ENGI 1313 Mechanics I

1. ENGI 1313 Mechanics I Lecture 40: Center of Gravity, Center of Mass and Geometric Centroid

2. 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

3. 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

4. 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

5. ~ ~ x1 z1 x z Center of Gravity (cont.) • Resultant Weight • Coordinates • Key Property L/2 w4 w3 xG w2 w1 WR

6. ~ ~ ~ zn z2 z1 Center of Gravity (cont.) • Generalized Formulae Moment about y-axis Moment about x-axis “Moment” about x-axis or y-axis

7. 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

8. Center of Mass (cont.) • Can the Center of Mass be Outside the Body? Fulcrum / Balance Center of Mass

9. Center of Gravity & Mass – Applications • Dynamics • Inertial terms • Vehicle roll-over and stability

10. 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

11. GC & CM GC & CM GC & CM Median Lines Geometric Centroid (cont.) • Common Geometric Shapes • Solid structure or frame elements

12. Composite Body • Find center of gravity or geometric centroid of complex shape based on knowledge of simpler geometric forms

13. 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.

14. Example 40-01 (cont.) • Center of Mass

15. 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.

16. Example 40-02 (cont.) • Assume Unit Thickness • Ignore bend radii • Center-to-center distance • Centroid Equations

17. ~ x1 = 7.5mm Example 40-02 (cont.) • Centroid Equations 1

18. ~ y5 = 25mm Example 40-02 (cont.) • Centroid Equations 5

19. ~ ~ y6 = 65mm x6 = 15mm Example 40-02 (cont.) 6 • Centroid Equations

20. 4 Example 40-02 (cont.) 6 3 • Centroid Equations 7 5 1 2

21. Example 40-02 (cont.) • Centroid Equations 24.4 mm 40.6 mm

22. 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.

23. Example 40-03 (cont.) • Center of Gravity

24. Example 40-03 (cont.) • Center of Gravity

25. Example 40-03 (cont.) • Center of Gravity

26. 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

27. References • Hibbeler (2007) • http://wps.prenhall.com/esm_hibbeler_engmech_1