Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

1. Engineering 36 Chp09: Centerof Gravity Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. Introduction: Center of Gravity • The earth exerts a gravitational force on each of the particles forming a body. • These forces can be replaced by a SINGLE equivalent force equal to the weight of the body and applied at the CENTER OF GRAVITY (CG) for the body • The CENTROID of an AREA is analogous to the CG of a body. • The concept of the FIRST MOMENT of an AREA is used to locate the centroid

3. Given a Massive Body in 3D Space Total Mass – General Case • Divide the Body in to Very Small Volumes → dV • Each dVn is located at position (xn,yn,zn) • The DENSITY, ρ, can be a function of POSITION →

4. Total Mass – General Case • Now since m = ρ•V, then the incremental mass, dm • Integrate dm over the entire body to obtain the total Mass, M

5. Total WEIGHT – General Case • Recall that W = Mg • Using the the previous expression for M • Now Define SPECIFIC WEIGHT, γ, as ρ•g →

6. Uniform Density Case • Consider a body with UNIFORM DENSITY; i.e. • Then M & W→

7. Center of Mass Location • Use MOMENTS to Locate Center of Mass/Gravity • Recall Defintion of a MOMENT • In the General Center-of-Mass Case • LeverArm ≡ Position Vector, rn, or its components • Intensity ≡ Incremental Mass, dmn

8. Center of Mass Location • RM in Component form • Now Define the IncrementalMoment, dΩn LeverArm Intensity

9. Center of Mass Location • Integrating dΩ to Find Ω for the entire body

10. Center of Mass Location • Now Equate Ω to RM•Mg • Canceling g, and equatingComponents yields, for example, in the X-Dir LeverArm Intensity

11. Center of Mass Location • Divide out the Total Intensity, M, to Isolate the Overall X-directed Lever Arm, XM • And the Similar expressions for the other CoOrd Directions

12. Centroid of an Area Center of Gravity of a 2D Body • Centroid of a Line • Taking Incremental Plate Areas, Forming the Ωx & Ωy, Along With the Fz=W Yields the Expression for the Equivalent POINT of W application • Note ΩUnits = In-lb or N-m

13. Centroid of an Area Centroids of Areas & Lines • Centroid of a Line • Wire of Uniform Thickness •   Specific Weight • a  X-Sec Area • dW = a(dL) • For Plate of Uniform Thickness •   Specific Weight • t  Plate Thickness • dW =  tdA

14. An area is symmetric with respect to an axis BB’ if for every point P there exists a point P’ such that PP’ is perpendicular to BB’ and the Area is divided into equal parts by BB’. • If an area possesses two lines of symmetry, its centroid lies at their INTERSECTION. • An area is symmetric with respect to a center O if for every element dA at (x,y) there exists an area dA’ of equal area at (−x, −y). First Moments of Areas & Lines • The first moment of an area with respect to a line of symmetry is ZERO. • If an area possesses a line of SYMMETRY, its centroid LIES on THAT AXIS • The centroid of the area coincides with the center of symmetry, O.

15. Centroids of Common Area Shapes

16. Centroids of Common Line Shapes • Recall that for a SMALL Angle, α

17. Composite Plates and Areas • Composite plates • Composite area

18. For the plane area shown, determine the first moments with respect to the x and y axes, and the location of the centroid. Solution Plan Divide the area into a triangle, rectangle, semicircle, and a circular cutout Calculate the first moments of each area w/ respect to the axes Find the total area and first moments of the triangle, rectangle, and semicircle. Subtract the area and first moment of the circular cutout Calc the coordinates of the area centroid by dividing the NET first moment by the total area Example: Composite Plate

20. Example: Composite Plate • Find the total area and first moments of the triangle, rectangle, and semicircle. Subtract the area and first moment of the circular cutout

21. Find the coordinates of the area centroid by dividing the first moment totals by the total area Solution Example: Composite Plate

22. Double integration to find the first moment may be avoided by defining dA as a thin rectangle or strip. Centroids by Strip Integration

23. Example: Centroid by Integration • Solution Plan • Determine Constant k • Calculate the Total Area • Using either vertical or horizontal STRIPS, perform a single integration to find the first moments • Evaluate the centroid coordinates by dividing the Total 1st Moment by Total Area. • Determine by direct integration the location of the centroid of a parabolic spandrel.

25. Solution Determine Constant k Example: Centroid by Integration • Calculate the Total Area

26. Example: Centroid by Integration • Solution • Calc the 1st Moments, Ωi • Using vertical strips, perform a single integration to find the first moments.

27. Example: Centroid by Integration • Finally the Answers • Evaluate the centroid coordinates • Divide Ωx and Ωyby the total Area

28. Theorems of Pappus-Guldinus • Surface of revolution is generated by rotating a plane curve about a fixed axis. • Area of a surface of revolution is equal to the length of the generating curve, L, times the distance traveled by the centroid through the rotation.

29. Theorems of Pappus-Guldinus • Body of revolution is generated by rotating a plane area about a fixed axis. • Volume of a body of revolution is equal to the generating area, A, times the distance traveled by the centroid through the rotation.

30. The outside diameter of a pulley is 0.8 m, and the cross section of its rim is as shown. Knowing that the pulley is made of steel and that the density of steel,  = 7850 kg/m3, determine the mass and weight of the rim. Solution Plan Apply the theorem of Pappus-Guldinus to evaluate the volumes of revolution for the rectangular rim section and the inner cutout section. Multiply by density and acceleration of gravity to get the mass and weight. Example:Pappus-Guldinus

31. Example: P-G • Apply Pappus-Guldinus to Sections I & II • Subtract: I-II

32. WhiteBoard Work Find theAreal &LinealCentroids

33. Engineering 36 Appendix Bruce Mayer, PE Registered Electrical & Mechanical EngineerBMayer@ChabotCollege.edu