PowerPoint PresentationHomework #7Exam 2Moments of AreaExampleMoments of Basic ShapesPolar MomentParallel Axis TheoremRadius of GyrationSlide 10ME 221 Exam 2 TopicsSlide 12Slide 13ME 221 Lecture 19 1ME 221 StaticsLecture #19Sections 9.1 - 9.6Exam #2 ReviewME 221 Lecture 19 2Homework #7•Chapter 4 problems:–52, 53 & 54•Chapter 9 problems:–2, 3, 4, 11, 23, 29, 32–Use double integration for 2, 3, 4 & 11•Due Wednesday, October 23ME 221 Lecture 19 3Exam 2•Friday, October 18•More on the exam later in the hourME 221 Lecture 19 4Moments of Area•Characterizes the way area is distributed about the centroid•Moment of inertia–Always a positive value with units of in4, mm4, ft4, m4, etc.dAyxr2 2 and xx yyA AI y dA I x dA ME 221 Lecture 19 5ExampledxdyxyxydA=dxdydIx=y2dAdIy=x2dAdxxydA=(a-x)dydIx=y2dAdA=ydxdIy=x2dAdxdyxyause horizontal element for value in xuse vertical element for value in yME 221 Lecture 19 6Moments of Basic Shapes•Rectanglexy2 22 22b hb hxxI y dy dx 223 32 2=3 3bbh hdx 3=12bh•Circularxy 220 0sinRxx yyI I r rdr d 24 2140sinR d 414RCommon values found in table 9.1, p451ME 221 Lecture 19 7Polar MomentThe polar moment is the second moment about the z-axisxyr23 4120 0ROzJ r dr d R Note that: Ixx + Iyy = JOzyyxxozIIdAyxdArJ )(222ME 221 Lecture 19 8Parallel Axis TheoremThe centroid of the area MUSTMUST be one of the axes used in the parallel axis theorem.ydxx’C22xx xxc yyy yyc xI I AdI I Ad ME 221 Lecture 19 9Radius of GyrationAn alternate, equivalent way to represent the moment of an area; ;yyxx Ozx y zII Jk k kA A A ME 221 Lecture 19 10ExampleME 221 Lecture 19 11ME 221 Exam 2 TopicsC Chapter 3 Calculating moments Cross products Moment of a force about an axis Moment of a couple Equivalent force systemsME 221 Lecture 19 12ME 221 Exam 2 TopicsC Chapter 4 Center of mass Centroids of lines, areas and volumes Centroids of composite bodies Distributed loads on beamsME 221 Lecture 19
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