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

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

Center of Gravity (cont.) Generalized Formulae Moment about y-axis z2 ~ Moment about x-axis zn ~ z1 ~ “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 Mass (cont.) Can the Center of Mass be Outside the Body? Fulcrum / Balance Center of Mass

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

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

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 Centroid Equations Ignore bend radii Center-to-center distance Centroid Equations

Example 40-02 (cont.) Centroid Equations 1 ~ x1 = 7.5mm # Area (mm2) 1 (15mm)(1mm) = 15mm2 15/2 = 7.5  (15)(7.5) = 112.5

Example 40-02 (cont.) Centroid Equations 5 ~ y5 = 25mm # Area (mm2) 5 (50mm)(1mm) = 50mm2  50/2 = 25 (50)(25) = 1250

Example 40-02 (cont.) Centroid Equations x6 = 15mm ~ 6 y6 = 65mm ~ # Area (mm2) 6 (30mm)(1mm) = 30mm2 15 50+30/2 = 65 450 1950

Example 40-02 (cont.) Centroid Equations 4 6 3 7 5 1 2 Sum 235mm2 # Area (mm2) 1 (15mm)(1mm) = 15mm2 15/2 = 7.5  112.5 2 30+15/2 = 37.5 562.5 3 50 750 4 (30mm)(1mm) = 30mm2 15+30/2 = 30 80 900 2400 5 (50mm)(1mm) = 50mm2 50/2 = 25 1250 6 15 50+30/2 = 65 450 1950 7 (80mm)(1mm) = 80mm2 45 80/2 = 40 3600 3200 Sum 235mm2 5737.5mm3 9550mm3

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

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 # Weight (lb) A 3.125 4 1 2 12.5 6.25 B 16.67 3 50  19.79 29.17 53.13 56.25

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