Statics

Chapter 9

Centroids and Centre

of Gravity

Eng. Iqbal Marie

iqbal@hu.edu.jo

Hibbeler, Engineering Mechanics: Statics,12e, Prentice Hall

Center of Gravity

The center of gravity G is a point

which locates the resultant weight of

a system of particles.

The weights of the particles is

considered to be a parallel force

system. The system of weights can be

replaced by a single weight acting tat

the Center of Gravity.

n

1i

iR WW

n

1i

i

i

n

1i

i

n

1i

i

i

n

1i

i

n

1i

i

i

n

1i

i

W

Wz~

z

W

Wy~

y

W

Wx~

x

n

1i

i

i

n

1i

i

n

1i

i

i

n

1i

i

n

1i

i

i

n

1i

i

m

mz~

z

m

my~

y

m

mx~

x

Center of Mass

Because the acceleration due to gravity, g, is constant, W=mg.

Therefore, when we treat an object as a particle, we can concentrate

all the mass at the same point G. The location of the c.g. and the

location of the centre of mass coincide, at the centroid of the

object.

Centroid of an Area

A

A

A

A

A

A

dA

dAz~

zdA

dAy~

ydA

dAx~

x

9.2 Center of Gravity and Centroid for a Body

The centroid of the area coincides with the center of symmetry.

Locate the Centroid of the area shown

A

A

A

A

A

A

dA

dAz~

zdA

dAy~

ydA

dAx~

x

Centroids of

Common

Shapes of

Areas

9.3 Centroids – Composite Areas

1. break up a shape into smaller, simpler shapes.

2. Find area of each shape

3. Locate the centroid of each shape from the given

axes

4.Then use the equations to find the centroid for the

whole area

Composite Bodies

Ai xi yi zi xiAi yiAi ziAi

1

2

Sum Ai Aixi Aiyi Aizi

i i

T

i i

T

1

1

x x AA

y y AA

Find the centroid of the given body

from the shown x- and y- axes

A2

A1

i i

T

i i

T

1

1

x x AA

y y AA

Body Area(mm2) x (mm) y(mm) x*Area (mm3) y*Area (mm3)

Triangle 3600 40 40 144000 144000

Square 12000 60 110 720000 1320000

Sum 15600 864000 1464000

centroid (x) 55.38 mm

centroid (y) 93.85 mm

For the plane area shown, determine the first moments with

respect to the x and y axes and the location of the centroid

with respect to the shown x-and y axes

23

33

mm1013.828

mm107.757

A

AxX mm 8.54X

23

33

mm1013.828

mm102.506

A

AyY mm 6.36Y

Force on a Submerged Surface

The pressure acts as a function of depth.

gH

R H

*width

*width

F y P

yg

1

* *width2

R d dg

A 3- by 3 ft gate is placed in a wall below water level as shown.

Determine the magnitude and location of the resultant of the forces

exerted by the water on the gate. (g =62.4 lb/ft3)

MULTIPLY by the width (3 ft)

1

2

374.4 lb/ft 3 ft 1123.2 lb

1936 lb/ft 374.4 lb/ft 3 ft

2

842.4 lb

F

F

1

2

3 ft5 ft 3.5 ft

2

3 ft5 ft 4.0 ft

3

x

x

1 2

Total force is 1965.6 lb at 3.71 ft

from the surface.

Force Area(lb) x (ft) x*Area (lb-ft)

Uniform 1123.2 3.5 3931.2

Triangular 842.4 4 3369.6

Sum 1965.60 7300.80

centroid (x) 3.71 ft

1965.6 lb

3.71 ft

3 ft