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ENGI 1313 Mechanics I

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ENGI 1313 Mechanics I

Lecture 40:Center of Gravity, Center of Mass and Geometric Centroid

Material Coverage for Final Exam

- Introduction (Ch.1: Sections 1.1–1.6)
- Force Vectors (Ch.2: Sections 2.1–2.9)
- Particle Equilibrium (Ch.3: Sections 3.1–3.4)
- Force System Resultants (Ch.4: Sections 4.1–4.10)
- Omit Wrench (p.174)

- Rigid Body Equilibrium (Ch.5: Sections 5.1–5.7)
- Structural Analysis (Ch.6: Sections 6.1–6.4 & 6.6)
- Friction (Ch.8: Sections 8.1–8.3)
- Center of Gravity and Centroid (Ch.9: Sections 9.1–9.3)
- Ignore problems involving closed-form integration
- Simple shapes such as square, rectangle, triangle and circle

Lecture 40 Objective

- to understand the concepts of center of gravity, center of mass, and geometric centroid
- to be able to determine the location of these points for a system of particles or a body

Center of Gravity

- Point locating the equivalent resultant weight of a system of particles or body
- Example: Solid Blocks
- Are both final configurations stable?

w5

w5

w3

w3

w2

w2

w1

w1

WR

WR

~

x1

z1

x

z

Center of Gravity (cont.)- Resultant Weight
- Coordinates
- Key Property

L/2

w4

w3

xG

w2

w1

WR

~

~

zn

z2

z1

Center of Gravity (cont.)- Generalized Formulae

Moment about y-axis

Moment about x-axis

“Moment” about x-axis

or y-axis

Center of Mass

- Point locating the equivalent resultant mass of a system of particles or body
- Generally coincides with center of gravity (G)
- Center of mass coordinates

Center of Gravity & Mass – Applications

- Dynamics
- Inertial terms
- Vehicle roll-over and stability

Geometric Centroid

- Point locating the geometric center of an object or body
- Homogeneous body
- Body with uniform distribution of density or specific weight
- Center of mass and center of gravity coincident
- Centroid only dependent on body dimensions and not weight terms

- Body with uniform distribution of density or specific weight

GC & CM

GC & CM

Median Lines

Geometric Centroid (cont.)- Common Geometric Shapes
- Solid structure or frame elements

Composite Body

- Find center of gravity or geometric centroid of complex shape based on knowledge of simpler geometric forms

Example 40-01

- Determine the location (x, y) of the 7-kg particle so that the three particles, which lie in the x−y plane, have a center of mass located at the origin O.

Example 40-01 (cont.)

- Center of Mass

Example 40-02

- A rack is made from roll-formed sheet steel and has the cross section shown. Determine the location (x,y) of the centroid of the cross section. The dimensions are indicated at the center thickness of each segment.

Example 40-02 (cont.)

- Assume Unit Thickness
- Ignore bend radii
- Center-to-center distance

- Centroid Equations

Example 40-03

- Two blocks of different materials are assembled as shown. The densities of the materials are: A = 150 lb/ft3 and A = 400 lb/ft3. The center of gravity of this assembly.

Example 40-03 (cont.)

- Center of Gravity

Example 40-03 (cont.)

- Center of Gravity

Example 40-03 (cont.)

- Center of Gravity

Chapter 9 Problems

- Understand principles for simple geometric shapes
- Rectangle, square, triangle and circle
- No closed form integration knowledge required

- Review
- Example 9.9 and 9.10
- Problems 9-44 to 9-61

- Omit
- Example 9.1 through 9.8
- Problems 9-1 through 9-43, 9-62, 9-67 to 9-83

References

- Hibbeler (2007)
- http://wps.prenhall.com/esm_hibbeler_engmech_1