Centroids
Transparency 1
5/4: Centroids of composite bodies
page 246
Uniform density
More dense towards right
Centroid
Center of
gravity
Center of gravity and centroid
Center of gravity: Only relevant when gravitational forces are involved. Suspend object from different points - the lines of action of the gravitational
forces intersect at the center of gravity
Center of mass: Coincides with center of gravity
in uniform, parallel gravitational field (on earth)
Centroid: Geometric center of an object. Coincides with center of mass when density is uniform
Centroids
Transparency 2
z
G1
G
W1
y1
x1
X
W
Y
yN
GN
y
WN
Z
xN
x
Take moments about y-axis:
W1x1 + W2x2 + . . . + WN xN = W X
m1gx1 + m2gx2 + . . . + mN gxN = M gX
m1x1+m2x2+. . .+mN xN = (m1+m2+. . .+mN )X
m1x1 + m2x2 + . . . + mN xN
X=
m1 + m2 + . . . + mN
Centroids
Transparency 3
Take moments about x-axis:
m1y1 + m2y2 + . . . + mN yN
Y =
m1 + m2 + . . . + mN
Rotate so that z-axis is horizontal, take moments:
m1z1 + m2z2 + . . . + mN zN
Z=
m1 + m2 + . . . + mN
Shorter notation:
Σmx
X=
Σm
Σmy
Y =
Σm
Σmz
Z=
Σm
m
Volumes: ρ =
V
When density is uniform throughout body:
ρV1x1 + ρV2x2 + . . . + ρVN xN
X =
ρV1 + ρV2 + . . . + ρVN
V1x1 + V2x2 + . . . + VN xN
=
V1 + V2 + . . . + VN
Shorter notation:
ΣV x
X=
ΣV
ΣV y
Y =
ΣV
ΣV z
Z=
ΣV
Centroids
Transparency 4
Areas
When thickness is uniform throughout body:
V1x1 + V2x2 + . . . + VN xN
X =
V1 + V2 + . . . + VN
dA1x1 + dA2x2 + . . . + dAN xN
=
dA1 + dA2 + . . . + dAN
A1x1 + A2x2 + . . . + AN xN
=
A1 + A2 + . . . + AN
Shorter notation:
ΣAx
X=
ΣA
ΣAy
Y =
ΣA
ΣAz
Z=
ΣA
Thin rods
When cross-sectional area is uniform:
V1x1 + V2x2 + . . . + VN xN
X =
V1 + V2 + . . . + VN
AL1x1 + AL2x2 + . . . + ALN xN
=
AL1 + AL2 + . . . + ALN
L1x1 + L2x2 + . . . + LN xN
=
L1 + L2 + . . . + LN
Centroids
Shorter notation:
ΣLx
X=
ΣL
Transparency 5
ΣLy
Y =
ΣL
ΣLz
Z=
ΣL
PROBLEMS: Centroids of areas
PR5/43: Determine the coordinates of the centroid of the trapezoidal area shown.
Centroids
Transparency 6
PR5/44: Calculate the y-coordinate of the centroid of the shaded area.
Centroids
Transparency 7
PR5/47: Calculate the coordinates of the centroid
of the shaded area.
Centroids
Transparency 8
PR5/48: Calculate the x- and y-coordinates of
the centroid of the shaded area.
Centroids
Transparency 9
Centroids from areas
PR5/63: Calculate the coordinates of the center
of mass of the bracket, which is constructed from
sheet metal of uniform thickness.
Answer:
X = 62.1 mm;
Y = 67.7 mm;
Z = −22.0 mm
Homework: SP5/6
Centroids
Transparency 10
Centroids from volumes
PR5/55: Consider the homogeneous hemisphere
of which the smaller hemispherical portion is removed. Determine the x-coordinate of the mass
center.
45
Answer: X =
R
112
Centroids
Transparency 11
Centroids from units of mass
PR5/53: The rigidly connected unit consists of a
2-kg circular disk, a 1.5-kg round shaft, and a 1kg square plate. Determine the z-coordinate of the
mass center of the unit.
Answer: Z = 70 mm
Homework: SP5/8
Centroids
Transparency 12
Centroids from lengths
PR5/52: Locate the mass center of the slender
rod bent into the shape shown.
Answer: X = 0 mm; Y = −58.3 mm
Centroids
Transparency 13
Different densities: use mass!
SP5/8: Locate the center of mass of the bracketand-shaft combination. The vertical face is made
from sheet metal which has a mass of 25 kg/m2.
The material of the horizontal base has a mass
of 40 kg/m2, and the steel shaft has a density of
7.83 Mg/m3.
Answer: X = 53.3 mm; Y = −45.7 mm
Centroids
Transparency 14
Centroids of general bodies
(page 227)
Composite body: Constituent parts are geometrical figures of which the centroids are known
General body: Boundaries of body are defined by
mathematical equations
y
Approximation
y = f ( x)
x= A
123
y = g( x)
•••
x= B
N
x
A1 x1 + A2 x2 + · · · AN xN
x≈
A1 + A2 + · · · AN
A1 y 1 + A2 y 2 + · · · AN y N
y ≈
A1 + A2 + · · · AN
Centroids
Transparency 15
Accurate
y = f ( x)
x= A
yc
x= B
dx
x
y = g( x)
xc = x
yc = (f (x) + g(x))/2
dA = (f (x) − g(x)) dx
Z
A = dA
x=
y =
Z
Z
xc dA
A
yc dA
A
Centroids
Transparency 16
Alternative
Accurate
y= B
x = g( y)
dy
yc = y
y
x = f ( y)
y= A
xc
xc = (f (y) + g(y))/2
dA = (f (y) − g(y)) dy
Z
A = dA
x=
y =
Z
Z
xc dA
A
yc dA
A
Centroids
Transparency 17
SP5/4: Locate the centroid of the area under the
curve x = ky 3 from x = 0 to x = a.
Answer:
4
x = a;
7
2
y= b
5
Centroids
Transparency 18
PR5/10: Determine the coordinates of the centroid of the shaded area in the figure (below left).
(First use dx and then verify with dy)
3a
;
y
=
Answer: x = 3b
5
8
PR5/11: Determine the coordinates of the centroid of the shaded area in the figure (above right).
(Use dx. Why?)
Answer: x = 1.443; y = 0.361 k
Centroids
Transparency 19
PR5/10: Determine the coordinates of the centroid of the shaded area.
(First use dx and then verify with dy)
3b
3a
Answer: x = ; y =
5
8
Centroids
Transparency 20
PR5/11: Determine the coordinates of the centroid of the shaded area.
(Use dx. Why?)
Answer: x = 1.443; y = 0.361 k
Centroids
Transparency 21
PR5/13: Determine the coordinates of the centroid of the shaded area. (Better to use dy − Why?)
2a
3b
Answer: x = ; y =
5
8
Centroids
Transparency 22
PR5/18 and PR5/24: Areas between curves
PR5/18: Determine the y-coordinate of the centroid of the shaded area.
11b
(Better to use dx − Why?)
Answer: y =
10
Centroids
Transparency 23
PR5/18 en PR5/24: Areas between curves
PR5/24: Determine the coordinates of the centroid of the shaded area. (Can use either dx or dy)
2a
b
Answer: x = ; y =
5
2