2. Chapter Objectives Chapter Outline
Concept of the center of gravity, center of mass, and the centroid
Determine the location of the center of gravity and centroid for a system of discrete particles and a body of arbitrary shape
Chapter Outline
Center of Gravity and Center of Mass for a System of Particles
Composite Bodies

3. 9.1 Center of Gravity and Center of Mass for a System of Particles
An object is composed of infinitesimal bodies and let each body has the weight dW
Let the body is in the 3D in the x, y, z frame
The total weight of all the bodies can be represented by a single force W at point G

4. 9.1 Center of Gravity and Center of Mass for a System of Particles
Center of Gravity (use equivalent force and moment)
Resultant weight of an object = the sum of all infinitestimal weights ofall bodies
Resultant moment about any axis =The sum of moments of all bodies about the axis
Let G is the location of the equivalent force, which is equal to

5. 9.1 Center of Gravity and Center of Mass for a System of Particles
In summary, for discrete quantity
For continuous quantity

6. 9.1 Center of Gravity and Center of Mass for a System of Particles
Center Mass
Using the relationship W = mg, consider the centre of mass with contant g
In general, the centre of mass is the centre of gravity
The centre of mass is usually used in Dynamics problems

7. 9.1 Center of Gravity and Center of Mass for a System of Particles
Centroid of a volume
For an object with uniformly distributed mass, hence or
Hence the centre of volume is

8. 9.1 Center of Gravity and Center of Mass for a System of Particles
Centroid of an Area
For an are with the same thickness thoughout the area
The area can be subdivide into dA

9. 9.1 Center of Gravity and Center of Mass for a System of Particles
Centroid of a Line
For an object with a line shape, consider a small length dL
From Pythagorus or

10. Example 9.1
Locate the centroid of the rod bent into the shape of a parabolic arc.

11. Example 9.1 Differential element
dL is 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 of a small element is located at

12. Example 9.1
Integrations

13. Example 9.2
Locate the centroid of the circular wire segment shown.

14. Example 9.2
Polar coordinate is more suitable since the arc is circular
Differential element
A circular arc is selected, it intersects the curve at
Area and Moment Arms
The differential length of the element
The centroid of a small element is located at
Integration

15. Example 9.3
Determine the distance measured from the x axis to the centroid of the area of the triangle shown.

16. Example 9.3 Differential element
A rectangular element with thickness dy, intersect at (x,y)
Area and Moment Arms
The area of the element
The centroid of a small element is located at
Integration

17. Example 9.4
Locate the centroid for the area of a quarter circle shown.

18. Example 9.4 Polar coordinate is used. Differential element
A triangle element with angle intersect at
Area and Moment Arms
The area of the element
The centroid of the triangle element is located at (Ex. 9.3)
Integration

19. Example 9.5
Locate the centroid for the area shown below.

20. Example 9.5 Solution I (Fig. a) Differential element
A rectangular element with thickness dx, has a height y.
Area and Moment Arms
The area of the element
The centroid is located at
Integration

21. Example 9.5 Solution II (Fig. b) Differential element
A rectangular element with thickness dy, has a length (1-x).
Area and Moment Arms
The area of the element
The centroid is located at
Integration

22. 9.2 Composite Bodies
is composed of objects with simple geometries such as triangles, rectangles, circles
Consider individual geometries to find the centroid

23. 9.2 Composite Bodies Analysis method Composite Parts
Devide a big object in to a series of simple geometries
For the void or spaces, the geometries have negative values
Moment Arms
Locate the frame and then find the centroid for each simple geometry
Summations
Find the centroid the previous centroid equaiton

24. Example 9.10
Locate the centroid of the plate area.

25. Solution Composite Parts Plate divided into 3 segments.
Area of small rectangle considered “negative”.

26. Solution
Moment Arm
Location of the centroid for each piece is determined and indicated in the diagram.
Summations