EM025 CHAPTER 2 ENGINEERING MECHANICS

Solution :
FBD
• Idealized model of the ramp.
• Center of gravity located at the midpoint since the ramp is

approximately uniform.

101

 M A = 0;
− T cos 20 (2sin 30 ) + T sin 20 (2 cos 30 ) +1600(1.5cos 30 ) = 0
T = 5985N

Since there are 2 cables supporting the ramp :
T = 5985
T/2 = 2992.5N

102

Problem 5 :
Determine the reactions on the member A, B, C.

103

Solution :

104

2.5 CENTROID AND CENTER OF GRAVITY

a) Describe the concept of centroid and center of gravity.
b) Determine the centroid of basic shapes

CENTRE OF GRAVITY

• Define: It is the point at which the whole weight of the body
may be assumed to be concentrated.

• If an object rotates when thrown, the centre of gravity is also
act as the centre of rotation.

• When an object is suspended so that it can move freely, its
centre of gravity is always directly below the point of
suspension.

• An object can be balanced on a sharp point placed directly
beneath its centre of gravity.

106

• It is represented by CG or simply G or C.
• A body is having only one CG for all positions of the body.
• Examples of center of gravity for regular-shaped objects :

• For some objects the CG is not located on the actual object :

107

CENTROID FOR A BODY

• Defines: the geometric center of an object. Its applies to plane
areas.

• The centre of mass, centre of gravity and the centroid only
coincide if the object is uniform in material density .

• Geometric centre of a two-dimensional region is the average
position of all the points in the shape.

• Calculation of centroid falls within three distinct categories, as a
line, an area or a volume.

108

DIFERRENCE BETWEEN
CENTROID AND CENTER OF GRAVITY

109

IMPORTANT POINTS
• "Centre Of Gravity" applies to the bodies with mass and weight,

while "Centroid" applies to plane areas.

• The centroid represents the geometric center of a body.

• Centre of gravity of a body is the point through which the
resultant gravitational force (weight) of the body acts for any
orientation of the body.

• In some cases the centroid is located at a point that is not on the
object (a ring).

• Centre of gravity is the point at which a object can be suspended

and be in perfect equilibrium 110

CENTROID LOCATION FORMULA

Shape ẋẏ Area

Rectangular/Square b.h

h bh b.h
22 2

b

Triangle

b h
3 3

Circle π.r2

Center of Center of OR 111
circle circle
π.d2
4

Example 1: 112
Locate the centroid of the plane area shown.

y
20 mm 30 mm

36 mm

24 mm
x

Solution : Step by step

y 30
20

36 Several points should be emphasized
when solving these types of problems.

24
x

y
20 + 10

C1 C2 Decide how to construct the given
30 area from common shapes.

24 + 12 113
x
10

Solution : Step by step

y
20 + 10

C1 C2 Construct a table containing areas and
30 respective coordinates of the centroids.
24 + 12
10 x

A, mm2 x, mm y, mm xA, mm3 yA, mm3

1 20 x 60 =1200 10 30 12,000 36,000
36 16,200 19,440
2 (1/2) x 30 x 36 =540 30 28,200
55,440
S 1740

1 1 114
20 + 3 30 = 30 24 + 3 36 = 36

Solution : Step by step

y Then XS A = S xA
20 + 10 X (1740) = 28,200

or X = 16.21 mm

C1 C2 and YS A = S yA
30 Y (1740) = 55,440
24 + 12
10 x or Y = 31.9 mm

A, mm2 x, mm y, mm xA, mm3 yA, mm3

1 20 x 60 =1200 10 30 12,000 36,000
2 (1/2) x 30 x 36 =540 30 36 16,200 19,440
S 1740 28,200 55,440
115

Example 2:
Determine the centroid x and y distances for the composite area

shown.

116

Solution

2 1
3 (9) = 6 3+3 3

117

Solution

118

Example 3:
Locate the centroid of the plate area. Radius of the circle

is 0.5m.

119

Solution

Segment A (m2) x (m) y (m) xA (m3) yA (m3)
4.5 4.5
1 4.5 1 1 -13.5 13.5
2 9 -1.5 1.5 5 -4
3 -2 -2.5 2 0.79
4 - 0.79 -1 1 -3.21 - 0.79
Sum 10.71 13.21

x =  ~x A = − 3.21 = −0.3m 120
 A 10.71

y =  ~yA = 13.21 = 1.23m
 A 10.71