1. Lecture 40: Center of Gravity, Center of Mass and Geometric Centroid
ENGI 1313 Mechanics I
Lecture 40: Center of Gravity, Center of Mass and Geometric Centroid

2. Material Coverage for Final Exam
Introduction (Ch.1: Sections 1.1–1.6)
Force Vectors (Ch.2: Sections 2.1–2.9)
Particle Equilibrium (Ch.3: Sections 3.1–3.4)
Force System Resultants (Ch.4: Sections 4.1–4.10)
Omit Wrench (p.174)
Rigid Body Equilibrium (Ch.5: Sections 5.1–5.7)
Structural Analysis (Ch.6: Sections 6.1–6.4 & 6.6)
Friction (Ch.8: Sections 8.1–8.3)
Center of Gravity and Centroid (Ch.9: Sections 9.1–9.3)
Ignore problems involving closed-form integration
Simple shapes such as square, rectangle, triangle and circle

3. Lecture 40 Objective
to understand the concepts of center of gravity, center of mass, and geometric centroid
to be able to determine the location of these points for a system of particles or a body

4. Center of Gravity
Point locating the equivalent resultant weight of a system of particles or body
Example: Solid Blocks
Are both final configurations stable? w5 w5 w3 w3 w2 w2 w1 w1 WR WR

5. Center of Gravity (cont.)
Resultant Weight
Coordinates
Key Property
L/2 w4 w3 xG w2 z w1 z1 ~ x1 ~ WR x

6. Center of Gravity (cont.)
Generalized Formulae
Moment about y-axis z2 ~
Moment about x-axis zn ~ z1 ~
“Moment” about x-axis
or y-axis

7. Center of Mass
Point locating the equivalent resultant mass of a system of particles or body
Generally coincides with center of gravity (G)
Center of mass coordinates

8. Center of Mass (cont.) Can the Center of Mass be Outside the Body?
Fulcrum / Balance
Center of Mass

9. Center of Gravity & Mass – Applications
Dynamics
Inertial terms
Vehicle roll-over and stability

10. Geometric Centroid
Point locating the geometric center of an object or body
Homogeneous body
Body with uniform distribution of density or specific weight
Center of mass and center of gravity coincident
Centroid only dependent on body dimensions and not weight terms

11. Geometric Centroid (cont.)
Common Geometric Shapes
Solid structure or frame elements
GC & CM
GC & CM
GC & CM
Median Lines

12. Composite Body
Find center of gravity or geometric centroid of complex shape based on knowledge of simpler geometric forms

13. Example 40-01
Determine the location (x, y) of the 7-kg particle so that the three particles, which lie in the x−y plane, have a center of mass located at the origin O.

14. Example (cont.)
Center of Mass

15. Example 40-02
A rack is made from roll-formed sheet steel and has the cross section shown. Determine the location (x,y) of the centroid of the cross section. The dimensions are indicated at the center thickness of each segment.

16. Example 40-02 (cont.) Assume Unit Thickness Centroid Equations
Ignore bend radii
Center-to-center distance
Centroid Equations

17. Example 40-02 (cont.) Centroid Equations 1 ~ x1 = 7.5mm # Area (mm2) 1
(15mm)(1mm) = 15mm2
15/2 = 7.5 
(15)(7.5) = 112.5

18. Example 40-02 (cont.) Centroid Equations 5 ~ y5 = 25mm # Area (mm2) 5
(50mm)(1mm) = 50mm2 
50/2 = 25
(50)(25) = 1250

19. Example 40-02 (cont.) Centroid Equations x6 = 15mm ~ 6 y6 = 65mm ~ #
Area (mm2) 6
(30mm)(1mm) = 30mm2 15
50+30/2 = 65
450
1950

20. Example 40-02 (cont.) Centroid Equations 4 6 3 7 5 1 2 Sum 235mm2#
Area (mm2) 1
(15mm)(1mm) = 15mm2
15/2 = 7.5 
112.5 2
30+15/2 = 37.5
562.5 3 50
750 4
(30mm)(1mm) = 30mm2
15+30/2 = 30 80
900
2400 5
(50mm)(1mm) = 50mm2
50/2 = 25
1250 6 15
50+30/2 = 65
450
1950 7
(80mm)(1mm) = 80mm2 45
80/2 = 40
3600
3200
Sum
235mm2
5737.5mm3
9550mm3

21. Example (cont.)
Centroid Equations
24.4 mm
40.6 mm

22. Example 40-03
Two blocks of different materials are assembled as shown. The densities of the materials are: A = 150 lb/ft3 and A = 400 lb/ft3. The center of gravity of this assembly.

23. Example (cont.)
Center of Gravity

24. Example 40-03 (cont.) Center of Gravity # Weight (lb) A 3.125 4 1 2
12.5
6.25 B
16.67 3 50 
19.79
29.17
53.13
56.25

25. Example (cont.)
Center of Gravity

26. Chapter 9 Problems Understand principles for simple geometric shapes
Rectangle, square, triangle and circle
No closed form integration knowledge required
Review
Example 9.9 and 9.10
Problems 9-44 to 9-61
Omit
Example 9.1 through 9.8
Problems 9-1 through 9-43, 9-62, 9-67 to 9-83

27. References Hibbeler (2007)