St_Mod13_F10

St_Mod13_F10 - Module 13: Centroid and Center of...

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Unformatted text preview: Module 13: Centroid and Center of Gravity November 15, 2010 Module Content: 1. Center of gravity, center of mass, and centroid are similar concepts which describe specific loca=ons in a body. The share similar approaches to their calcula=on. 2. Centroid of an area is par=cularly important for structural analysis, because we oYen use it to describe the cross sec=on of a beam. 3. Composite bodies are analyzed using a par=cle discre=za=on. Module Reading, Problems, and Demo: Reading: Chapter 9 Problems: Fund. Prob. 9.8, Prob. 9.54 Demo: none Homework PlaTorm: hUp://www.masteringengineering.com Course Blog: hUp://pages.shan=.virginia.edu/sta=cs2010 MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 1 Introducing Chapter 9 ­ ­Centroid and CG • the center of gravity of a body is the loca=on through which the resultant force (“weight”) of the body acts MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 2 Review: Center of Mass of a Par=cle System • say we had a system of par=cles, n of them, and we wanted to find the center of mass • this would be a summa=on process, over both force and moment MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 3 Review: Center of Mass of a Par=cle System • say we had a system of par=cles, n of them, and we wanted to find the center of mass • this would be a summa=on process, over both force and moment force balance MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 3 Review: Center of Mass of a Par=cle System • say we had a system of par=cles, n of them, and we wanted to find the center of mass • this would be a summa=on process, over both force and moment force balance moment balance MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 3 Review: Center of Mass of a Par=cle System • say we had a system of par=cles, n of them, and we wanted to find the center of mass • this would be a summa=on process, over both force and moment force balance moment balance MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 3 Review: Center of Mass of a Par=cle System • say we had a system of par=cles, n of them, and we wanted to find the center of mass • this would be a summa=on process, over both force and moment force balance moment balance MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 3 Review: Center of Mass of a Par=cle System • say we had a system of par=cles, n of them, and we wanted to find the center of mass • this would be a summa=on process, over both force and moment force balance moment balance MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 3 Theory: A Con=nuous System • for a con=nuous system, the summa=ons become integrals, and: force balance W= dW x= MAE 2300 Sta=cs © E. J. Berger, 2010 xdW ˜ y dW ˜ zW = xdW ˜ ; y= dW xW = yW = moment balance z dW ˜ y dW ˜ ; z= dW z dW ˜ dW 13 ­ 4 Ques=on: Center of Mass vs Center of Gravity? force balance W= dW x= MAE 2300 Sta=cs xdW ˜ y dW ˜ zW = xdW ˜ ; y= dW xW = yW = moment balance z dW ˜ y dW ˜ ; z= dW z dW ˜ dW © E. J. Berger, 2010 13 ­ 5 Theory: Centroids of Areas and Volumes x= xdV ˜ ; y= dV y dV ˜ ; z= dV z dV ˜ dV the centroid is the geometric center of a volume or area MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 6 Example: Centroid of an Area • when we write the centroid integral, we need to think carefully about how we write dV (volume) or dA (area) • consider the examples below... MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 7 Example 1: Centroid of a Rectangle MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 8 Example 2: Centroid of a Rectangle MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 9 Theory: Composite Bodies • composite bodies have complicated geometries which are typically built up from simpler shapes to form useful shapes for specific applica=ons • in engineering, the basic building blocks are usually circles and rectangles • in the analysis, we treat each simple shape as a par=cle MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 10 Theory: Composite Bodies • composite bodies have complicated geometries which are typically built up from simpler shapes to form useful shapes for specific applica=ons • in engineering, the basic building blocks are usually circles and rectangles • in the analysis, we treat each simple shape as a par=cle MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 10 Theory: Composite Bodies • composite bodies have complicated geometries which are typically built up from simpler shapes to form useful shapes for specific applica=ons • in engineering, the basic building blocks are usually circles and rectangles • in the analysis, we treat each simple shape as a par=cle MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 10 Theory: Composite Bodies • composite bodies have complicated geometries which are typically built up from simpler shapes to form useful shapes for specific applica=ons • in engineering, the basic building blocks are usually circles and rectangles • in the analysis, we treat each simple shape as a par=cle MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 10 Example 9.10: Composite Body (p. 473) MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 11 Example 9.10: Composite Body (p. 473) = MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 11 Example 9.10: Composite Body (p. 473) = MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 11 Example 9.10: Composite Body (p. 473) = MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 11 Example 9.10: Composite Body (p. 473) = - MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 11 Example 9.10: Composite Body (p. 473) = - MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 11 Example 9.10: Composite Body (p. 473) MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 11 Example 9.10: Composite Body (p. 473) MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 11 Example 9.10: Composite Body (p. 473) MAE 2300 Sta=cs © E. J. Berger, 2010 13 ­ 11 ...
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This note was uploaded on 02/09/2012 for the course MAE 2300 taught by Professor Staff during the Fall '10 term at UVA.

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