Ch.09 Center of Gravity and Centroid
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Transcript of Ch.09 Center of Gravity and Centroid
3/25/2013
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09. Center of Gravity and Centroid
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.01 Center of Gravity and Centroid
Chapter Objectives
• To discuss the concept of the center of gravity, center of mass,
and the centroid
• To show how to determine the location of the center of gravity
and centroid for a system of discrete particles and a body of
arbitrary shape
• To use the theorems of Pappus and Guldinus for finding the
surface area and volume for a body having axial symmetry
• To present a method for finding the resultant of a general
distributed loading and show how it applies to finding the
resultant force of a pressure loading caused by a fluid
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.02 Center of Gravity and Centroid
§1. Center of Gravity, Center of Mass, and the Centroid of a Body
- How can we determine these weights and their locations?
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.03 Center of Gravity and Centroid
- To design the structure
for supporting a water
tank, we will need to
know the weights of
the tank and water as
well as the locations
where the resultant
forces representing these
distributed loads are
acting
§1. Center of Gravity, Center of Mass, and the Centroid of a Body
Center of Gravity
- Center of gravity: the point at which the entire weight of a body
may be considered as concentrated so that if supported at this
point the body would remain in equilibrium in any position
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.04 Center of Gravity and Centroid
- A body is composed of an infinite
number of particles of differential size
and each of these particles have a
weight 𝑑𝑊
- These weights will form a parallel
force system, and the resultant of this
system is the total weight of the body,
which passes through a single point
called the center of gravity, 𝐺
§1. Center of Gravity, Center of Mass, and the Centroid of a Body
- The location of the center of gravity 𝐺 with respect to the 𝑥,𝑦,𝑧
𝑥 = 𝑥 𝑑𝑊
𝑑𝑊, 𝑦 =
𝑦 𝑑𝑊
𝑑𝑊, 𝑧 =
𝑧 𝑑𝑊
𝑑𝑊
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.05 Center of Gravity and Centroid
- The weight of the body: the sum of
the weights of all of its particles
𝑊 = 𝑑𝑊
- The location of the center of gravity
𝐺(𝑥 , 𝑦 , 𝑧 ) 𝑥 𝑊 = 𝑥 𝑑𝑊
𝑦 𝑊 = 𝑦 𝑑𝑊
𝑧 𝑊 = 𝑧 𝑑𝑊
§1. Center of Gravity, Center of Mass, and the Centroid of a Body
Center of Mass of a Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.06 Center of Gravity and Centroid
- The location of the center of mass 𝐶𝑚
with respect to the 𝑥,𝑦,𝑧 axes
Substitute 𝑑𝑊 = 𝑔𝑑𝑚 to
𝑥 = 𝑥 𝑑𝑊
𝑑𝑊,𝑦 =
𝑦 𝑑𝑊
𝑑𝑊, 𝑧 =
𝑧 𝑑𝑊
𝑑𝑊
⟹ 𝑥 = 𝑥 𝑑𝑚
𝑑𝑚, 𝑦 =
𝑦 𝑑𝑚
𝑑𝑚, 𝑧 =
𝑧 𝑑𝑚
𝑑𝑚
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§1. Center of Gravity, Center of Mass, and the Centroid of a Body
Center of a Volume
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.07 Center of Gravity and Centroid
- The center of a volume or centroid 𝐶
or geometric center of the body with
respect to the 𝑥,𝑦,𝑧 axes
Substitute 𝑑𝑚 = 𝜌𝑑𝑉 to
𝑥 = 𝑥 𝑑𝑚
𝑑𝑚, 𝑦 =
𝑦 𝑑𝑚
𝑑𝑚, 𝑧 =
𝑧 𝑑𝑚
𝑑𝑚
⟹ 𝑥 = 𝑥 𝑑𝑉
𝑉
𝑑𝑉
𝑉
, 𝑦 = 𝑦 𝑑𝑉
𝑉
𝑑𝑉
𝑉
, 𝑧 = 𝑧 𝑑𝑉
𝑉
𝑑𝑉
𝑉
§1. Center of Gravity, Center of Mass, and the Centroid of a Body
Center of an Area
- If an area lies in the 𝑥 − 𝑦 plane and is bounded by the curve
𝑦 = 𝑓(𝑥), then its Centroid 𝐶 𝑥 , 𝑦
𝑥 = 𝑥 𝑑𝐴
𝐴
𝑑𝐴
𝐴
, 𝑦 = 𝑦 𝑑𝐴
𝐴
𝑑𝐴
𝐴
- These integrals can be evaluated by performing a single integration
if we use a rectangular strip for the differential area element
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.08 Center of Gravity and Centroid
§1. Center of Gravity, Center of Mass, and the Centroid of a Body
Center of a Line
- If a line segment lies within the 𝑥 − 𝑦 plane and it can be
described by a thin curve 𝑦 = 𝑓(𝑥), then its Centroid 𝐶 𝑥 , 𝑦
𝑥 = 𝑥 𝑑𝐿
𝐿
𝑑𝐿
𝐿
, 𝑦 = 𝑦 𝑑𝐿
𝐿
𝑑𝐿
𝐿
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.09 Center of Gravity and Centroid
§1. Center of Gravity, Center of Mass, and the Centroid of a Body
- Example 9.1 Locate the centroid of the rod bent into the
shape of a parabolic arc
Solution
Differential Element: arbitrary point (𝑥,𝑦)
Area: 𝑥 = 𝑦2 → 𝑑𝑥/𝑑𝑦 = 2𝑦
𝑑𝐿 = (𝑑𝑥)2+(𝑑𝑦)2= 2𝑦 2 + 1𝑑𝑦
Moment Arms: 𝑥 = 𝑥, 𝑦 = 𝑦
Integrations
𝑥 = 𝑥 𝑑𝐿
𝐿
𝑑𝐿
𝐿
= 𝑥 4𝑦2 +1𝑑𝑦
1
0
4𝑦2 +1𝑑𝑦1
0
= 𝑦2 4𝑦2 +1𝑑𝑦
1
0
4𝑦2 +1𝑑𝑦1
0
=0.6063
1.479= 0.41𝑚
𝑦 = 𝑦 𝑑𝐿
𝐿
𝑑𝐿
𝐿
= 𝑦 4𝑦2 +1𝑑𝑦
1
0
4𝑦2 +1𝑑𝑦1
0
=0.8484
1.479= 0.574𝑚
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.10 Center of Gravity and Centroid
§1. Center of Gravity, Center of Mass, and the Centroid of a Body
- Example 9.2 Locate the centroid of the circular wire segment
Solution
Polar coordinates will be used to solve this
problem since the arc is circular
Differential Element: arbitrary point (𝑅,𝜃)
Area: 𝑑𝐿 = 𝑅𝑑𝜃
Moment Arms: 𝑥 = 𝑅𝑐𝑜𝑠𝜃, 𝑦 = 𝑅𝑠𝑖𝑛𝜃
Integrations
𝑥 = 𝑥 𝑑𝐿
𝐿
𝑑𝐿
𝐿
= 𝑅𝑐𝑜𝑠𝜃𝑅𝑑𝜃
𝜋/2
0
𝑅𝑑𝜃𝜋/2
0
= 𝑐𝑜𝑠𝜃𝑑𝜃
𝜋/2
0
𝑅 𝑑𝜃𝜋/2
0
=2𝑅
𝜋
𝑦 = 𝑦 𝑑𝐿
𝐿
𝑑𝐿
𝐿
= 𝑅𝑠𝑖𝑛𝜃𝑅𝑑𝜃
𝜋/2
0
𝑅𝑑𝜃𝜋/2
0
= 𝑐𝑜𝑠𝜃𝑑𝜃
𝜋/2
0
𝑅 𝑑𝜃𝜋/2
0
=2𝑅
𝜋
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.11 Center of Gravity and Centroid
§1. Center of Gravity, Center of Mass, and the Centroid of a Body
- Example 9.3 Determine the distance 𝑦 measured from the 𝑥
axis to the centroid of the area of the triangle
Solution
Differential Element
Arbitrary element (𝑥,𝑑𝑦) at 𝑦
Area: 𝑑𝐴 = 𝑥𝑑𝑦
=𝑏
ℎ(ℎ − 𝑦)𝑑𝑦
Moment Arms: 𝑦 = 𝑦
Integrations
𝑦 = 𝑦 𝑑𝐴
𝐴
𝑑𝐴
𝐴
= 𝑦
𝑏ℎ
ℎ − 𝑦 𝑑𝑦ℎ
0
𝑏ℎ
ℎ − 𝑦 𝑑𝑦ℎ
0
=
16𝑏ℎ2
12𝑏ℎ
=1
3ℎ
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.12 Center of Gravity and Centroid
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§1. Center of Gravity, Center of Mass, and the Centroid of a Body
- Example 9.4 Locate the centroid for the area of a quarter circle
Solution
Polar coordinates will be used
Differential Element (𝑂,𝑅,𝑑𝜃)
Area: 𝑑𝐴 =1
2𝑅 𝑅𝑑𝜃 =
1
2𝑅2𝑑𝜃
Moment Arms: 𝑥 =2
3𝑅𝑐𝑜𝑠𝜃, 𝑦 =
2
3𝑅𝑠𝑖𝑛𝜃
𝑥 = 𝑥 𝑑𝐴
𝐴
𝑑𝐴
𝐴
=
23𝑅𝑐𝑜𝑠𝜃
𝑅2
2𝑑𝜃
𝜋/2
0
𝑅2
2𝑑𝜃
𝜋/2
0
=
23𝑅 𝑐𝑜𝑠𝜃𝑑𝜃
𝜋/2
0
𝑑𝜃𝜋/2
0
=4𝑅
3𝜋
𝑦 = 𝑦 𝑑𝐴
𝐴
𝑑𝐴
𝐴
=
23𝑅𝑠𝑖𝑛𝜃
𝑅2
2𝑑𝜃
𝜋/2
0
𝑅2
2𝑑𝜃
𝜋/2
0
=
23𝑅 𝑠𝑖𝑛𝜃𝑑𝜃
𝜋/2
0
𝑑𝜃𝜋/2
0
=4𝑅
3𝜋
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.13 Center of Gravity and Centroid
Integrations
§1. Center of Gravity, Center of Mass, and the Centroid of a Body
- Example 9.5 Locate the centroid for the area
Solution 1
Differential Element
Arbitrary element (𝑦,𝑑𝑥) at 𝑥
Area: 𝑑𝐴 = 𝑦𝑑𝑥
Moment Arms: 𝑥 = 𝑥, 𝑦 = 𝑦/2
Integration
𝑥 = 𝑥 𝑑𝐴
𝐴
𝑑𝐴
𝐴
= 𝑥𝑦𝑑𝑥
1
0
𝑦𝑑𝑥1
0
= 𝑥3𝑑𝑥
1
0
𝑥2𝑑𝑥1
0
=0.250
0.333= 0.75𝑚
𝑦 = 𝑦 𝑑𝐴
𝐴
𝑑𝐴
𝐴
= (𝑦/2)𝑦𝑑𝑥
1
0
𝑦𝑑𝑥1
0
= (𝑥2/2)𝑥2𝑑𝑥
1
0
𝑥2𝑑𝑥1
0
=0.100
0.333= 0.3𝑚
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.14 Center of Gravity and Centroid
§1. Center of Gravity, Center of Mass, and the Centroid of a Body
- Example 9.5 Locate the centroid for the area
Solution 2
Differential Element
Arbitrary element (1 − 𝑥,𝑑𝑦) at (𝑥, 𝑦)
Area: 𝑑𝐴 = (1 − 𝑥)𝑑𝑦
Moment Arms: 𝑥 = 𝑥 +1−𝑥
2=
1+𝑥
2 𝑦 = 𝑦
Integration
𝑥 = 𝑥 𝑑𝐴
𝐴
𝑑𝐴
𝐴
= [ 1 + 𝑥 /2](1 − 𝑥)𝑑𝑦
1
0
(1 − 𝑥)𝑑𝑦1
0
=0.250
0.333= 0.75𝑚
𝑦 = 𝑦 𝑑𝐴
𝐴
𝑑𝐴
𝐴
= 𝑦(1− 𝑥)𝑑𝑦
1
0
(1− 𝑥)𝑑𝑦1
0
= 𝑦 − 𝑦
32 𝑑𝑦
1
0
1 − 𝑦 𝑑𝑦1
0
=0.100
0.333= 0.3𝑚
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.15 Center of Gravity and Centroid
§1. Center of Gravity, Center of Mass, and the Centroid of a Body
- Example 9.6 Locate the centroid of the semi-elliptical area
Solution
Differential Element
Arbitrary element (𝑑𝑥,𝑦)
Area: 𝑑𝐴 = 𝑦𝑑𝑥
Moment Arms: 𝑥 = 𝑥, 𝑦 = 𝑦/2
Integration
𝑥 = 0
𝑦 = 𝑦 𝑑𝐴
𝐴
𝑑𝐴
𝐴
=
𝑦2(1− 𝑥)𝑑𝑦
2
−2
𝑦𝑑𝑥2
−2
= 1 −
𝑥2
4𝑑𝑥
2
−2
1 −𝑥2
4𝑑𝑥
2
−2
=4/3
𝜋= 0.424
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.16 Center of Gravity and Centroid
§1. Center of Gravity, Center of Mass, and the Centroid of a Body
- Example 9.7 Locate the 𝑦 centroid for
the paraboloid of revolution
Solution
Differential Element
Arbitrary element: thin disk
Volume: 𝑑𝑉 = 𝜋𝑧2 𝑑𝑦
Moment Arms: 𝑦 = 𝑦
Integration
𝑥 = 0
𝑦 = 𝑦 𝑑𝑉
𝑉
𝑑𝑉
𝑉
= 𝑦(𝜋𝑧2)𝑑𝑦
100
0
(𝜋𝑧2)𝑑𝑦100
0
=100𝜋 𝑦2𝑑𝑦
100
0
100𝜋 𝑦𝑑𝑦100
0
= 66.7𝑚𝑚
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.17 Center of Gravity and Centroid
§1. Center of Gravity, Center of Mass, and the Centroid of a Body
- Example 9.8 Determine the location of the center of mass of
the cylinder if its density varies directly with the distance from
its base, i.e., 𝜌 = 200𝑧𝑘𝑔/𝑚3
Solution
Differential Element
Arbitrary element: disk element thickness 𝑑𝑧
Volume: 𝑑𝑉 = 𝜋0.52𝑑𝑧
Moment Arms: 𝑧 = 𝑧
Integration
𝑥 = 0, 𝑦 = 0
𝑧 = 𝑧 𝑑𝑉
𝑉
𝜌𝑑𝑉
𝑉
= 𝑧 200𝑧 [𝜋 0.52 ]𝑑𝑧
1
0
200𝑧 [𝜋 0.52 ]𝑑𝑧1
0
= 𝑧2𝑑𝑧
1
0
𝑧𝑑𝑧1
0
= 0.667𝑚
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.18 Center of Gravity and Centroid
3/25/2013
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Fundamental Problems
- F9.1 Determine the centroid (𝑥 , 𝑦 ) of the shaded area
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.19 Center of Gravity and Centroid
Fundamental Problems
- F9.2 Determine the centroid (𝑥 , 𝑦 ) of the shaded area
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.20 Center of Gravity and Centroid
Fundamental Problems
- F9.3 Determine the centroid 𝑦 of the shaded area
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.21 Center of Gravity and Centroid
Fundamental Problems
- F9.4 Locate the center mass 𝑥 of the straight rod if its mass
per unit length is given by 𝑚 = 𝑚0(1 + 𝑥2/𝐿2)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.22 Center of Gravity and Centroid
Fundamental Problems
- F9.5 Locate the centroid 𝑦 of the homogeneous solid formed
by revolving the shaded area about the 𝑦 axis
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.23 Center of Gravity and Centroid
Fundamental Problems
- F9.6 Locate the centroid 𝑧 of the homogeneous solid formed
by revolving the shaded area about the 𝑧 axis
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.24 Center of Gravity and Centroid
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§2. Composite Body
- A composite body consists of a series of
connected “simpler” shaped bodies, which
may be rectangular, triangular,
semicircular, …
- Such a body can often be sectioned or
divided into its composite parts and,
provided the weight and location of the
center of gravity of each of these parts are
known
- We can then eliminate the need for
integration to determine the center of
gravity for the entire body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.25 Center of Gravity and Centroid
§2. Composite Body
- The center of gravity of 𝐺
𝑥 =∑𝑥 𝑊
∑𝑊, 𝑦 =
∑𝑦 𝑊
∑𝑊, 𝑧 =
∑𝑧 𝑊
∑𝑊
𝑥 , 𝑦 , 𝑧 : the coordinates of the center of gravity 𝐺 of the
composite body
𝑊: the weights of the composite parts of the body
𝑥 , 𝑦 , 𝑧 : the coordinates of the center of gravity of each
composite part of the body
- The Centroid for composite lines, areas and volumes can be
found using relations analogous to the above one
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.26 Center of Gravity and Centroid
§2. Composite Body
- Example 9.9 Locate the centroid of the wire
Solution
𝑥 =∑𝑥 𝐿
∑𝐿=
11310
248.5= 45.5𝑚𝑚, 𝑦 =
∑𝑦 𝐿
∑𝐿=
−5600
248.5= −22.5𝑚𝑚,
𝑧 =∑𝑧 𝑊
∑𝑊=
−200
248.5= −0.805𝑚𝑚
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.27 Center of Gravity and Centroid
§2. Composite Body
- Example 9.9 Locate the centroid of the plate area
Solution
Composite Parts: divide the wire into 3 segments
Moment Arms: the centroid are located as indicated
Summary
𝑥 =∑𝑥 𝐴
∑𝐴=
−4
11.5= −0.348𝑚
𝑦 =∑𝑦 𝐴
∑𝐴=
14
11.5= 1.22𝑚
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.28 Center of Gravity and Centroid
§2. Composite Body
- Example 9.10 Locate the center of mass of the assembly. The
conical frustum has a density of and the
hemisphere has a density of There is a 25-
𝑚𝑚-radius cylindrical hole in the center of the
frustum
Solution
Composite Parts
(1) + (2) − (3) − (4)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.29 Center of Gravity and Centroid
§2. Composite Body
Composite Parts: (1) + (2) − (3) − (4)
Moment Arms: the centroid are located as indicated
Summary
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.30 Center of Gravity and Centroid
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§2. Composite Body
Composite Parts: (1) + (2) − (3) − (4)
Moment Arms: the centroid are located as indicated
Summary
The center of mass
𝑥 = 0
𝑦 = 0
𝑧 =∑𝑧 𝑚
∑𝑚=
45.815
3.142= 14.6𝑚𝑚
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.31 Center of Gravity and Centroid
Fundamental Problems
- F9.7 Locate the centroid (𝑥 , 𝑦 , 𝑧 ) of the wire bent in the shape
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.32 Center of Gravity and Centroid
Fundamental Problems
- F9.8 Locate the centroid 𝑦 of the beam’s cross-sectional area
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.33 Center of Gravity and Centroid
Fundamental Problems
- F9.9 Locate the centroid 𝑦 of the beam’s cross sectional area
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.34 Center of Gravity and Centroid
Fundamental Problems
- F9.10 Locate the centroid (𝑥 , 𝑦 ) of the cross-sectional area
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.35 Center of Gravity and Centroid
Fundamental Problems
- F9.11 Locate the center of mass (𝑥 , 𝑦 , 𝑧 ) of the homogeneous
solid block
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.36 Center of Gravity and Centroid
3/25/2013
7
Fundamental Problems
- F9.12 Locate the center of mass (𝑥 , 𝑦 , 𝑧 ) of the homogeneous
solid block
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics – Statics 9.37 Center of Gravity and Centroid