1. Engineering Mechanics: Statics
Chapter 9:
Center of Gravity and Centroid
Engineering Mechanics: Statics

2. Chapter Objectives
To discuss the concept of the center of gravity, center of mass, and the centroid.
To show how to determine the location of the center of gravity and centroid for a system of discrete particles and a body of arbitrary shape.
To use the theorems of Pappus and Guldinus for finding the area and volume for a surface of revolution.
To present a method for finding the resultant of a general distributed loading and show how it applies to finding the resultant of a fluid.

3. Chapter Outline
Center of Gravity and Center of Mass for a System of Particles
Center of Gravity and Center of Mass and Centroid for a Body
Composite Bodies
Theorems of Pappus and Guldinus
Resultant of a General Distributed Loading
Fluid Pressure

4. 9.1 Center of Gravity and Center of Mass for a System of Particles
Locates the resultant weight of a system of particles
Consider system of n particles fixed within a region of space
The weights of the particles
comprise a system of parallel
forces which can be replaced
by a single (equivalent) resultant
weight having defined point G
of application

5. 9.1 Center of Gravity and Center of Mass for a System of Particles
Resultant weight = total weight of n particles
Sum of moments of weights of all the particles about x, y, z axes = moment of resultant weight about these axes
Summing moments about the x axis,
Summing moments about y axis,

6. 9.1 Center of Gravity and Center of Mass for a System of Particles
Although the weights do not produce a moment about z axis, by rotating the coordinate system 90° about x or y axis with the particles fixed in it and summing moments about the x axis,
Generally,

7. 9.1 Center of Gravity and Center of Mass for a System of Particles
Where represent the coordinates of the center of gravity G of the system of particles,
represent the coordinates of each particle in the system and represent the resultant sum of the weights of all the particles in the system.
These equations represent a balance between the sum of the moments of the weights of each particle and the moment of resultant weight for the system.

8. 9.1 Center of Gravity and Center of Mass for a System of Particles
Center Mass
Provided acceleration due to gravity g for every particle is constant, then W = mg
By comparison, the location of the center of gravity coincides with that of center of mass
Particles have weight only when under the influence of gravitational attraction, whereas center of mass is independent of gravity

9. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Center Mass
A rigid body is composed of an infinite number of particles
Consider arbitrary particle
having a weight of dW

10. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Center Mass
γ represents the specific weight and dW = γdV

11. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Density ρ, or mass per unit volume, is related to γ by γ = ρg, where g = acceleration due to gravity
Substitute this relationship into this equation to determine the body’s center of mass

12. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Defines the geometric center of object
Its location can be determined from equations used to determine the body’s center of gravity or center of mass
If the material composing a body is uniform or homogenous, the density or specific weight will be constant throughout the body
The following formulas are independent of the body’s weight and depend on the body’s geometry

13. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Volume
Consider an object subdivided into volume elements dV, for location of the centroid,

14. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Area
For centroid for surface area of an object, such as plate and shell, subdivide the area into differential elements dA

15. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Line
If the geometry of the object such as a thin rod or wire, takes the form of a line, the balance of moments of differential elements dL about each of the coordinate system yields

16. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Line
Choose a coordinate system that simplifies as much as possible the equation used to describe the object’s boundary
Example
Polar coordinates are appropriate for area with circular boundaries

17. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Symmetry
The centroids of some shapes may be partially or completely specified by using conditions of symmetry
In cases where the shape has an axis of symmetry, the centroid of the shape must lie along the line
Example
Centroid C must lie along the
y axis since for every element
length dL, it lies in the middle

18. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Symmetry
For total moment of all the elements about the axis of symmetry will therefore be cancelled
In cases where a shape has 2 or 3 axes of symmetry, the centroid lies at the intersection of these axes

19. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Procedure for Analysis
Differential Element
Select an appropriate coordinate system, specify the coordinate axes, and choose an differential element for integration
For lines, the element dL is represented as a differential line segment
For areas, the element dA is generally a rectangular having a finite length and differential width

20. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Procedure for Analysis
Differential Element
For volumes, the element dV is either a circular disk having a finite radius and differential thickness, or a shell having a finite length and radius and a differential thickness
Locate the element at an arbitrary point (x, y, z) on the curve that defines the shape

21. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Procedure for Analysis
Size and Moment Arms
Express the length dL, area dA or volume dV of the element in terms of the curve used to define the geometric shape
Determine the coordinates or moment arms for the centroid of the center of gravity of the element

22. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Procedure for Analysis
Integrations
Substitute the formations and dL, dA and dV into the appropriate equations and perform integrations
Express the function in the integrand and in terms of the same variable as the differential thickness of the element
The limits of integrals are defined from the two extreme locations of the element’s differential thickness so that entire area is covered during integration

23. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Example 9.1
Locate the centroid of
the rod bent into the
shape of a parabolic arc.

24. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Differential element
Located on the curve at the arbitrary point (x, y)
Area and Moment Arms
For differential length of the element dL
Since x = y2 and then dx/dy = 2y
The centroid is located at

25. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Integrations

26. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Example 9.2
Locate the centroid of
the circular wire
segment.

27. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Differential element
A differential circular arc is selected
This element intersects the curve at (R, θ)
Length and Moment Arms
For differential length of the element
For centroid,

28. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Integrations

29. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Example 9.3
Determine the distance
from the x axis to the
centroid of the area
of the triangle

30. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Differential element
Consider a rectangular element having thickness dy which intersects the boundary at (x, y)
Length and Moment Arms
For area of the element
Centroid is located y distance from the x axis

31. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Integrations

32. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Example 9.4
Locate the centroid for
the area of a quarter
circle.

33. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Method 1
Differential element
Use polar coordinates for circular boundary
Triangular element intersects at point (R,θ)
Length and Moment Arms
For area of the element
Centroid is located y distance from the x axis

34. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Integrations

35. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Method 2
Differential element
Circular arc element having thickness of dr
Element intersects the
axes at point (r,0) and
(r, π/2)

36. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Area and Moment Arms
For area of the element
Centroid is located y distance from the x axis

37. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Integrations

38. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Example 9.5
Locate the centroid of
the area.

39. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Method 1
Differential element
Differential element of thickness dx
Element intersects curve at point (x, y), height y
Area and Moment Arms
For area of the element
For centroid

40. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Integrations

41. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Method 2
Differential element
Differential element of thickness dy
Element intersects curve at point (x, y)
Length = (1 – x)

42. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Area and Moment Arms
For area of the element
Centroid is located y distance from the x axis

43. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Integrations

44. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Example 9.6
Locate the centroid of
the shaded are bounded
by the two curves
y = x
and y = x2.

45. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Method 1
Differential element
Differential element of thickness dx
Intersects curve at point (x1, y1) and (x2, y2), height y
Area and Moment Arms
For area of the element
For centroid

46. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Integrations

47. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Method 2
Differential element
Differential element of thickness dy
Element intersects curve at point (x1, y1) and (x2, y2)
Length = (x1 – x2)

48. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Area and Moment Arms
For area of the element
Centroid is located y distance from the x axis

49. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Integrations

50. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Example 9.7
Locate the centroid for the
paraboloid of revolution,
which is generated by
revolving the shaded area
about the y axis.

51. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Method 1
Differential element
Element in the shape of a thin disk, thickness dy, radius z
dA is always perpendicular to the axis of revolution
Intersects at point (0, y, z)
Area and Moment Arms
For volume of the element

52. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
For centroid
Integrations

53. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Method 2
Differential element
Volume element in the form of thin cylindrical shell, thickness of dz
dA is taken parallel to the axis of revolution
Element intersects the
axes at point (0, y, z) and
radius r = z

54. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Area and Moment Arms
For area of the element
Centroid is located y distance from the x axis

55. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Integrations

56. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Example 9.8
Determine the location of
the center of mass of the
cylinder if its density
varies directly with its
distance from the base
ρ = 200z kg/m3.

57. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
View Free Body Diagram
Solution
For reasons of material symmetry
Differential element
Disk element of radius 0.5m and thickness dz since density is constant for given value of z
Located along z axis at point (0, 0, z)
Area and Moment Arms
For volume of the element

58. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
For centroid
Integrations

59. 9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Not possible to use a shell element for integration since the density of the material composing the shell would vary along the shell’s height and hence the location of the element cannot be specified