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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 Statics, Force, J.00 Berger, sta=cs, MAE00

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