Centroids. Centroid Principles Objects center of gravity or center of mass Graphically labeled as.
6161103 9.2 center of gravity and center of mass and centroid for a body
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Transcript of 6161103 9.2 center of gravity and center of mass and centroid for a body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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 � Consider arbitrary particle
having a weight of dW
∫∫
∫∫
∫∫ ===
dW
dWzz
dW
dWyy
dW
dWxx
~;
~;
~
9.2 Center of Gravity and Center of Mass and Centroid for a Body9.2 Center of Gravity and Center of Mass and Centroid for a Body
Center Mass
� γ represents the specific weight and dW = γdV
∫∫∫ dVzdVydVx γγγ ~~~
∫
∫
∫
∫
∫
∫===
V
V
V
V
V
V
dV
dVz
zdV
dVy
ydV
dVx
xγ
γ
γ
γ
γ
γ ~
;
~
;
~
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Center of Mass
� Density ρ, or mass per unit volume, is related to γ by γ = ρg, where g = acceleration due to gravitygravity
� Substitute this relationship into this equation to determine the body’s center of mass
∫
∫
∫
∫
∫
∫===
V
V
V
V
V
V
dV
dVz
zdV
dVy
ydV
dVx
xγ
γ
γ
γ
γ
γ ~
;
~
;
~
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Centroid� 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 masscenter 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
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Centroid
Volume
� Consider an object subdivided into volume elements dV, for location of the centroid, elements dV, for location of the centroid,
∫
∫
∫
∫
∫
∫===
V
V
V
V
V
V
dV
dVz
zdV
dVy
ydV
dVx
x
~
;
~
;
~
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Centroid
Area
� For centroid for surface area of an object, such as plate and shell, subdivide the area such as plate and shell, subdivide the area into differential elements dA
∫
∫
∫
∫
∫
∫===
A
A
A
A
A
A
dA
dAz
zdA
dAy
ydA
dAx
x
~
;
~
;
~
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Centroid
Line
� If the geometry of the object such as a thin rod or wire, takes the form of a line, the balance of or wire, takes the form of a line, the balance of moments of differential elements dL about each of the coordinate system yields
∫
∫
∫
∫
∫
∫===
L
L
L
L
L
L
dL
dLz
zdL
dLy
ydL
dLx
x
~
;
~
;
~
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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
ExampleExample� Polar coordinates are appropriate for area
with circular boundaries
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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 � 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
Symmetry
� For total moment of all the elements about the axis of symmetry will therefore be cancelled
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
00~ ==∫ xdLx
� In cases where a shape has 2 or 3 axes of symmetry, the centroid lies at the intersection of these axes
00 ==∫ xdLx
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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 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
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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 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
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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 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
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Procedure for AnalysisIntegrations� Substitute the formations and dL, dA and dV into
the appropriate equations and perform integrationsintegrations
� 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
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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.shape of a parabolic arc.
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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 ArmsArea and Moment Arms
� For differential length of the element dL
� Since x = y2 and then dx/dy = 2y
� The centroid is located at
( ) ( )
( )
yyxx
dyydL
dydy
dxdydxdL
==
+=
+
=+=
~,~
12
1
2
222
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Integrations
dyyydyyxdLx
x L1414
~
1
1
022
1
1
02 +
=+
==∫∫
∫
∫
m
dyy
dyyy
dL
dLy
y
m
dyydyydLx
L
L
L
574.0479.18484.0
14
14~
410.0479.16063.0
1414
1
02
1
02
1
021
02
==
+
+==
==
+=
+==
∫∫
∫
∫
∫∫∫
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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.segment.
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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)� This element intersects the curve at (R, θ)
Length and Moment Arms
� For differential length of the element
� For centroid,
θθ
θ
sin~cos~ RyRx
RddL
==
=
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Integrations
( ) θθθθππ
dRdRRdLx
x Lcoscos
~ 2/
022/
0 ===∫∫∫
( )
π
θ
θθ
θ
θθ
π
θθ
π
π
π
π
ππ
R
dR
dR
dR
dRR
dL
dLy
y
R
dRdRdLx
L
L
L
L
2
sinsin~
2
2/
0
2/
02
2/
0
2/
0
2/
0
02/
0
0
=
===
=
===
∫∫
∫∫
∫
∫
∫∫
∫∫
∫
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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 centroid of the area
of the triangle
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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 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
( )
yy
dyyhhb
xdydA
=
−==
~
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Integrations
( )~0
dyyhh
bydAy
y
h
A−
==∫∫
( )
32161 2
0
0
h
bh
bh
dyyhh
bh
dAy
h
A
A
==
−==
∫
∫∫
∫
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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.circle.
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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� 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
θθ
θθ
sin32~cos
32~
2)cos)((
21 2
RyRx
dR
RRdA
==
==
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Integrations
θθθθ
π
ππcos
32
2cos
32~
2/
2/
02
2/
0
2dRd
RRdAx
x A
=
==∫∫
∫
∫
π
θ
θθ
θ
θθ
π
θθ
π
π
π
π
ππ
34
sin32
2
2sin
32~
34
2
2/
0
2/
0
2/
0
2
2/
0
2
2/
02/
0
2
R
d
dR
dR
dR
R
dA
dAy
y
R
ddRdA
x
A
A
A
=
=
==
=
===
∫
∫
∫
∫∫
∫
∫∫∫
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Method 2
Differential element
� Circular arc element � Circular arc element having thickness of dr
� Element intersects the
axes at point (r,0) and
(r, π/2)
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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
( )
ππ
π
/2~/2~
4/2
ryrx
drrdA
==
=
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Integrations
ππ
ππ
24
22~
02
0 drr
r
drrr
dA
dAx
xR
R
R
R
A =
==
∫∫
∫
∫∫
∫
π
ππ
ππ
π
ππ
34
242
422~
34
242
0
02
0
0
00
R
drr
drr
drr
drrr
dA
dAy
y
R
drrdrrdA
x
R
R
R
R
A
A
RR
A
=
=
==
=
===
∫∫
∫
∫∫
∫
∫∫∫
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Example 9.5
Locate the centroid of
the area.
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Method 1
Differential element
� Differential element of thickness dx� 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
2/~~ yyxx
ydxdA
==
=
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Integrations
dxxdxxydAx
x A
~ 1
031
0 ===∫∫∫
m
dxx
dxxx
dxy
dxyy
dA
dAy
y
m
dxxdxydAx
A
A
A
A
3.0333.0100.0
)2/()2/(~
75.0333.0250.0
1
02
1
022
1
0
1
0
1
02
01
0
0
==
===
==
===
∫∫
∫∫
∫
∫
∫∫
∫∫
∫
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Method 2
Differential element
Differential element � Differential element of thickness dy
� Element intersects curve at point (x, y)
� Length = (1 – x)
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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
( )
yy
xxxx
dyxdA
=
+=
−+=
−=
~2
12
1~
1
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Integrations
( )[ ]( )
( )
( )
( )
dyydxxx
dA
dAx
x A1
21
12/1~
1
1
01
1
0−
=−−
==
∫
∫
∫∫
∫
∫
( ) ( )
( )
( )
( )( )
m
dyy
dyyy
dxx
dxxy
dA
dAy
y
m
dyydxxdAx
A
A
A
3.0333.0100.0
11
1~
75.0333.0250.0
11
1
0
1
02/3
1
0
1
0
1
0
1
0
==
−
−=
−
−==
==
−=
−==
∫∫
∫∫
∫
∫
∫∫∫
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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 the shaded are bounded
by the two curves
y = x
and y = x2.
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Method 1
Differential element
� Differential element of thickness dx� 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
xx
dxyydA
=
−=
~
)( 12
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Integrations
( ) ( )dxxxxdxyyxdAx
x A
~ 1
021
0 12 −=
−==
∫∫∫ ( )
( )
( )( )
m
dxxx
dxxxx
dxyy
dxyyx
dAx
A
A
5.0
61121
1
02
01
0 12
0 12
==
−
−=
−
−==
∫∫
∫∫
∫
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Method 2
Differential element
� Differential element � Differential element of thickness dy
� Element intersects curve at point (x1, y1) and (x2, y2)
� Length = (x1 – x2)
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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
( )
22~ 2121
2
21
xxxxxx
dyxxdA
+=
−
+=
−=
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Integrations
( )[ ]( ) ( )[ ]( )dyyyyydxxxxxdAx 2/2/~ 11
−+−+ ∫∫∫ ( )[ ]( )
( )
( )[ ]( )( )
( )( )
mdxyy
dyyy
dxyy
dyyyyy
dxxx
dxxxxx
dA
dAx
x
A
A
5.0
61
121
21
2/2/
1
0
1
0
2
1
0
01
0 21
0 2121
==−
−=
−
−+=
−
−+==
∫
∫
∫∫
∫∫
∫
∫
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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 which is generated by
revolving the shaded area
about the y axis.
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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 zradius 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
( )dyzdV 2π=
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
� For centroid
Integrationsyy~ =
Integrations
( )( )
mm
dyy
dyy
dyz
dyzy
dV
dVy
y
V
V
7.66
100
100~
100
0
100
0
2
100
0
2
100
0
2
=
===
∫∫
∫∫
∫
∫
π
π
π
π
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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 � 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
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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
( )
( ) ( ) 2/1002/100~
10022
yyyy
dzyzrdA
+=−+=
−= ππ
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
Integrations
( )[ ] ( )
( )
dzyzydVy
y V10022/100
~ 100
0−+
==∫
∫
∫ π
( )
( )( )
m
dzzz
dzzz
dzyzdVy
V
7.66
101002
1010
1002
100
0
22
100
0
444
100
0
=
−
−=
−==
∫∫
∫∫
−
−
π
π
π
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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 the center of mass of the
cylinder if its density
varies directly with its
distance from the base
ρ = 200z kg/m3.
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
For reasons of material symmetry
Differential element
yx 0==
� 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
( ) dzdV 25.0π=
9.2 Center of Gravity and Center of Mass and Centroid for a Body
9.2 Center of Gravity and Center of Mass and Centroid for a Body
Solution
� For centroid
Integrations
zz~ =
Integrations
( )
( )
mzdz
dzz
dzz
dzzz
dV
dVz
z
V
V
667.0
5.0)200(
5.0)200(~
~
1
0
1
02
1
0
2
1
0
2
==
==
∫∫
∫∫
∫
∫
π
π
ρ
ρ
9.2 Center of Gravity and Center of Mass and Centroid for a Body
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 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