W- Composite Bodies

CENTER OF CENTER OF GRAVITY AND GRAVITY AND

CENTROID CENTROID Composite BodiesComposite Bodies

LEARNING OBJECTIVESLEARNING OBJECTIVES

Be able to determine the location of Be able to determine the location of the center of gravity, center of mass, the center of gravity, center of mass, or the centroid using the method of or the centroid using the method of composite bodies composite bodies

PRE-REQUISITE KNOWLEDGEPRE-REQUISITE KNOWLEDGE

Units of measurementUnits of measurement Concepts of center of gravity, center Concepts of center of gravity, center

of mass and centroidof mass and centroid

CONCEPT OF A COMPOSITE BODY

Many industrial objects can be considered as composite bodies made up of a series of connected “simpler” shaped parts or holes, like a rectangle, triangle, and semicircle. Knowing the location of the centroid or the center of gravity of the simpler shaped part, we can easily determine the location of the points for the more complex composite body.

EXAMPLE 1EXAMPLE 1

Determine the location of the centroid of Determine the location of the centroid of the cross sectionthe cross section

SOLUTION 1SOLUTION 1Separate the cross section into three segments :

1

2

3

Calculate the area and the moment of the area for each segment:

Segment Area, in.2 y, in. y(A), in.3

1 12 3 362 24 1 243 12 3 36

Area= 48 y(A)= 96

SOLUTION 1 SOLUTION 1 (alternative)(alternative)

Separate the cross section into two segments

Calculate the area and the moment of the area for each segment

Segment Area (A) (in)2 Y (in) Y(A) (in)3

1 96 3 288

2 -48 4 192

Σ(A) 48 Σy(A) 96

Σy(A)/ Σ(A) = 2 in

12

Centroid of Trapezoid Centroid of Trapezoid (review)(review)

b

h

(a-b)/2b

1 2 3

a(a-b)/2

Centroid of Trapezoid Centroid of Trapezoid (review)(review)

b

h

(a-b)/2b

1 2 3

a(a-b)/2

EXAMPLE 2EXAMPLE 2

Determine the centroid of the above cross sectionDetermine the centroid of the above cross section

SOLUTION 2SOLUTION 2

Separate the cross section into four segments

1

2

Calculate the area and the moment of the area for each segment

Σy(A)/ Σ(A) = 135 mm

34

Segment Area (A) (mm)2 Y (mm) Y(A) (mm)3

1 72000 40 2880000

2 36000 260 9360000

3 18000 200 3600000

4 18000 200 3600000

Σ(A) 144000 Σy(A) 19440000

2

SOLUTION 2 SOLUTION 2 (alternative)(alternative)

Separate the cross section into two segments

1

2

Calculate the area and the moment of the area for each segment :

Segment Area (A) (mm)2

Y (mm) Y(A) (mm)3

1 72000 40 2880000

2 72000 230 16560000

Σ(A) 144000 Σy(A) 19440000Σy(A)/ Σ(A) = 135 mm

QUESTION

A composite body in this section refers to a body made of ____.

A) carbon fibers and an epoxy matrix

B) steel and concrete

C) a collection of “simple” shaped parts or holes

D) a collection of “complex” shaped parts or holes

The composite method for determining the location of the center of gravity of a composite body requires _______.

A)integration

B) differentiation

C) simple arithmetic

D) all of the above

QUESTION

Based on the typical centroid information available in handbooks, what are the minimum number of segments you will have to consider for determing the centroid of the given area?

A) 1 B) 2

C) 3 D) 4

3cm 1 cm

1 cm

3cm

QUESTION

A storage box is tilted up to clean the rug underneath the box. It is tilted up by pulling the handle C, with edge A remaining on the ground. What is the maximum possible angle of tilt (measured between the bottom AB and the ground) before the box tips over?

A) 30° B) 45 °

C) 60 ° D) 90°

30º

G

C

AB

QUESTION

A rectangular area has semicircular and triangular cuts as shown. What is the minimum number of pieces that can be used in determining the location of the centroid?

A) Two

B) Three

C) Four

D) Five 2cm 2cm

2cm

4cm

x

y

QUESTION

For determining the centroid of the area, two square segments are considered; square ABCD and square DEFG. What are the coordinates (x, y ) of the centroid of square DEFG?

A) (1, 1) m

B) (1.25, 1.25) m

C) (0.5, 0.5 ) m

D) (1.5, 1.5) m C

A1m

1m

y

E

FG

B x

1m 1m

D

QUESTION

What are the coordinates (X, Y) of the centroid?

A) (1, 1) m

B) (1.1, 1.1) m

C) (0.83, 0.83 ) m

D) Cannot be determined

C

A1m

1m

y

E

FG

B x

1m 1m

D

QUESTION

QUESTION QUESTION Which of the following is the location of the centroid in the x-direction of the area below?

A) 3.5 in

B) 4.0 in

C) 4.5 in

D) 5.0 in

x

y

1

2

Which of the following is the location of the centroid in the y-direction of the area below?

A) 5.5 in

B) 6.0 in

C) 6.5 in

D) 7.0 in

x

y

1

2

QUESTIONQUESTION

Which of the following is the moment of inertia Ix about the neutral axis x’? Remember y = 6.5”

A) 177 in4

B) 248 in4

C) 275 in4

D) 291 in4

x

y

1

2

QUESTION QUESTION

Which of the following is the location of the centroid in the x-direction of the given area?

A) 5.5 in

B) 6.0 in

C) 6.5 in

D) 7.0 in

QUESTIONQUESTION

x

2” 2”4” 4”

12”

20”

3”

y

Which of the following is the location of the centroid in the y-direction of the rectangular cross-section with three squares 2”x 2” holes below?

A) 10.0 in

B) 10.4 in

C) 11.5 in

D) 12.1 in

QUESTION QUESTION

x

2” 2”4” 4”

12”

20”

3”

y

Which of the following is the moment of inertia Ix about the neutral axis of the rectangular cross-section with three squares 2”x2” holes below? Remember y = 10.37”

A) 5277 in4

B) 6633 in4

C) 7377 in4

D) 7996 in4

QUESTION QUESTION

x

2” 2”4” 4”

12”

20”

3”

y

x

2” 2”4” 4”

12”

20”

3”

y