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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)

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ENGI 1313 Mechanics I

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Engi 1313 mechanics i

ENGI 1313 Mechanics I

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


Material coverage for final exam

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

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

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


Center of gravity cont

~

~

x1

z1

x

z

Center of Gravity (cont.)

  • Resultant Weight

  • Coordinates

  • Key Property

L/2

w4

w3

xG

w2

w1

WR


Center of gravity cont1

~

~

~

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

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 mass cont

Center of Mass (cont.)

  • Can the Center of Mass be Outside the Body?

Fulcrum / Balance

Center of Mass


Center of gravity mass applications

Center of Gravity & Mass – Applications

  • Dynamics

    • Inertial terms

    • Vehicle roll-over and stability


Geometric centroid

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


Geometric centroid cont

GC & CM

GC & CM

GC & CM

Median Lines

Geometric Centroid (cont.)

  • Common Geometric Shapes

    • Solid structure or frame elements


Composite body

Composite Body

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


Example 40 01

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

Example 40-01 (cont.)

  • Center of Mass


Example 40 02

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

Example 40-02 (cont.)

  • Assume Unit Thickness

    • Ignore bend radii

    • Center-to-center distance

  • Centroid Equations


Example 40 02 cont1

~

x1 = 7.5mm

Example 40-02 (cont.)

  • Centroid Equations

1


Example 40 02 cont2

~

y5 = 25mm

Example 40-02 (cont.)

  • Centroid Equations

5


Example 40 02 cont3

~

~

y6 = 65mm

x6 = 15mm

Example 40-02 (cont.)

6

  • Centroid Equations


Example 40 02 cont4

4

Example 40-02 (cont.)

6

3

  • Centroid Equations

7

5

1

2


Example 40 02 cont5

Example 40-02 (cont.)

  • Centroid Equations

24.4 mm

40.6 mm


Example 40 03

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

Example 40-03 (cont.)

  • Center of Gravity


Example 40 03 cont1

Example 40-03 (cont.)

  • Center of Gravity


Example 40 03 cont2

Example 40-03 (cont.)

  • Center of Gravity


Chapter 9 problems

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

References

  • Hibbeler (2007)

  • http://wps.prenhall.com/esm_hibbeler_engmech_1


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