Engineering Mechanics: Statics Chapter 9: Center of Gravity and Centroid Chapter 9: Center of...
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Transcript of Engineering Mechanics: Statics Chapter 9: Center of Gravity and Centroid Chapter 9: Center of...
Engineering Mechanics: StaticsEngineering Mechanics: Statics
Chapter 9: Center of Gravity and
Centroid
Chapter 9: Center of Gravity and
Centroid
Chapter ObjectivesChapter Objectives
To discuss the concept of the center of gravity, center of mass, and the centroid.
To show how to determine the location of the center of gravity and centroid for a system of discrete particles and a body of arbitrary shape.
To use the theorems of Pappus and Guldinus for finding the area and volume for a surface of revolution.
To present a method for finding the resultant of a general distributed loading and show how it applies to finding the resultant of a fluid.
Chapter OutlineChapter Outline
Center of Gravity and Center of Mass for a System of Particles
Center of Gravity and Center of Mass and Centroid for a Body
Composite BodiesTheorems of Pappus and GuldinusResultant of a General Distributed
LoadingFluid Pressure
9.1 Center of Gravity and Center of Mass for a System of Particles9.1 Center of Gravity and Center of Mass for a System of Particles
Center of Gravity Locates the resultant weight of a system of
particles Consider system of n particles fixed within
a region of space The weights of the particles
comprise a system of parallel forces which can be replaced by a single (equivalent) resultant weight having defined point G of application
9.1 Center of Gravity and Center of Mass for a System of Particles9.1 Center of Gravity and Center of Mass for a System of Particles
Center of Gravity Resultant weight = total weight of n
particles
Sum of moments of weights of all the particles about x, y, z axes = moment of resultant weight about these axes
Summing moments about the x axis,
Summing moments about y axis,nnR
nnR
R
WyWyWyWy
WxWxWxWx
WW
~...~~
~...~~
2211
2211
9.1 Center of Gravity and Center of Mass for a System of Particles9.1 Center of Gravity and Center of Mass for a System of Particles
Center of Gravity Although the weights do not
produce a moment about z axis, by rotating the coordinate system 90° about x or y axis with the particles fixed in it and summing moments about the x axis,
Generally, mmz
zmmy
ymmx
x
WzWzWzWz nnR
~,
~;
~
~...~~2211
9.1 Center of Gravity and Center of Mass for a System of Particles9.1 Center of Gravity and Center of Mass for a System of Particles
Center of Gravity Where represent the coordinates of the
center of gravity G of the system of particles, represent the coordinates of each particle in the
system and represent the resultant sum of the weights of all the particles in the system.
These equations represent a balance between the sum of the moments of the weights of each particle and the moment of resultant weight for the system.
zyx ,,
W
zyx ~,~,~
9.1 Center of Gravity and Center of Mass for a System of Particles9.1 Center of Gravity and Center of Mass for a System of Particles
Center Mass Provided acceleration due to gravity g for
every particle is constant, then W = mg
By comparison, the location of the center of gravity coincides with that of center of mass
Particles have weight only when under the influence of gravitational attraction, whereas center of mass is independent of gravity
mmz
zmmy
ymmx
x
~,
~;
~
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 A rigid body is composed of an infinite
number of particles 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 Body
9.2 Center of Gravity and Center of Mass and Centroid
for a BodyCenter Mass γ represents the specific weight and dW =
γdV
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 Body9.2 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 gravity
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
x
~
;
~
;
~
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
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 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 Body9.2 Center of Gravity and Center of Mass and Centroid for a Body
CentroidVolume Consider an object subdivided into volume
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 Body9.2 Center of Gravity and Center of Mass and Centroid for a Body
CentroidArea For centroid for surface area of an
object, 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 Body9.2 Center of Gravity and Center of Mass and Centroid for a Body
CentroidLine If the geometry of the object such as a thin
rod 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 Body9.2 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
Example Polar coordinates are appropriate for
area with circular boundaries
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
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 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
In cases where a shape has 2 or 3 axes of symmetry, the centroid lies at the intersection of these axes
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
00~ xdLx
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
Procedure for AnalysisDifferential Element Select an appropriate coordinate system,
specify 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 Body9.2 Center of Gravity and Center of Mass and Centroid for a Body
Procedure for AnalysisDifferential Element For volumes, the element dV is either a
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 Body9.2 Center of Gravity and Center of Mass and Centroid for a Body
Procedure for AnalysisSize and Moment Arms Express the length dL, area dA or
volume dV 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 Body9.2 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 integrations
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 Body9.2 Center of Gravity and Center of Mass and Centroid for a Body
Example 9.1Locate the centroid of the rod bent into the shape of a parabolic arc.
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
SolutionDifferential element 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 is located at
yyxx
dyydL
dydydx
dydxdL
~,~
12
1
2
222
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
SolutionIntegrations
m
dyy
dyyy
dL
dLy
y
m
dyy
dyyy
dyy
dyyx
dL
dLx
x
L
L
L
L
574.0479.18484.0
14
14~
410.0479.16063.0
14
14
14
14~
1
02
1
02
1
02
1
022
1
02
1
02
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
Example 9.2Locate the centroid of the circular wire segment.
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
SolutionDifferential element A differential circular arc is selected This element intersects the curve at (R,
θ)Length and Moment Arms For differential length of the element
For centroid,
sin~cos~ RyRx
RddL
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
SolutionIntegrations
R
dR
dR
dR
dRR
dL
dLy
y
R
dR
dR
dR
dRR
dL
dLx
x
L
L
L
L
2
sinsin~
2
coscos~
2/
0
2/
02
2/
0
2/
0
2/
0
2/
02
2/
0
2/
0
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
Example 9.3Determine the distance from the x axis to the centroid of the area of the triangle
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
SolutionDifferential element Consider a rectangular element having
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 Body9.2 Center of Gravity and Center of Mass and Centroid for a Body
SolutionIntegrations
32161
~
2
0
0
h
bh
bh
dyyhhb
dyyhhby
dA
dAy
yh
h
A
A
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
Example 9.4Locate the centroid
for the area of a quarter circle.
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
SolutionMethod 1Differential element 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
sin32~cos
32~
2)cos)((
21 2
RyRx
dR
RRdA
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
SolutionIntegrations
34
sin32
2
2sin
32~
34
cos32
2
2cos
32~
2/
0
2/
0
2/
0
2
2/
0
2
2/
0
2/
0
2/
0
2
2/
0
2
R
d
dR
dR
dR
R
dA
dAy
y
R
d
dR
dR
dR
R
dA
dAx
x
A
A
A
A
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
SolutionMethod 2Differential 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 Body9.2 Center of Gravity and Center of Mass and Centroid for a Body
SolutionArea and Moment Arms For area of the element
Centroid is located y distance from the x axis
/2~/2~
4/2
ryrx
drrdA
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
SolutionIntegrations
34
242
422~
34
242
422~
0
02
0
0
0
02
0
0
R
drr
drr
drr
drrr
dA
dAy
y
R
drr
drr
drr
drrr
dA
dAx
x
R
R
R
R
A
A
R
R
R
R
A
A
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
Example 9.5Locate the centroid of the area.
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
SolutionMethod 1Differential element Differential element of thickness dx Element intersects curve at point (x, y),
height yArea and Moment Arms For area of the element
For centroid2/~~ yyxx
ydxdA
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
SolutionIntegrations
m
dxx
dxxx
dxy
dxyy
dA
dAy
y
m
dxx
dxx
dxy
dxxy
dA
dAx
x
A
A
A
A
3.0333.0100.0
)2/()2/(~
75.0333.0250.0
~
1
02
1
022
1
0
1
0
1
02
1
03
1
0
1
0
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
SolutionMethod 2Differential 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 Body9.2 Center of Gravity and Center of Mass and Centroid for a Body
SolutionArea 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 Body9.2 Center of Gravity and Center of Mass and Centroid for a Body
SolutionIntegrations
m
dyy
dyyy
dxx
dxxy
dA
dAy
y
m
dyy
dyy
dxx
dxxx
dA
dAx
x
A
A
A
A
3.0333.0100.0
11
1~
75.0333.0250.0
1
121
1
12/1~
1
0
1
02/3
1
0
1
0
1
0
1
01
0
1
0
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
Example 9.6Locate the centroid of 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 Body9.2 Center of Gravity and Center of Mass and Centroid for a Body
SolutionMethod 1Differential element Differential element of thickness dx Intersects curve at point (x1, y1) and (x2, y2),
height yArea and Moment Arms For area of the element
For centroidxx
dxyydA
~
)( 12
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
SolutionIntegrations
m
dxxx
dxxxx
dxyy
dxyyx
dA
dAx
x
A
A
5.0
61
121
~
1
02
1
02
1
0 12
1
0 12
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
SolutionMethod 2Differential 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 Body9.2 Center of Gravity and Center of Mass and Centroid for a Body
SolutionArea 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 Body9.2 Center of Gravity and Center of Mass and Centroid for a Body
SolutionIntegrations
m
dxyy
dyyy
dxyy
dyyyyy
dxxx
dxxxxx
dA
dAx
x
A
A
5.0
61
121
21
2/2/~
1
0
1
0
2
1
0
1
01
0 21
1
0 2121
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
Example 9.7Locate the centroid for the paraboloid of revolution, 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 Body9.2 Center of Gravity and Center of Mass and Centroid for a Body
SolutionMethod 1Differential element Element in the shape of a thin disk, thickness
dy, radius 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 2
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
Solution For centroid
Integrations
mm
dyy
dyy
dyz
dyzy
dV
dVy
y
yy
V
V
7.66
100
100~
~
100
0
100
0
2
100
0
2
100
0
2
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
SolutionMethod 2Differential element 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 Body9.2 Center of Gravity and Center of Mass and Centroid for a Body
SolutionArea and Moment Arms For area of the element
Centroid is located y distance from the x axis
2/1002/100~
10022
yyyy
dzyzrdA
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
SolutionIntegrations
m
dzzz
dzzz
dzyz
dzyzy
dV
dVy
y
V
V
7.66
101002
1010
1002
10022/100~
100
0
22
100
0
444
100
0
100
0
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
Example 9.8Determine the location of 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 Body9.2 Center of Gravity and Center of Mass and Centroid for a Body
SolutionFor reasons of material symmetry
Differential element 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
yx
25.0
0
View Free Body Diagram
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
Solution For centroid
Integrations
mzdz
dzz
dzz
dzzz
dV
dVz
z
zz
V
V
667.0
5.0)200(
5.0)200(~
~
~
1
0
1
02
1
0
2
1
0
2
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
Solution Not possible to use a
shell element for integration 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