pe2113-chapter 9 - center of gravity and centroid_draft

7/23/2019 PE2113-Chapter 9 - Center of Gravity and Centroid_draft

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Chapter 9 –

Center ofGravity and Centroid



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Objectives

• To discuss the concept of the center ofgravity, center of mass, and the centroid

•

To show how to determine the location of thecenter of gravity and centroid for a system of

discrete particle and a body of arbitrary

shape.



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9.1 Center of Gravity

• The center of gravity, G is a point which locates the resultant

weight (not mass!!!) of a system of particles.• The weights of the particles comprise of a system of parallel forces

which can be replaced by a single (equivalent) resultant weight

having the defined point G of application.

•

The generalized formulas for determining the center of gravity aregiven as described in next page:



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• The above equations are easily remembered if it is kept

in mind that they simply represent a balance between

the sum of the moments of the weight of each particle of

the system and the moment of the resultant weight for

the system

OR(described later)



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Center of Gravity

• The above equations are easily remembered if it

is kept in mind that they simply represent abalance between the sum of the moments of the

weight of each particle of the system and the

moment of the resultant weight for the system



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Center of Mass

• If the acceleration due to gravity g for every particle inthe previous slide is constant , then W = mg and thecenter of gravity equation becomes the equation above.These are the equations for center of mass.

• By comparison, then, the location of the center of gravity

coincides with that of center of mass if g is constant.

• However, center of gravity is not independent ofgravity whereas center of mass is independent ofgravity.



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Center of Mass

• If the acceleration due to gravity g for every particle inthe previous slide is constant, then W = mg and thecenter of gravity equation becomes the equation above.These are the equations for center of mass.

• By comparison, then, the location of the center of gravity

coincides with that of center of mass if g is constant.

• However, center of gravity is not independent ofgravity whereas center of mass is independent ofgravity.



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Center of Gravity for a Rigid-Body

• A rigid body is composed of an infinite number of particles and so if

the principles used to determine the equation for the CG of discrete particles are applied to the system of particles composing a rigid body ,

it becomes necessary to use integration rather than a discrete

summation of the terms. The resulting equations are given as follows:



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Center of Mass of a Rigid Body

• For the center of mass of a rigid body, thedensity, ,or mass per unit volume is

used. This density is related to specific

weight , , by the equation = g .• Substituting g into Eqn. 9.4, and

assuming g is constant, the center of

mass is determined.



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Centroid (Volume, Area, Line)• The centroid C is a point which defines geometric center of an

object. Its location can be determined from formulas similar tothose used to determine the body’s center of gravity or center of

mass.

• In particular, if the material composing a body is uniform or

homogeneous, the density or specific weight will be constant throughout the body, and therefore this term will factor out of the

integrals and cancel out of Eqn. 9.4.

•

The resulting formulas define the centroid of the body since theyare independent of the body weight and instead depend only on

the body geometry.

• Three specific cases are considered: Volume, Area, and Line



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Centroid (Volume, Area, Line)

Volume

Area

Line



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Solv ing Centro id Prob lems

• In order to use Eqns. 9.4 to 9.7, it is best

to choose a coordinate system that

simplifies as much as possible the

equations used to describe the object’s

boundary.

• The terms x , y , and z in the equations

refer to the “moment arms ” or coordinates

of the center of gravity or centroid of thedifferential element used.



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601



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Exercise Pro. 9.7, 9.8



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• Solve Pro. 9-28

• Exercise Pro. 9-32

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pe2113-chapter 9 - center of gravity and centroid_draft

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