Statics

1. ENGINEERING STATICS
Session 2020
Chapter
5 Distributed Forces:
Center of Mass and Centroid

2. Weight of the body
14"
𝟑𝟎°
𝑁1
𝑁2
W
• Where does the
weight of the body
act?
• Is there any
procedure to find its
position?
𝑥
𝑦
𝑁3
2"

3. Center of Gravity (Experimental)
It is the point on a body where the whole weight is assumed to be concentrated.
A 3-dimensional body having
a mass 𝒎 and suspended
from a support to be in
Equilibrium
𝒘
𝐺
𝑑𝑚 = Mass of subatomic
particle of element
𝑔 = Acceleration due to
gravity in a uniform
Gravitational field of
earth
Earth
A body under earth
gravitational field

4. Center of Gravity (Mathematical)
Principle of Moment
The moment of the resultant gravitational force 𝑾 about any axis equals the sum of
moments about the same axis of the gravitational forces 𝒅𝑾 acting on all particles treated
as infinitesimal elements of the body.
The resultant of gravitational forces (Weight) is given by
𝑊 = 𝑑𝑊
Applying the moment principle about an axis (y-axis) gives
𝑥 𝑊 = 𝑥 𝑑𝑊
Moment of sum Sum of moments
𝑥 =
𝑥 𝑑𝑊
𝑊
𝑥 =
𝑥 𝑑𝑊
𝑊
𝑦 =
𝑦 𝑑𝑊
𝑊
𝑧 =
𝑧 𝑑𝑊
𝑊
Coordinates of
Center of Gravity
Putting the value 𝑊 = 𝑚𝑔 and taking its derivative as
𝑑𝑊 = 𝑔 𝑑𝑚
𝑥 =
𝑥 𝑑𝑚
𝑚
𝑦 =
𝑦 𝑑𝑚
𝑚
𝑧 =
𝑧 𝑑𝑚
𝑚
Coordinates of
Center of Mass
For vector representation with the Position vectors
𝑟 = 𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘 and 𝑟 = 𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘
𝑟 =
𝑟 𝑑𝑚
𝑚

5. Centroids
The center of mass relationship for a three-dimensional body is given as
The density of a body is its mass per unit volume 𝜌 =
𝑚
𝑉
, thus,
the mass of a differential element 𝑑𝑚 = 𝜌𝑑𝑉
𝑥 =
𝑥 𝜌 𝑑𝑉
𝜌 𝑑𝑉
𝑦 =
𝑦 𝜌 𝑑𝑉
𝜌 𝑑𝑉
𝑧 =
𝑧 𝜌 𝑑𝑉
𝜌 𝑑𝑉
Coordinates of Center of Mass when
density is not homogenous
𝑥 =
𝑥 𝑑𝑉
𝑉
𝑦 =
𝑦 𝑑𝑉
𝑉
𝑧 =
𝑧 𝑑𝑉
𝑉
Coordinates of Center of Mass when
density is homogenous
Coordinates of Centroid of body
𝑥 =
𝑥 𝑑𝑚
𝑚
𝑦 =
𝑦 𝑑𝑚
𝑚
𝑧 =
𝑧 𝑑𝑚
𝑚

6. Centroids and First moment of Line, Area and Volume
For a wire of length 𝐿, a small cross-sectional area 𝐴 having
density 𝜌, the body approximate a line segment having mass
𝑚 = 𝜌𝐴𝐿. If 𝐴 and 𝜌 are constant over length, then centroid is
calculated by taking 𝑑𝑚 = 𝜌𝐴𝑑𝐿
𝑥 =
𝑥 𝑑𝐿
𝐿
𝑦 =
𝑦 𝑑𝐿
𝐿
𝑧 =
𝑧 𝑑𝐿
𝐿
1. Centroids of Line
𝑥 =
𝑥 𝑑𝐴
𝐴
𝑦 =
𝑦 𝑑𝐴
𝐴
𝑧 =
𝑧 𝑑𝐴
𝐴
𝑥 =
𝑥 𝑑𝑉
𝑉
𝑦 =
𝑦 𝑑𝑉
𝑉
𝑧 =
𝑧 𝑑𝑉
𝑉
2. Centroids of Area
3. Centroids of Volume
For a shell of area 𝐴 , a constant thickness 𝑡 having
homogenous density 𝜌, the body approximate a surface area
having mass 𝑚 = 𝜌𝐴𝑡. If 𝑡 and 𝜌 are constant over area, then
centroid is calculated by taking 𝑑𝑚 = 𝜌𝑡𝑑𝐴
For a body of volume 𝑉 having homogenous density 𝜌, the
mass of the generalized 3D body is 𝑚 = 𝜌𝑉 and the centroid
is calculated by taking 𝑑𝑚 = 𝜌𝑑𝑉

7. First Moments of Areas and 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 is divided
into two 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.

8. Key Concepts for selecting integration element
Generally, we choose the coordinate system which best
matches the boundaries of the figure.
1. Choice of Coordinates
Rectangular Coordinates Polar Coordinates
Higher-order terms may always be dropped compared with
lower-order terms. Thus, the first-order term 𝑑𝐴 = 𝑦 𝑑𝑥 is
kept, and the second-order triangular area 𝑑𝐴 =
1
2
𝑑𝑥 𝑑𝑦 is
discarded.
2. Discarding higher-order terms
Whenever possible, a first-order differential element should be
selected in preference to a higher-order element so that only
one integration will be required to cover the entire figure.
3. Order of Element
𝑥 𝑑𝐴 = 𝑥 𝑙 𝑑𝑦 𝑥 𝑑𝑥 𝑑𝑦
Generally, we choose the coordinate system which best
matches the boundaries of the figure.
4. Symmetry of Shape
For a first- or second order differential element, it is essential to
use the coordinate of the centroid of the element for the
moment arm.
5. Centroidal Coordinates of Element

9. Example 5/1
Centroid of a circular arc. Locate the centroid of a
circular arc as shown in the figure.
Choosing the axis of symmetry as the 𝑥-axis makes y = 0.
Solution
A small first-order differential arc segment having length
𝒅𝑳 = 𝒓𝒅𝜽 is chosen. 𝑑𝐿 = 𝑟𝑑𝜃
differential arc segment
The total length of the circular arc is 𝑳 = 𝟐𝜶𝒓 is chosen.
𝑦
𝑥
𝑥 = 𝑟𝑐𝑜𝑠𝜃
𝜃
𝑑𝜃

10. Example 5/2
Centroid of a triangular area. Determine the distance
y from the base of a triangle of altitude h to the centroid
of its area.
The x-axis is taken to coincide with the base.
Solution
A small first-order differential strip having area 𝒅𝑨 = 𝒙𝒅𝒚
is chosen.
The total area of the triangle (
1
2
base × ℎ𝑒𝑖𝑔ℎ𝑡) is 𝑨 =
1
2
𝒃𝒉
is chosen.
𝑥
ℎ − 𝑦
=
𝑏
ℎ

11. Example 5/3 (a)
Centroid of the area of a circular sector.
Locate the centroid of the area of a circular
sector with respect to its vertex.
Solution I
A small first-order differential circular ring having radius 𝑟0 and
thickness 𝑑𝑟0 has an area 𝒅𝑨 = 𝟐𝜶𝑟0𝒅𝑟0is chosen.
The total area of the sector is 𝑨 =
𝟐𝜶
2𝝅
𝝅𝒓𝟐
is chosen.
Choosing the axis of symmetry as the 𝑥-axis
makes y = 0.
The 𝑥-coordinate of centroid of element can be
calculated

12. Example 5/3 (b)
Solution II
A small first-order differential triangle
with neglecting higher-order terms, having
an area 𝒅𝑨 =
𝟏
𝟐
𝒓(𝒓𝒅𝜽) is chosen.
The 𝑥-coordinate of centroid of element
can be calculated by using (
2
3
of the
altitude of triangle) as 𝑥𝑐 =
2
3
𝑟𝑐𝑜𝑠𝜃

13. Centroids of Common Shapes of Areas

14. Centroids of Common Shapes of Lines

15. Composite Plates and Areas
• Composite plates






W
y
W
Y
W
x
W
X
• Composite area






A
y
A
Y
A
x
A
X

16. Sample Problem 5.4
For the plane area shown, determine
the first moments with respect to the
x and y axes and the location of the
centroid.
SOLUTION:
• Divide the area into a triangle, rectangle,
and semicircle with a circular cutout.
• Compute the coordinates of the area
centroid by dividing the first moments by
the total area.
• Find the total area and first moments of
the triangle, rectangle, and semicircle.
Subtract the area and first moment of the
circular cutout.
• Calculate the first moments of each area
with respect to the axes.

17. Sample Problem 5.4
3
3
3
3
mm
10
7
.
757
mm
10
2
.
506






y
x
Q
Q
• Find the total area and first moments of the
triangle, rectangle, and semicircle. Subtract the
area and first moment of the circular cutout.

18. Sample Problem 5.4
2
3
3
3
mm
10
13.828
mm
10
7
.
757







A
A
x
X
mm
8
.
54

X
2
3
3
3
mm
10
13.828
mm
10
2
.
506







A
A
y
Y
mm
6
.
36

Y
• Compute the coordinates of the area
centroid by dividing the first moments by
the total area.