3/25/2013 1 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 - 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, §1. Center of Gravity, Center of Mass, and the Centroid of a Body - The location of the center of gravity with respect to the ,,= , = , = 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 = - The location of the center of gravity (, ,) = = = §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 with respect to the ,,axes Substitute = to = , = ,= ⟹ = , = ,=
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3/25/2013
1
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