Contributed by:

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

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

• 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

• 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

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

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

• 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

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

n1 , 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

~ xi wi ~ yi wi ~ zi wi

x i

n 1 , y i

n1 , 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

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

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

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

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

~

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

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

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

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

•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

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

• 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

• 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

• 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

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

• 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