1. 9.3 Composite Bodies
Consists of a series of connected “simpler”
shaped bodies, which may be rectangular,
triangular or semicircular
A body can be sectioned or divided into its
composite parts
Provided the weight and location of the center of
gravity of each of these parts are known, the
need for integration to determine the center of
gravity for the entire body can be neglected

2. 9.3 Composite Bodies
Accounting for finite number of weights
∑ ~W
x ∑ ~W
y ∑ ~W
z
x= y= z=
∑W ∑W ∑W
Where
x, y, z represent the coordinates of the center
of gravity G of the composite body
~, ~, ~
x y z represent the coordinates of the center
of gravity at each composite part of the body
∑W represent the sum of the weights of all
the composite parts of the body or total weight

3. 9.3 Composite Bodies
When the body has a constant density or
specified weight, the center of gravity coincides
with the centroid of the body
The centroid for composite lines, areas, and
volumes can be found using the equation
∑~W
x ∑~W
y ∑~W
z
x= y= z=
∑W ∑W ∑W
However, the W’s are replaced by L’s, A’s and V’s
respectively

4. 9.3 Composite Bodies
Procedure for Analysis
Composite Parts
Using a sketch, divide the body or object into
a finite number of composite parts that have
simpler shapes
If a composite part has a hole, or a
geometric region having no material,
consider it without the hole and treat the
hole as an additional composite part having
negative weight or size

5. 9.3 Composite Bodies
Procedure for Analysis
Moment Arms
Establish the coordinate axes on the
sketch and determine the coordinates
of the center of gravity or centroid of
each part

6. 9.3 Composite Bodies
Procedure for Analysis
Summations
Determine the coordinates of the
center of gravity by applying the center
of gravity equations
If an object is symmetrical about an
axis, the centroid of the objects lies on
the axis

7. 9.3 Composite Bodies
Example 9.9
Locate the centroid of the wire.

8. 9.3 Composite Bodies
Solution
Composite Parts
Moment Arms
Location of the centroid for each piece is
determined and indicated in the diagram

9. 9.3 Composite Bodies
Solution
Summations
Segment L x (mm) y (mm) z (mm) xL yL zL
(mm) (mm2) (mm2) (mm2)
1 188.5 60 -38.2 0 11 310 -7200 0
2 40 0 20 0 0 800 0
3 20 0 40 -10 0 800 -200
Sum 248.5 11 310 -5600 -200

10. 9.3 Composite Bodies
Solution
Summations
∑ ~L 11310
x
x= = = 45.5mm
∑L 248.5
∑ ~L − 5600
y
y= = = −22.5mm
∑L 248.5
∑ ~L − 200
z
z= = = −0.805mm
∑ L 248.5

11. 9.3 Composite Bodies
Example 9.10
Locate the centroid of the plate area.

12. 9.3 Composite Bodies
Solution
Composite Parts
Plate divided into 3 segments
Area of small rectangle considered “negative”

13. 9.3 Composite Bodies
Solution
Moment Arm
Location of the centroid for each piece is
determined and indicated in the diagram

14. 9.3 Composite Bodies
Solution
Summations
Segment A (mm2) x (mm) y (mm) xA (mm3) yA (mm3)
1 4.5 1 1 4.5 4.5
2 9 -1.5 1.5 -13.5 13.5
3 -2 -2.5 2 5 -4
Sum 11.5 -4 14

15. 9.3 Composite Bodies
Solution
Summations
∑ ~A − 4
x
x= = = −0.348mm
∑ A 11.5
∑ ~A 14
y
y= = = 1.22mm
∑ A 11.5

16. 9.3 Composite Bodies
Example 9.11
Locate the center of mass of the
composite assembly. The conical
frustum has a density of
ρc = 8Mg/m3 and the hemisphere
has a density of ρh = 4Mg/m3.
There is a 25mm radius
cylindrical hole in the center.

17. 9.3 Composite Bodies
Solution
Composite Parts
Assembly divided into 4 segments
Area of 3 and 4 considered “negative”

18. 9.3 Composite Bodies
Solution
Moment Arm
Location of the centroid for each piece is
determined and indicated in the diagram
Summations
Because of symmetry,
x = y=0

19. 9.3 Composite Bodies
Solution
Summations
Segment m (kg) z (mm) zm (kg.mm)
1 4.189 50 209.440
2 1.047 -18.75 -19.635
3 -0.524 125 -65.450
4 -1.571 50 -78.540
Sum 3.141 45.815

20. 9.3 Composite Bodies
Solution
Summations
∑ ~m 45.815
z
z= = = 14.6mm
∑m 3.141