Centroid and Center of Gravity

Contributed by:

Jonathan James Thu, Jan 20, 2022 08:04 PM UTC

Center of gravity, Center of mass, Procedure for analysis, Centroid

1. ME 221 Statics
Lecture #7
Sections 4.1 – 4.3
ME221 Lecture 7 1

2. Homework #3
• Chapter 3 problems:
– 48, 55, 57, 61, 62, 65 & 72
• Chapter 4 problems:
– 2, 4, 10, 11, 18, 24, 39 & 43
– Must use integration methods to solve
• Due Monday, June 7
ME221 Lecture 7 2

3. Quiz #4
• Monday, June 7
ME221 Lecture 7 3

4. Distributed Forces (Loads);
Centroids & Center of Gravity
• The concept of distributed loads will be
introduced
• Center of mass will be discussed as an
important application of distributed loading
– mass, (hence, weight), is distributed throughout
a body; we want to find the “balance” point
ME221 Lecture 7 4

5. Distributed Loads
Two types of distributed loads exist:
– forces that exist throughout the body
• e. g., gravity acting on mass
• these are called “body forces”
– forces arising from contact between two bodies
• these are called “contact forces”
ME221 Lecture 7 5

6. Contact Distributed Load
• Snow on roof, tire on road, bearing on race,
liquid on container wall, ...
ME221 Lecture 7 6

7. Center of Gravityz
z
w3(x˜3,y
˜ 3,z˜3) w5(x˜5,y
˜ 5,z˜5)
˜ 1,z˜1)
w2(x˜2,y
˜ 2,z˜2) w4(x˜4,y
˜ 4,z˜4) y
y z
x
y
x
x
The weights of the n particles comprise a system of parallel forces. We can
replace them with an equivalent force w located at G(x,y,z), such that:
~ ~ ~ ~ ~
x w=x w
1 1+x w
2 2 +x w
3 3+x w
4 4 +x 5w5
ME221 Lecture 7 7

8. n n n
 ~ xi wi  ~ yi wi  ~ zi wi
x i
 n 1 , y i
 n1 , z i
 n 1
 wi  wi  wi
i 1 i 1 i 1
Where ~ ~
x, y, ~z are the coordinates of each
point. Point G is called the center of gravity
which is defined as the point in the space where
all the weight is concentrated.
ME221 Lecture 7 8

9. CG in Discrete Sense
20 10
?? ?? ??
Where do we hold the bar to balance it?
Find the point where the system’s weight may
be balanced without the use of a moment.
ME221 Lecture 7 9

10. Discrete Equations
y
r dw
Define a reference frame
z x
~
xi wi ~
x dw

x  x
 wi  dw
ME221 Lecture 7 10

11. Center of Mass
The total mass is given by M
M m i
i
Mass center is defined by
i mi xi i mi yi i mi zi
xc.m.  ; y c .m .  ; z c . m. 
M M M
ME221 Lecture 7 11

12. Continuous Equations
Take our volume, dV, to be infinitesimal.
Summing over all volumes becomes an integral.
1
M  dV

VV
1 1 1
xc.m.   xdV ; y c.m.  ydV ; zc.m. 
 zdV

VV VV VV
Note that dm = dV . Center of gravity deals with forces and
gdm is used in the integrals.
ME221 Lecture 7 12

13. If  is constant
~
x dv ~
y dv ~
z dv
x , y , z
dv dv dv
•These coordinates define the geometric center
of an object (the centroid)
•In case of 2-D, the geometric center can be
defined using a differential element dA
~
x dA ~
y dA ~
z dA
x , y , z
dA dA dA
ME221 Lecture 7 13

14. If the geometry of an object takes the form
of a line (thin rod or wire), then the
centroid may be defined as:
~
x dL ~
y dL ~
z dL
x , y , z
dL dL dL
ME221 Lecture 7 14

15. Procedure for Analysis
1-Differential element
Specify the coordinate axes and choose an appropriate
differential element of integration.
•For a line, the differential element is dl
•For an area, the differential element dA is generally a
rectangle having a finite height and differential width.
•For a volume, the element dv is either a circular disk having a
finite radius and differential thickness or a shell having a finite
length and radius and differential thickness.
ME221 Lecture 7 15

16. 2- Size
Express the length dl, dA, or dv of the element in terms of
the coordinate used to define the object.
3-Moment Arm
Determine the perpendicular distance from the coordinate
axes to the centroid of the differential element.
4- Equation
Substitute the data computed above in the appropriate
ME221 Lecture 7 16

17. Symmetry Conditions
•The centroid of some objects may be partially or
completely specified by using the symmetry conditions
•In the case where the shape of the object has an axis of
symmetry, then the centroid will be located along that line of
y
x
In this case, the centroid is located along the y-axis
ME221 Lecture 7 17

18. In cases of more than one axis of symmetry, the
centroid will be located at the intersection of these axes.
ME221 Lecture 7 18

19. Centroid of an Area
• Geometric center of the area
– Average of the first moment over the entire area
1 1
xc  xdA yc  ydA
AA AA
– Where: A dA
A
ME221 Lecture 7 19

20. Centroid of an Area
• Is then defined as an integral over the area.
• Integration of areas may be accomplished by the use
of either single integrals or double integrals.
ME221 Lecture 7 20

21. Centroid of a Volume
• Geometric center of the volume
– Average of the first moment over entire volume
1 1 1
xc  xdV yc  ydV zc  zdV
VV VV VV
1
– In vector notation: rc  rdV
VV
ME221 Lecture 7 21

22. Examples
ME221 Lecture 7 22

23. Homework Assignments 4, 5 & 6
• Combination of hand-calculated and computer-generated solutions (MatLab). 15
points possible for each.
• Must register for 1 of 2 MatLab sessions on Wednesdays
June 9, 16 & 23 (12:40-2:30pm or 5:00-6:50pm).
• ME221 Wednesday lectures will be 10:20am to 11:10am.
• Will be assigned to a group with 2 ME221 & 2 CSE131
students.
• Group members will receive the same grade for MatLab part.
ME221 Lecture 7 23

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