Example 1. Find the centroid of
a right triangle of base b and
height h.
Chapter 5 – Distributed Forces: Center of Gravity
Centroid
• The average position of an area, volume, or line
Consider an Area:
Example 3. Find the centroid.
Example 2. Find the coordinates of
the centroid of a quarter circle of
radius R.
Composite area - composed of simpler sub-areas
Example 4: Problem 5.1
SOLUTION:
• Divide the area into a triangle, rectangle,
and semicircle with a circular cutout.
• Calculate the first moments of each area
with respect to the axes.
For the plane area shown, determine
the first moments with respect to the
x and y axes and the location of the
centroid.
• Find the total area and first moments of
the triangle, rectangle, and semicircle.
Subtract the area and first moment of the
circular cutout.
• Compute the coordinates of the area
centroid by dividing the first moments by
the total area.
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Sample Problem 5.1
Sample Problem 5.1
• Compute the coordinates of the area
centroid by dividing the first moments by
the total area.
X =∑
x A + 757.7 × 10 3 mm 3
=
∑ A 13.828 × 103 mm 2
X = 54.8 mm
• Find the total area and first moments of the
triangle, rectangle, and semicircle. Subtract the
area and first moment of the circular cutout.
Q x = +506.2 × 10 mm
3
Y =∑
yA
∑A
3
=
+ 506.2 × 10 3 mm 3
13.828 × 10 3 mm 2
Q y = +757.7 × 10 3 mm 3
Y = 36.6 mm
Distributed Loads
Find the resultant of the distributed load.
Load Intensity: w = Force per unit length
Resultant: a) magnitude = area under w vs.x diagram
b) passes through the centroid of the area
Example 4. Find the
reactions at A and B.
Centroids of Volumes
x=
∫
V
x dV
∫ dV
V
;
y=
∫
V
y dV
∫ dV
V
; z=
∫
V
z dV
∫ dV
V
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Ex. 5. Find the coordinates
of the centroid of a right
circular cone.
Example 6. Find the coordinates of the centroid of the arc.
Centroid of a Line
x=
∫ x dL ;
∫ dL
L
L
y=
∫
L
y dL
∫ dL
; z=
L
Composite Volume
∫ z dL
∫ dL
L
L
Pappus-Guldinus Theorems
Second Theorem:
First Theorem:
•Given: line L,
with centroid
•Rotate L about the
x-axis Æ Surface S
•Find S
•Given: area A, with centroid
•Rotate A about the x-axis Æ Volume
•Find the volume V.
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Ex. 7. Use Pappus-Guldinus Theorems
to find the surface are and volume of a
cone.
Center of Mass
Convert Liquid Pressure Problem into Distributed Load Problem:
p = ρgh
w = ρgBh
-Both increase linearly with depth
-Both compressive and perpendicular to area
-B = width of the plate.
Ex. 8. Use P-G
Theorems to find
centroid of
a) semicircular line,
b) semicircular area
Liquid Pressure
Example: The fresh water channel is contained by a 4-m wide plate
hanging freely hinged at A. If the gate is designed to open when water
reaches 0.8 m, what must be the weight W of the gate.
Ans: W = 39, 250 N
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