johnston, Jr.
N E 9 Distributed CHAPTER VECTOR MECHANCS FOR ENGNEERS: STATCS Ferdinand P. Beer E. Russell Johnston, Jr. Lecture Notes: J. Walt Oler Teas Tech Universit Forces: Moments of nertia Contents ntroduction
More informationSTATICS. Moments of Inertia VECTOR MECHANICS FOR ENGINEERS: Seventh Edition CHAPTER. Ferdinand P. Beer
00 The McGraw-Hill Companies, nc. All rights reserved. Seventh E CHAPTER VECTOR MECHANCS FOR ENGNEERS: 9 STATCS Ferdinand P. Beer E. Russell Johnston, Jr. Distributed Forces: Lecture Notes: J. Walt Oler
More informationME 141. Lecture 8: Moment of Inertia
ME 4 Engineering Mechanics Lecture 8: Moment of nertia Ahmad Shahedi Shakil Lecturer, Dept. of Mechanical Engg, BUET E-mail: sshakil@me.buet.ac.bd, shakil679@gmail.com Website: teacher.buet.ac.bd/sshakil
More information3/31/ Product of Inertia. Sample Problem Sample Problem 10.6 (continue)
/1/01 10.6 Product of Inertia Product of Inertia: I xy = xy da When the x axis, the y axis, or both are an axis of symmetry, the product of inertia is zero. Parallel axis theorem for products of inertia:
More informationMoments and Product of Inertia
Moments and Product of nertia Contents ntroduction( 绪论 ) Moments of nertia of an Area( 平面图形的惯性矩 ) Moments of nertia of an Area b ntegration( 积分法求惯性矩 ) Polar Moments of nertia( 极惯性矩 ) Radius of Gration
More informationSOLUTION Determine the moment of inertia for the shaded area about the x axis. I x = y 2 da = 2 y 2 (xdy) = 2 y y dy
5. Determine the moment of inertia for the shaded area about the ais. 4 4m 4 4 I = da = (d) 4 = 4 - d I = B (5 + (4)() + 8(4) ) (4 - ) 3-5 4 R m m I = 39. m 4 6. Determine the moment of inertia for the
More informationSTATICS. Distributed Forces: Moments of Inertia VECTOR MECHANICS FOR ENGINEERS: Eighth Edition CHAPTER. Ferdinand P. Beer E. Russell Johnston, Jr.
007 The McGraw-Hill Companies, nc. All rights reserved. Eighth E CHAPTER 9 VECTOR MECHANCS FOR ENGNEERS: STATCS Ferdinand P. Beer E. Russell Johnston, Jr. Lecture Notes: J. Walt Oler Texas Tech University
More information10 3. Determine the moment of inertia of the area about the x axis.
10 3. Determine the moment of inertia of the area about the ais. m m 10 4. Determine the moment of inertia of the area about the ais. m m 10 3. Determine the moment of inertia of the shaded area about
More informationStatics: Lecture Notes for Sections 10.1,10.2,10.3 1
Chapter 10 MOMENTS of INERTIA for AREAS, RADIUS OF GYRATION Today s Objectives: Students will be able to: a) Define the moments of inertia (MoI) for an area. b) Determine the MoI for an area by integration.
More informationMoments of Inertia. Notation:
RCH 1 Note Set 9. S015abn Moments of nertia Notation: b d d d h c Jo O = name for area = name for a (base) width = calculus smbol for differentiation = name for a difference = name for a depth = difference
More informationENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 3 Solutions [Multiple Integration; Lines of Force]
ENGI 44 Advanced Calculus for Engineering Facult of Engineering and Applied Science Problem Set Solutions [Multiple Integration; Lines of Force]. Evaluate D da over the triangular region D that is bounded
More informationENGI 4430 Multiple Integration Cartesian Double Integrals Page 3-01
ENGI 4430 Multiple Integration Cartesian Double Integrals Page 3-01 3. Multiple Integration This chapter provides only a very brief introduction to the major topic of multiple integration. Uses of multiple
More informationPROBLEM Area of Problem I = 471,040 mm xy 2(471,040) 252,757 1,752,789 = or θ m = and = (1,002,773 ± 885,665) mm
PROBLEM 9.88 F the area indicated, determine the ientation of the principal aes at the igin the cresponding values of the moments of inertia. Area of Problem 9.75. From Problem 9.8: Problem 9.75: = 5,757
More informationMECHANICS OF MATERIALS
CHAPTER MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Teas Tech Universit Transformations of Stress and Strain 006 The McGraw-Hill Companies,
More informationChapter 5 Equilibrium of a Rigid Body Objectives
Chapter 5 Equilibrium of a Rigid Bod Objectives Develop the equations of equilibrium for a rigid bod Concept of the free-bod diagram for a rigid bod Solve rigid-bod equilibrium problems using the equations
More informationChapter 10: Moments of Inertia
Chapter 10: Moments of Inertia Chapter Objectives To develop a method for determining the moment of inertia and product of inertia for an area with respect to given x- and y-axes. To develop a method for
More informationMoment of Inertia and Centroid
Chapter- Moment of nertia and Centroid Page- 1. Moment of nertia and Centroid Theory at a Glance (for ES, GATE, PSU).1 Centre of gravity: The centre of gravity of a body defined as the point through which
More informationEMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 4 Pure Bending
EA 3702 echanics & aterials Science (echanics of aterials) Chapter 4 Pure Bending Pure Bending Ch 2 Aial Loading & Parallel Loading: uniform normal stress and shearing stress distribution Ch 3 Torsion:
More information2. Supports which resist forces in two directions. Fig Hinge. Rough Surface. Fig Rocker. Roller. Frictionless Surface
4. Structural Equilibrium 4.1 ntroduction n statics, it becomes convenient to ignore the small deformation and displacement. We pretend that the materials used are rigid, having the propert or infinite
More information10.5 MOMENT OF INERTIA FOR A COMPOSITE AREA
10.5 MOMENT OF NERTA FOR A COMPOSTE AREA A composite area is made by adding or subtracting a series of simple shaped areas like rectangles, triangles, and circles. For example, the area on the left can
More information[4] Properties of Geometry
[4] Properties of Geometr Page 1 of 6 [4] Properties of Geometr [4.1] Center of Gravit and Centroid [4.] Composite Bodies [4.3] Moments of Inertia [4.4] Composite reas and Products of Inertia [4] Properties
More informationMECHANICS OF MATERIALS
00 The McGraw-Hill Companies, Inc. All rights reserved. T Edition CHAPTER MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Teas Tech Universit
More informationBEAMS: SHEAR AND MOMENT DIAGRAMS (FORMULA)
LETURE Third Edition BEMS: SHER ND MOMENT DGRMS (FORMUL). J. lark School of Engineering Department of ivil and Environmental Engineering 1 hapter 5.1 5. b Dr. brahim. ssakkaf SPRNG 00 ENES 0 Mechanics
More informationEXERCISES Chapter 15: Multiple Integrals. Evaluating Integrals in Cylindrical Coordinates
08 Chapter 5: Multiple Integrals EXERCISES 5.6 Evaluating Integrals in Clindrical Evaluate the clindrical coordinate integrals in Eercises 6... 3. 4. 5. 6. Changing Order of Integration in Clindrical The
More informationPHY 5246: Theoretical Dynamics, Fall Assignment # 9, Solutions. y CM (θ = 0) = 2 ρ m
PHY 546: Theoretical Dnamics, Fall 5 Assignment # 9, Solutions Graded Problems Problem (.a) l l/ l/ CM θ x In order to find the equation of motion of the triangle, we need to write the Lagrangian, with
More informationME 201 Engineering Mechanics: Statics
ME 0 Engineering Mechanics: Statics Unit 9. Moments of nertia Definition of Moments of nertia for Areas Parallel-Axis Theorem for an Area Radius of Gyration of an Area Moments of nertia for Composite Areas
More informationSET-I SECTION A SECTION B. General Instructions. Time : 3 hours Max. Marks : 100
General Instructions. All questions are compulsor.. This question paper contains 9 questions.. Questions - in Section A are ver short answer tpe questions carring mark each.. Questions 5- in Section B
More informationQ1. If (1, 2) lies on the circle. x 2 + y 2 + 2gx + 2fy + c = 0. which is concentric with the circle x 2 + y 2 +4x + 2y 5 = 0 then c =
Q1. If (1, 2) lies on the circle x 2 + y 2 + 2gx + 2fy + c = 0 which is concentric with the circle x 2 + y 2 +4x + 2y 5 = 0 then c = a) 11 b) -13 c) 24 d) 100 Solution: Any circle concentric with x 2 +
More informationProperties of surfaces II: Second moment of area
Properties of surfaces II: Second moment of area Just as we have discussing first moment of an area and its relation with problems in mechanics, we will now describe second moment and product of area of
More informationCHAPTER 4 Stress Transformation
CHAPTER 4 Stress Transformation ANALYSIS OF STRESS For this topic, the stresses to be considered are not on the perpendicular and parallel planes only but also on other inclined planes. A P a a b b P z
More informationSpecial Mathematics Notes
Special Mathematics Notes Tetbook: Classroom Mathematics Stds 9 & 10 CHAPTER 6 Trigonometr Trigonometr is a stud of measurements of sides of triangles as related to the angles, and the application of this
More informationStress and Strain ( , 3.14) MAE 316 Strength of Mechanical Components NC State University Department of Mechanical & Aerospace Engineering
(3.8-3.1, 3.14) MAE 316 Strength of Mechanical Components NC State Universit Department of Mechanical & Aerospace Engineering 1 Introduction MAE 316 is a continuation of MAE 314 (solid mechanics) Review
More informationSecond Moments or Moments of Inertia
Second Moments or Moments of Inertia The second moment of inertia of an element of area such as da in Figure 1 with respect to any axis is defined as the product of the area of the element and the square
More informationTime : 3 hours 02 - Mathematics - July 2006 Marks : 100 Pg - 1 Instructions : S E CT I O N - A
Time : 3 hours 0 Mathematics July 006 Marks : 00 Pg Instructions :. Answer all questions.. Write your answers according to the instructions given below with the questions. 3. Begin each section on a new
More informationIntegrals in cylindrical, spherical coordinates (Sect. 15.7)
Integrals in clindrical, spherical coordinates (Sect. 15.7 Integration in spherical coordinates. Review: Clindrical coordinates. Spherical coordinates in space. Triple integral in spherical coordinates.
More informationDynamics and control of mechanical systems
JU 18/HL Dnamics and control of mechanical sstems Date Da 1 (3/5) 5/5 Da (7/5) Da 3 (9/5) Da 4 (11/5) Da 5 (14/5) Da 6 (16/5) Content Revie of the basics of mechanics. Kinematics of rigid bodies coordinate
More informationChapter 2 GEOMETRIC ASPECT OF THE STATE OF SOLICITATION
Capter GEOMETRC SPECT OF THE STTE OF SOLCTTON. THE DEFORMTON ROUND PONT.. Te relative displacement Due to te influence of external forces, temperature variation, magnetic and electric fields, te construction
More informationENGI Multiple Integration Page 8-01
ENGI 345 8. Multiple Integration Page 8-01 8. Multiple Integration This chapter provides only a very brief introduction to the major topic of multiple integration. Uses of multiple integration include
More informationTHE COMPOUND ANGLE IDENTITIES
TRIGONOMETRY THE COMPOUND ANGLE IDENTITIES Question 1 Prove the validity of each of the following trigonometric identities. a) sin x + cos x 4 4 b) cos x + + 3 sin x + 2cos x 3 3 c) cos 2x + + cos 2x cos
More informationCHAPTER SIXTEEN. = 4 x y + 6 x y + 3 x y + 4 x y = 17 x y = 31(0.1)(0.2) = f(x i, y i) x y = 7 x y + 10 x y + 6 x y + 8 x y = 31 x y. x = 0.
CHAPTE SIXTEEN 6. SOLUTIONS 5 Solutions for Section 6. Eercises. Mark the values of the function on the plane, as shown in Figure 6., so that ou can guess respectivel at the smallest and largest values
More informationy=1/4 x x=4y y=x 3 x=y 1/3 Example: 3.1 (1/2, 1/8) (1/2, 1/8) Find the area in the positive quadrant bounded by y = 1 x and y = x3
Eample: 3.1 Find the area in the positive quadrant bounded b 1 and 3 4 First find the points of intersection of the two curves: clearl the curves intersect at (, ) and at 1 4 3 1, 1 8 Select a strip at
More informationME 243. Lecture 10: Combined stresses
ME 243 Mechanics of Solids Lecture 10: Combined stresses Ahmad Shahedi Shakil Lecturer, Dept. of Mechanical Engg, BUET E-mail: sshakil@me.buet.ac.bd, shakil6791@gmail.com Website: teacher.buet.ac.bd/sshakil
More informationStatics: Lecture Notes for Sections
0.5 MOMENT OF INERTIA FOR A COMPOSITE AREA A composite area is made by adding or subtracting a series of simple shaped areas like rectangles, triangles, and circles. For example, the area on the left can
More informationLecture 6: Distributed Forces Part 2 Second Moment of Area
Lecture 6: Distributed Forces Part Second Moment of rea The second moment of area is also sometimes called the. This quantit takes the form of The phsical representation of the above integral can be described
More informationTwo small balls, each of mass m, with perpendicular bisector of the line joining the two balls as the axis of rotation:
PHYSCS LOCUS 17 summation in mi ri becomes an integration. We imagine the body to be subdivided into infinitesimal elements, each of mass dm, as shown in figure 7.17. Let r be the distance from such an
More informationAMB121F Trigonometry Notes
AMB11F Trigonometry Notes Trigonometry is a study of measurements of sides of triangles linked to the angles, and the application of this theory. Let ABC be right-angled so that angles A and B are acute
More informationAn angle in the Cartesian plane is in standard position if its vertex lies at the origin and its initial arm lies on the positive x-axis.
Learning Goals 1. To understand what standard position represents. 2. To understand what a principal and related acute angle are. 3. To understand that positive angles are measured by a counter-clockwise
More informationMTE 119 STATICS LECTURE MATERIALS FINAL REVIEW PAGE NAME & ID DATE. Example Problem F.1: (Beer & Johnston Example 9-11)
Eample Problem F.: (Beer & Johnston Eample 9-) Determine the mass moment of inertia with respect to: (a) its longitudinal ais (-ais) (b) the y-ais SOLUTION: a) Mass moment of inertia about the -ais: Step
More informationMathematics, Algebra, and Geometry
Mathematics, Algebra, and Geometry by Satya http://www.thesatya.com/ Contents 1 Algebra 1 1.1 Logarithms............................................ 1. Complex numbers........................................
More informationragsdale (zdr82) HW7 ditmire (58335) 1 The magnetic force is
ragsdale (zdr8) HW7 ditmire (585) This print-out should have 8 questions. Multiple-choice questions ma continue on the net column or page find all choices efore answering. 00 0.0 points A wire carring
More informationWYSE ACADEMIC CHALLENGE State Math Exam 2009 Solution Set. 2. Ans E: Function f(x) is an infinite geometric series with the ratio r = :
WYSE ACADEMIC CHALLENGE State Math Eam 009 Solution Set 40. Ans A: ( C( 40,8 ) * C( 3,8 ) * C( 4,8 ) * C( 6,8 ) * C( 8,8 )) / 5 = 0.00084. Ans E: Function f() is an infinite geometric series with the ratio
More informationSTATICS VECTOR MECHANICS FOR ENGINEERS: Distributed Forces: Centroids and Centers of Gravity. Tenth Edition CHAPTER
Tenth E CHAPTER 5 VECTOR MECHANICS FOR ENGINEERS: STATICS Ferdinand P. Beer E. Russell Johnston, Jr. David F. Mazurek Lecture Notes: John Chen California Polytechnic State University Distributed Forces:
More informationMath 221 Examination 2 Several Variable Calculus
Math Examination Spring Instructions These problems should be viewed as essa questions. Before making a calculation, ou should explain in words what our strateg is. Please write our solutions on our own
More informationSVKM s NMIMS. Mukesh Patel School of Technology Management & Engineering, Vile Parle, Mumbai
Mukesh Patel School of Technolog Management & Engineering Page SVKM s NMIMS Mukesh Patel School of Technolog Management & Engineering, Vile Parle, Mumbai- 456 Tutorial Manual Academic Year : 4-5 Program:
More informationMathematics Trigonometry: Unit Circle
a place of mind F A C U L T Y O F E D U C A T I O N Department of Curriculum and Pedagog Mathematics Trigonometr: Unit Circle Science and Mathematics Education Research Group Supported b UBC Teaching and
More informationSOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours)
SOLUTIONS TO THE 18.02 FINAL EXAM BJORN POONEN December 14, 2010, 9:00am-12:00 (3 hours) 1) For each of (a)-(e) below: If the statement is true, write TRUE. If the statement is false, write FALSE. (Please
More informationIdentifying second degree equations
Chapter 7 Identifing second degree equations 71 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +
More informationHandout 6: Rotational motion and moment of inertia. Angular velocity and angular acceleration
1 Handout 6: Rotational motion and moment of inertia Angular velocity and angular acceleration In Figure 1, a particle b is rotating about an axis along a circular path with radius r. The radius sweeps
More informationThe region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed.
Section A ln. Let g() =, for > 0. ln Use the quotient rule to show that g ( ). 3 (b) The graph of g has a maimum point at A. Find the -coordinate of A. (Total 7 marks) 6. Let h() =. Find h (0). cos 3.
More informationLone Star College-CyFair Formula Sheet
Lone Star College-CyFair Formula Sheet The following formulas are critical for success in the indicated course. Student CANNOT bring these formulas on a formula sheet or card to tests and instructors MUST
More informationAPPM 1360 Final Exam Spring 2016
APPM 36 Final Eam Spring 6. 8 points) State whether each of the following quantities converge or diverge. Eplain your reasoning. a) The sequence a, a, a 3,... where a n ln8n) lnn + ) n!) b) ln d c) arctan
More informationOutline. Organization. Stresses in Beams
Stresses in Beams B the end of this lesson, ou should be able to: Calculate the maimum stress in a beam undergoing a bending moment 1 Outline Curvature Normal Strain Normal Stress Neutral is Moment of
More informationContents. Dynamics and control of mechanical systems. Focuses on
Dnamics and control of mechanical sstems Date Da (/8) Da (3/8) Da 3 (5/8) Da 4 (7/8) Da 5 (9/8) Da 6 (/8) Content Review of the basics of mechanics. Kinematics of rigid bodies - coordinate transformation,
More informationConsider a cross section with a general shape such as shown in Figure B.2.1 with the x axis normal to the cross section. Figure B.2.1.
ppendix B rea Properties of Cross Sections B.1 Introduction The area, the centroid of area, and the area moments of inertia of the cross sections are needed in slender bar calculations for stress and deflection.
More informationVector Analysis 1.1 VECTOR ANALYSIS. A= Aa A. Aa, A direction of the vector A.
1 Vector nalsis 1.1 VECTR NYSIS Introduction In general, electromagnetic field problem involves three space variables, as a result of which the solutions tend to become complex. This can be overcome b
More informationCHAPTER 2. Copyright McGraw-Hill Education. Permission required for reproduction or display.
CHAPTER 2 PROBLEM 2.1 Two forces are applied as shown to a hook. Determinee graphicall the magnitude and direction of their resultant using (a) the parallelogram law, (b) the triangle rule. (a) Parallelogram
More informationMathematics 5 SN TRIGONOMETRY PROBLEMS 2., which one of the following statements is TRUE?, which one of the following statements is TRUE?
Mathematics 5 SN TRIGONOMETRY PROBLEMS 1 If x 4 which one of the following statements is TRUE? A) sin x > 0 and cos x > 0 C) sin x < 0 and cos x > 0 B) sin x > 0 and cos x < 0 D) sin x < 0 and cos x
More information14. Rotational Kinematics and Moment of Inertia
14. Rotational Kinematics and Moment of nertia A) Overview n this unit we will introduce rotational motion. n particular, we will introduce the angular kinematic variables that are used to describe the
More informationI xx + I yy + I zz = (y 2 + z 2 )dm + (x 2 + y 2 )dm. (x 2 + z 2 )dm + (x 2 + y 2 + z 2 )dm = 2
9196_1_s1_p095-0987 6/8/09 1:09 PM Page 95 010 Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copright laws as the currentl 1 1. Show that the
More informationJUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 13 (Second moments of a volume (A)) A.J.Hobson
JUST THE MATHS UNIT NUMBER 13.13 INTEGRATION APPLICATIONS 13 (Second moments of a volume (A)) by A.J.Hobson 13.13.1 Introduction 13.13. The second moment of a volume of revolution about the y-axis 13.13.3
More information2 4πε ( ) ( r θ. , symmetric about the x-axis, as shown in Figure What is the electric field E at the origin O?
p E( r, θ) = cosθ 3 ( sinθ ˆi + cosθ ˆj ) + sinθ cosθ ˆi + ( cos θ 1) ˆj r ( ) ( p = cosθ sinθ ˆi + cosθ ˆj + sinθ cosθ ˆi sinθ ˆj 3 r where the trigonometric identit ( θ ) vectors ˆr and cos 1 = sin θ
More informationAREAS, RADIUS OF GYRATION
Chapter 10 MOMENTS of INERTIA for AREAS, RADIUS OF GYRATION Today s Objectives: Students will be able to: a) Define the moments of inertia (MoI) for an area. b) Determine the MoI for an area by integration.
More information1-1 Locate the centroid of the plane area shown. 1-2 Determine the location of centroid of the composite area shown.
Chapter 1 Review of Mechanics of Materials 1-1 Locate the centroid of the plane area shown 650 mm 1000 mm 650 x 1- Determine the location of centroid of the composite area shown. 00 150 mm radius 00 mm
More informationMATHEMATICS 200 December 2013 Final Exam Solutions
MATHEMATICS 2 December 21 Final Eam Solutions 1. Short Answer Problems. Show our work. Not all questions are of equal difficult. Simplif our answers as much as possible in this question. (a) The line L
More informationMECHANICS OF MATERIALS REVIEW
MCHANICS OF MATRIALS RVIW Notation: - normal stress (psi or Pa) - shear stress (psi or Pa) - normal strain (in/in or m/m) - shearing strain (in/in or m/m) I - area moment of inertia (in 4 or m 4 ) J -
More informationMoments of Inertia (7 pages; 23/3/18)
Moments of Inertia (7 pages; 3/3/8) () Suppose that an object rotates about a fixed axis AB with angular velocity θ. Considering the object to be made up of particles, suppose that particle i (with mass
More informationKINEMATIC RELATIONS IN DEFORMATION OF SOLIDS
Chapter 8 KINEMATIC RELATIONS IN DEFORMATION OF SOLIDS Figure 8.1: 195 196 CHAPTER 8. KINEMATIC RELATIONS IN DEFORMATION OF SOLIDS 8.1 Motivation In Chapter 3, the conservation of linear momentum for a
More informationWhere, m = slope of line = constant c = Intercept on y axis = effort required to start the machine
(ISO/IEC - 700-005 Certified) Model Answer: Summer 07 Code: 70 Important Instructions to examiners: ) The answers should be examined by key words and not as word-to-word as given in the model answer scheme.
More information1.1. Rotational Kinematics Description Of Motion Of A Rotating Body
PHY 19- PHYSICS III 1. Moment Of Inertia 1.1. Rotational Kinematics Description Of Motion Of A Rotating Body 1.1.1. Linear Kinematics Consider the case of linear kinematics; it concerns the description
More informationAPPLIED MECHANICS I Resultant of Concurrent Forces Consider a body acted upon by co-planar forces as shown in Fig 1.1(a).
PPLIED MECHNICS I 1. Introduction to Mechanics Mechanics is a science that describes and predicts the conditions of rest or motion of bodies under the action of forces. It is divided into three parts 1.
More informationUniversity of Pretoria Department of Mechanical & Aeronautical Engineering MOW 227, 2 nd Semester 2014
Universit of Pretoria Department of Mechanical & Aeronautical Engineering MOW 7, nd Semester 04 Semester Test Date: August, 04 Total: 00 Internal eaminer: Duration: hours Mr. Riaan Meeser Instructions:
More informationRigid Body Dynamics, SG2150 Solutions to Exam,
KTH Mechanics 011 10 Calculational problems Rigid Body Dynamics, SG150 Solutions to Eam, 011 10 Problem 1: A slender homogeneous rod of mass m and length a can rotate in a vertical plane about a fied smooth
More informationAnswer Explanations. The SAT Subject Tests. Mathematics Level 1 & 2 TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE
The SAT Subject Tests Answer Eplanations TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE Mathematics Level & Visit sat.org/stpractice to get more practice and stud tips for the Subject Test
More informationPrincipal Stresses, Yielding Criteria, wall structures
Principal Stresses, Yielding Criteria, St i thi Stresses in thin wall structures Introduction The most general state of stress at a point may be represented by 6 components, x, y, z τ xy, τ yz, τ zx normal
More informationstorage tank, or the hull of a ship at rest, is subjected to fluid pressure distributed over its surface.
Hydrostatic Forces on Submerged Plane Surfaces Hydrostatic forces mean forces exerted by fluid at rest. - A plate exposed to a liquid, such as a gate valve in a dam, the wall of a liquid storage tank,
More informationMOI (SEM. II) EXAMINATION.
Problems Based On Centroid And MOI (SEM. II) EXAMINATION. 2006-07 1- Find the centroid of a uniform wire bent in form of a quadrant of the arc of a circle of radius R. 2- State the parallel axis theorem.
More informationFE Sta'cs Review. Torch Ellio0 (801) MCE room 2016 (through 2000B door)
FE Sta'cs Review h0p://www.coe.utah.edu/current- undergrad/fee.php Scroll down to: Sta'cs Review - Slides Torch Ellio0 ellio0@eng.utah.edu (801) 587-9016 MCE room 2016 (through 2000B door) Posi'on and
More informationProf. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
50 Module 4: Lecture 1 on Stress-strain relationship and Shear strength of soils Contents Stress state, Mohr s circle analysis and Pole, Principal stressspace, Stress pathsin p-q space; Mohr-Coulomb failure
More informationSolutions to the Exercises of Chapter 4
Solutions to the Eercises of Chapter 4 4A. Basic Analtic Geometr. The distance between (, ) and (4, 5) is ( 4) +( 5) = 9+6 = 5 and that from (, 6) to (, ) is ( ( )) +( 6 ( )) = ( + )=.. i. AB = (6 ) +(
More informationPRACTICE PAPER 6 SOLUTIONS
PRACTICE PAPER 6 SOLUTIONS SECTION A I.. Find the value of k if the points (, ) and (k, 3) are conjugate points with respect to the circle + y 5 + 8y + 6. Sol. Equation of the circle is + y 5 + 8y + 6
More informationFundamentals of Applied Electromagnetics. Chapter 2 - Vector Analysis
Fundamentals of pplied Electromagnetics Chapter - Vector nalsis Chapter Objectives Operations of vector algebra Dot product of two vectors Differential functions in vector calculus Divergence of a vector
More informationReview problems for the final exam Calculus III Fall 2003
Review problems for the final exam alculus III Fall 2003 1. Perform the operations indicated with F (t) = 2t ı 5 j + t 2 k, G(t) = (1 t) ı + 1 t k, H(t) = sin(t) ı + e t j a) F (t) G(t) b) F (t) [ H(t)
More informationMath 223 Final. July 24, 2014
Math 223 Final July 24, 2014 Name Directions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Total 1. No books, notes, or evil looks. You may use a calculator to do routine arithmetic computations. You may not use your
More informationMULTIVARIABLE INTEGRATION
MULTIVARIABLE INTEGRATION (SPHERICAL POLAR COORDINATES) Question 1 a) Determine with the aid of a diagram an expression for the volume element in r, θ, ϕ. spherical polar coordinates, ( ) [You may not
More informationPhysics 1A Lecture 10B
Physics 1A Lecture 10B "Sometimes the world puts a spin on life. When our equilibrium returns to us, we understand more because we've seen the whole picture. --Davis Barton Cross Products Another way to
More informationPhysicsAndMathsTutor.com
1. A uniform circular disc has mass 4m, centre O and radius 4a. The line POQ is a diameter of the disc. A circular hole of radius a is made in the disc with the centre of the hole at the point R on PQ
More informationGG612 Lecture 3. Outline
GG612 Lecture 3 Strain 11/3/15 GG611 1 Outline Mathema8cal Opera8ons Strain General concepts Homogeneous strain E (strain matri) ε (infinitesimal strain) Principal values and principal direc8ons 11/3/15
More informationMATH 52 MIDTERM 1. April 23, 2004
MATH 5 MIDTERM April 3, Student ID: Signature: Instructions: Print your name and student ID number and write your signature to indicate that you accept the honor code. During the test, you may not use
More informationMathematics Paper 1 (Non-Calculator)
H National Qualifications CFE Higher Mathematics - Specimen Paper F Duration hour and 0 minutes Mathematics Paper (Non-Calculator) Total marks 60 Attempt ALL questions. You ma NOT use a calculator. Full
More information