ENGR-36_Lec-23_Center_of_Gravity
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Transcript ENGR-36_Lec-23_Center_of_Gravity
Engineering 36
Chp09: Center
of Gravity
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
[email protected]
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
[email protected] • ENGR-36_Lec-232_Center_of_Gravity.pptx
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
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
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Total Mass – General Case
dVn at xn , yn , zn
Given a Massive
Body in 3D Space
Divide the Body in to
Very Small Volumes
→ dV
Each dVn is located at
position (xn,yn,zn)
r xniˆ yn ˆj zn kˆ
The DENSITY, ρ, can
be a function of
POSITION → n xn , yn , zn
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Bruce Mayer, PE
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Total Mass – General Case
Now since m = ρ•V, then the
incremental mass, dm
dVn at xn , yn , zn
dmn xn , yn , zn dVn
r xniˆ yn ˆj zn kˆ
Integrate dm over the entire
body to obtain the total
Mass, M
M
dm
body
n
x , y , z dV
n
n
n
volume
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n
Bruce Mayer, PE
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Total WEIGHT – General Case
Recall that W = Mg
dVn at xn , yn , zn
Using the the previous
expression for M
W Mg
gdm
n
body
g x , y , z dV
n
n
n
r xniˆ yn ˆj zn kˆ
n
volume
x , y , z g dV
n
n
n
n
volume
x , y , z g dV
n
n
n
n
volume
Now Define SPECIFIC
WEIGHT, γ, as ρ•g →
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W
x , y , z g dV
n
n
n
n
volume
x , y , z dV
n
n
n
volume
Bruce Mayer, PE
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n
Uniform Density Case
Consider a body with
UNIFORM DENSITY; i.e.
dVn at xn , yn , zn
xn , yn , zn xm , ym , zn
xn , yn , zn xm , ym , zn
r xniˆ yn ˆj zn kˆ
Then M & W→
M
dV
dV
n
volume
W
n
volume
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dV
dV
n
V
n
V
volume
volume
Bruce Mayer, PE
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Center of Mass Location
Use MOMENTS to Locate
Center of Mass/Gravity
dVn at xn , yn , zn
Recall Defintion of a
MOMENT
r xniˆ yn ˆj zn kˆ
Moment leverArm Intensity
In the General Center-of-Mass Case
• LeverArm ≡ Position Vector, rn, or its components
• Intensity ≡ Incremental Mass, dmn
RM LeverArms Intensities
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Intensities
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Center of Mass Location
RM in Component form
dVn at xn , yn , zn
RM X M iˆ YM ˆj ZM kˆ
r xniˆ yn ˆj zn kˆ
Now Define the Incremental
Moment, dΩn
ˆ
ˆ
ˆ
dΩn xni yn j zn k gdVn
LeverArm
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Intensity
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Center of Mass Location
Integrating dΩ to Find Ω for
the entire body
Ω
dΩ
n
dVn at xn , yn , zn
r xniˆ yn ˆj zn kˆ
body
x iˆ y ˆj z kˆgdV
n
n
n
n
volume
Ω xiˆ y ˆj x kˆ
giˆ
ˆj y dV gkˆ z dV
x
dV
g
n n
n n
n n
all x
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all y
all z
Bruce Mayer, PE
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Center of Mass Location
Now Equate Ω to RM•Mg
dVn at xn , yn , zn
R M Mg Mg X M iˆ YM ˆj Z M kˆ
giˆ xn dVn g ˆj yn dVn gkˆ zn dVn
all x
all y
r xniˆ yn ˆj zn kˆ
all z
Canceling g, and equating
Components yields, for example, in the X-Dir
XMM
x dV
n
n
all x
LeverArm
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Intensity
Bruce Mayer, PE
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Center of Mass Location
Divide out the Total Intensity,
M, to Isolate the Overall
X-directed Lever Arm, XM
XM
dVn at xn , yn , zn
xn dVn
r xniˆ yn ˆj zn kˆ
all x
M
And the Similar expressions
for the other CoOrd Directions
yn dVn
zn dVn
YM
all y
M
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ZM
all x
M
Bruce Mayer, PE
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Center of Gravity of a 2D Body
Centroid of an Area
Taking Incremental Plate
Areas, Forming the Ωx &
Ωy, Along With the Fz=W
Yields the Expression for the
Equivalent POINT of W
application
• Note Ω Units = In-lb or N-m
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Centroid of a Line
x
x W x W
x dW
y
y W y W
y dW
Bruce Mayer, PE
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Centroids of Areas & Lines
Centroid of an Area
x W x dW
x At x t dA
x A x dA x
For Plate of Uniform
Thickness
•
•
•
Specific Weight
t Plate Thickness
dW = tdA
1st moment w.r.t. y axis
yA y dA y
1st moment w.r.t. to x axis
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Centroid of a Line
x W x dW
x La x a dL
Wire of
Uniform
Thickness
x L x dL
yL y dL
Bruce Mayer, PE
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•
Specific
Weight
•
a X-Sec
Area
•
dW = a(dL)
First Moments of Areas & Lines
• 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’.
• 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
• 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).
• The centroid of the area coincides
with the center of symmetry, O.
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Bruce Mayer, PE
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Centroids of Common Area Shapes
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Centroids of Common Line Shapes
Recall that for a SMALL Angle, α
sin
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xr
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Composite Plates and Areas
• Composite plates
X W xk Wk
Y W yk W k
X W x1W1 x 2W2 x3W3
• Composite area
X A xk Ak
Y A yk Ak
Y A y1 A1 y 2 A2 y 3 A3
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Bruce Mayer, PE
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Example: Composite Plate
Solution Plan
For the plane area
shown, determine the
first moments with
respect to the x and y
axes, and the location
of the centroid.
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• 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
Bruce Mayer, PE
[email protected] • ENGR-36_Lec-232_Center_of_Gravity.pptx
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
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Example: Composite Plate
Find the total area and first moments of the
x
triangle, rectangle, and semicircle. Subtract
the area and first moment of the circular cutout y
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757.7 103 mm3
506.2 103 mm3
Bruce Mayer, PE
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Example: Composite Plate
Solution
3
3
x
A
757
.
7
10
mm
X
A 13.828103 mm2
X 54.8 mm
Find the coordinates
of the area centroid
by dividing the
first moment totals
by the total area
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3
3
y
A
506
.
2
10
mm
Y
A 13.828103 mm2
Y 36.6 mm
Bruce Mayer, PE
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Centroids by Strip Integration
xA xdA x dxdy xel dA
yA ydA y dxdy yel dA
x A xel dA
x ydx
yA yel dA
y
ydx
2
x A xel dA
x A xel dA
ax
a x dy
2
yA yel dA
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• Double integration to find the first
moment may be avoided by defining
dA as a thin rectangle or strip.
y a x dy
2r
1 2
cos r d
3
2
yA yel dA
2r
1
sin r 2 d
3
2
Bruce Mayer, PE
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Example: Centroid by Integration
Solution Plan
Determine by direct
integration the
location of the
centroid of a
parabolic spandrel.
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• 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.
Bruce Mayer, PE
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Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
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Example: Centroid by Integration
Solution
• Determine Constant k
y k x 2 ; and y b when x a
b
2
bka k 2
a
b
a
y 2 x 2 or x 1 2 y1 2
a
b
• Calculate the Total Area
A dA Use VerticalStrips
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a
b x3
b 2
y dx 2 x dx 2
a
a 3 0
0
ab
3
Bruce Mayer, PE
a
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Example: Centroid by Integration
Solution
• Calc the 1st Moments, Ωi
b
x xel dA x y dx x 2 x 2 dx
a
0
a
a
b x
b 3
a 2b
2 x dx 2
a
4
a 4 0
0
a
4
Using vertical strips,
2
a
y
1
b
perform a single
y yel dA y dx 2 x 2 dx
2
2a
integration to find
0
a
2
5
the first moments.
b x
ab2
4
2a 5 0 10
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Bruce Mayer, PE
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Example: Centroid by Integration
Finally the Answers
xA x
2
ab a b
x
3
4
Evaluate the
centroid coordinates
• Divide Ωx and Ωy
by the total Area
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yA y
2
ab ab
y
3 10
3
x a
4
3
y b
10
Bruce Mayer, PE
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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.
A 2ydL 2 ydL
A 2 yL as
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yL y ydL
Bruce Mayer, PE
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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.
V dV 2ydA
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V 2 ydA 2 yA
Bruce Mayer, PE
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Example: Pappus-Guldinus
Solution Plan
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.
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• Apply the theorem of PappusGuldinus 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.
Bruce Mayer, PE
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Example: P-G
Apply Pappus-Guldinus
to Sections I & II
Subtract: I-II
3
9 3
m V 7.85 10 kg m 7.65 10 mm 10 m mm
W 589 N
W mg 60.0 kg 9.81 m s 2
3
3
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6
3
m 60.0 kg
Bruce Mayer, PE
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WhiteBoard Work
Find the
Areal &
Lineal
Centroids
Find for Area
XA & YA
Find for OutSide Line
XL & YL
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Bruce Mayer, PE
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Engineering 36
Appendix
dy
µx µs
sinh
dx
T0 T0
Bruce Mayer, PE
Registered Electrical & Mechanical Engineer
[email protected]
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Bruce Mayer, PE
[email protected] • ENGR-36_Lec-232_Center_of_Gravity.pptx