The center of gravity (CG)
is a point, often shown as G,
which locates the resultant
weight of a system of
particles or a solid body.
To be balanced, the woman’s
center of mass – the point at
which her weight effectively
acts – must be directly above
her hands.
Weight
G
Normal
Force
“STEPS TO DETERMINE THE
CENTROID OF AN AREA”
1. Choose an appropriate differential element dA.
Hint: Generally, if y is easily expressed in terms of x, use
a vertical rectangular element. If the converse is true,
then use a horizontal rectangular element.
2. Express dA in terms of the differentiating element dx/dy
3. Determine coordinates ( x , y) of the centroid of the
rectangular element in terms of the general point (x,y).
4. Express all the variables and integral limits in the
formula using either x or y depending on whether the
differential element is in terms of dx or dy, respectively,
and integrate.
Example 1:
Determine the coordinates of the centroid of
the figure shown below.
y-axis
1m
y = x3
1m
x-axis
Example 2:
Determine the coordinates of the centroid of the
quarter circle shown in the figure whose radius
is r.
y-axis
x2 + y2 = r2
x-axis
Example 3:
Locate the coordinates
of centroid in the
figure shown with
respect to the given
axes.
y-axis
0.5 in
2 in
y = 1/x
0.5 in
2 in
x-axis
Composite Area
 an area consisting of a combination of simple areas
The centroid of a composite
area can be determined
without integration if the
centroids of its parts are
known. The area in the
figure consists of a triangle, a
rectangle & a semicircle,
which we call parts 1, 2 & 3.
“STEPS TO DETERMINE THE CENTROID
OF A COMPOSITE AREA”
1. Choose the parts — try to divide the composite
area into parts whose centroids you know or can
easily determine.
2. Determine the values for the parts — determine
the centroid & the area of each part. Watch for
instances of symmetry that can simplify your task.
3. Calculate the centroid of the composite area using
AT x= A1x1 + A2x2 + Anxn
Example 4:
The dimensions of the Tsection of a cast-iron beam
are shown in the figure
below. How far is the
centroid of the area above
the base?
1’’
8’’
1’’
6’’
Example 5:
Determine the coordinates
of the centroid of the area
shown in the figure with
respect to the given axes.
y-axis
3’’
9’’
x-axis
Example 6:
Locate the coordinates of the centroid of the
shaded area shown in the figure with respect to
the given axes.
y-axis
12’’
6’’
x-axis
6’’
6’’
Example 7:
The centroid of the shaded area in the figure is
required to lie on the y-axis. Determine the
distance “b” that will fulfill this requirement.
y-axis
6’’
4’’
8’’
b
x-axis
Example 8:
Refer to the T-section
shown in the figure. To
what value should the “b”
width of the flange be so
that the centroid of the
area is 2.5 in. above the
base?
1’’
8’’
1’’
b
Example 9:
Locate the centroid of the shaded area in the
figure created by cutting a semicircle of diameter
r from a quarter circle of radius r.
y-axis
4
4
x-axis
Example 10:
Locate the centroid of the shaded area in the
figure shown with respect to the given axes.
y-axis
8
4.5’’
x-axis
Example 11:
A right triangle of sides b & h is related about an
axis coinciding with side h to generate a right
circular cone. Derive the volume generated.
y-axis
h
b
x-axis
Example 12:
A circle of radius r lies in the
XY plane with its center at a
distance a above the x-axis.
Revolving it about the x-axis
will generate a doughnutshape ring called a torus,
provided a is greater than r.
Compute the surface area &
volume of the torus.
r
a
r
Example 13:
Two blocks of different materials are assembled
as shown. The densities of the materials are:
y
3
ρA = 150 lb/ft and
ρB = 400 lb/ft3.
Find the center of
gravity of this
assembly.
x
z
Example 14:
Locate the center of gravity of the homogeneous
y
block assembly
having a hemispherical hole.
z
x