1. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Engineering 36
Chp09: Center of Gravity
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

2. Introduction: Center of Gravity
The earth exerts a gravitational force on each of the particles forming a body.
These forces can be replaced by a SINGLE equivalent force equal to the weight of the body and applied at the CENTER OF GRAVITY (CG) for the body
The CENTROID of an AREA is analogous to the CG of a body.
The concept of the FIRST MOMENT of an AREA is used to locate the centroid

3. Total Mass – General Case
Given a Massive Body in 3D Space
Divide the Body in to Very Small Volumes → dV
Each dVn is located at position (xn,yn,zn)
The DENSITY, ρ, can be a function of POSITION →

4. Total Mass – General Case
Now since m = ρ•V, then the incremental mass, dm
Integrate dm over the entire body to obtain the total Mass, M

5. Total WEIGHT – General Case
Recall that W = Mg
Using the the previous expression for M
Now Define SPECIFIC WEIGHT, γ, as ρ•g →

6. Uniform Density Case Consider a body with UNIFORM DENSITY; i.e.
Then M & W→

7. Center of Mass Location
Use MOMENTS to Locate Center of Mass/Gravity
Recall Defintion of a MOMENT
In the General Center-of-Mass Case
LeverArm ≡ Position Vector, rn, or its components
Intensity ≡ Incremental Mass, dmn

8. Center of Mass Location
RM in Component form
Now Define the Incremental Moment, dΩn
LeverArm
Intensity

9. Center of Mass Location
Integrating dΩ to Find Ω for the entire body

10. Center of Mass Location
Now Equate Ω to RM•Mg
Canceling g, and equating Components yields, for example, in the X-Dir
LeverArm
Intensity

11. Center of Mass Location
Divide out the Total Intensity, M, to Isolate the Overall X-directed Lever Arm, XM
And the Similar expressions for the other CoOrd Directions

12. Center of Gravity of a 2D Body
Centroid of an Area
Centroid of a Line
Taking Incremental Plate Areas, Forming the Ωx & Ωy, Along With the Fz=W Yields the Expression for the Equivalent POINT of W application
Note Ω Units = In-lb or N-m
Called the moment of area due to LEVER ARM, xi or yi, times a force, del-W

13. Centroids of Areas & Lines
Centroid of an Area
Centroid of a Line
Wire of Uniform Thickness
  Specific Weight
a  X-Sec Area
dW = a(dL)
For Plate of Uniform Thickness
  Specific Weight
t  Plate Thickness
dW =  tdA

14. First Moments of Areas & Lines
An area is symmetric with respect to an axis BB’ if for every point P there exists a point P’ such that PP’ is perpendicular to BB’ and the Area is divided into equal parts by BB’.
The first moment of an area with respect to a line of symmetry is ZERO.
If an area possesses two lines of symmetry, its centroid lies at their INTERSECTION.
If an area possesses a line of SYMMETRY, its centroid LIES on THAT AXIS
An area is symmetric with respect to a center O if for every element dA at (x,y) there exists an area dA’ of equal area at (−x, −y).
The centroid of the area coincides with the center of symmetry, O.

15. Centroids of Common Area Shapes

16. Centroids of Common Line Shapes
Recall that for a SMALL Angle, α

17. Composite Plates and Areas
Composite area

18. Example: Composite Plate
Solution Plan
Divide the area into a triangle, rectangle, semicircle, and a circular cutout
Calculate the first moments of each area w/ respect to the axes
Find the total area and first moments of the triangle, rectangle, and semicircle. Subtract the area and first moment of the circular cutout
Calc the coordinates of the area centroid by dividing the NET first moment by the total area
For the plane area shown, determine the first moments with respect to the x and y axes, and the location of the centroid.

20. Example: Composite Plate
Find the total area and first moments of the triangle, rectangle, and semicircle. Subtract the area and first moment of the circular cutout

21. Example: Composite Plate
Solution
Find the coordinates of the area centroid by dividing the first moment totals by the total area

22. Centroids by Strip Integration
Double integration to find the first moment may be avoided by defining dA as a thin rectangle or strip.
For POLAR case. The Area of the small triangle = ½*Base*Hgt = ½*[r*dθ]*r = ½*r2

23. Example: Centroid by Integration
Solution Plan
Determine Constant k
Calculate the Total Area
Using either vertical or horizontal STRIPS, perform a single integration to find the first moments
Evaluate the centroid coordinates by dividing the Total 1st Moment by Total Area.
Determine by direct integration the location of the centroid of a parabolic spandrel.

24. Ask for “k” when b =32 & x=2 => k = 32/2^2 = 32/4 = 8

25. Example: Centroid by Integration
Solution
Determine Constant k
Calculate the Total Area

26. Example: Centroid by Integration
Solution
Calc the 1st Moments, Ωi
Using vertical strips, perform a single integration to find the first moments.

27. Example: Centroid by Integration
Finally the Answers
Evaluate the centroid coordinates
Divide Ωx and Ωy by the total Area

28. Theorems of Pappus-Guldinus
Surface of revolution is generated by rotating a plane curve about a fixed axis.
Area of a surface of revolution is equal to the length of the generating curve, L, times the distance traveled by the centroid through the rotation.

29. Theorems of Pappus-Guldinus
Body of revolution is generated by rotating a plane area about a fixed axis.
Volume of a body of revolution is equal to the generating area, A, times the distance traveled by the centroid through the rotation.

30. Example: Pappus-Guldinus
Solution Plan
Apply the theorem of Pappus-Guldinus to evaluate the volumes of revolution for the rectangular rim section and the inner cutout section.
Multiply by density and acceleration of gravity to get the mass and weight.
The outside diameter of a pulley is 0.8 m, and the cross section of its rim is as shown. Knowing that the pulley is made of steel and that the density of steel,  = 7850 kg/m3, determine the mass and weight of the rim.

31. Example: P-G
Apply Pappus-Guldinus to Sections I & II
Subtract: I-II

32. Find the Areal & Lineal Centroids
WhiteBoard Work
Find the Areal & Lineal Centroids
See ST ENGR-36_Lab-15_Fa07_Lec-Notes.ppt

33. Registered Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Engineering 36
Appendix
Bruce Mayer, PE
Registered Electrical & Mechanical Engineer

35. S&T P7.2.56