CE 201- Statics Chapter 9 – Lecture 1. CENTER OF GRAVITY AND CENTROID The following will be...

CE 201- Statics

Chapter 9 – Lecture 1

CENTER OF GRAVITY AND CENTROID

The following will be studiedLocation of center of gravity (C. G.) and center of

mass for discrete particlesLocation of C. G. and center of mass for an

arbitrary-shaped bodyLocation of centroid or geometric center

Center of Gravity and Center of Mass for a System of Particles

• Consider ( n ) particles• Weight of particles are

parallel forces• Weight can be replaced by

a single resultant weight• The point of application of

the resultant weight is called the center of gravity (C. G.)

Resultant weight, WR = w

w1

w4wn

w3w2

WR

y

z

x

Moment of all weights about x, y, and z is equal to the moment of the resultant weight about the same axes.

My xRWR = x1w1 + x2w2 + ..+xnwn Mx yRWR = y1w1 + y2w2 + ..+ynwn

w1

w4wn

w3w2

WR

y

z

x

To find z, imagine rotating the system coordinates by 90 with particles are fixed in it.

w1

w4wn

w3w2

WR

y

z

x

w1

w4w

n

w3w2

WR

y

z

x

Mx zRWR = z1w1 + z2w2 + ……..+znwn

then, x = ( x W) / WR

y = ( y W) / WR

Z = ( z W) / WR

Note:x, y, and z for C. G. of the systemx, y, and z for C. G. of each particle

Center of Mass

W = mg

x = ( x m) / mR

y = ( y m) / mR

x = ( z m) / mR

The location of the center of gravity coincides with that of the center of mass

CENTER OF GRAVITY, CENTER OF MASS AND CETROID FOR A BODY

Center of Gravity

A rigid body is composed of a system of particles, where each particle has a differential weight (dW). Applying the same principles that were used with discrete particles, the following is obtained:

x = ( x dW) / ( dW)

y = ( y dW) / ( dW)

z = ( z dW) / ( dW)

x

y

z

dW

here, we use integration rather than summation due to differential weight (dW).

If dW = dV

Where

= specific weight (weight / volume)

V = volume of body

Then,

x = ( x dV) / ( dV)

y = ( y dV) / ( dV)

z = ( z dV) / ( dV)

x

y

z

dW

Center of Mass

Substitute = g into the previous equations ( = density (mass/volume), then:

x = ( x g dV) / ( g dV)

y = ( y g dV) / ( g dV)

z = ( z g dV) / ( g dV)

Centroid

Centroid is the geometric center of the object Centroid is independent of the weight Centroid is dependent of the body's geometry

Volume Centroid

•Subdivide the object into volume elements (dV)

•Compute the moments of the volume elements about the coordinate axes

x = ( x dV) / ( dV)

y = ( y dV) / ( dV)

z = ( z dV) / ( dV)

x

y

z

dV

y

x

c

z

Area Centroid

Subdivide the object into area elements (dA)

Compute the moments of the area elements about the coordinate axes

x = ( x dA) / ( dA)

y = ( y dA) / ( dA)

z = ( z dA) / ( dA)

x

y

z

dV

y

x

c

z

Line Centroid

Subdivide the line into elements (dA)

Compute the moments of the line elements about the coordinate axes

x = ( x dL) / ( dL) y = ( y dL) / ( dL) z = ( z dL) / ( dL)

Centroid could be located off the object in space

Centroid of some shapes may be specified by using the conditions of symmetry

x

y

z

dL

y

x

c

z

Line Centroid

If the shape has an axis of symmetry, then the centroid will be located along that axis

For every element dL having a distance ( y ), there is an element dL having a distance ( -y )

So, y = 0

x

y

c

dL

dL

The same thing can apply if the shape has more than one axis of symmetry. The centroid lies at the intersection of the axe.

Procedure for Analysis

To determine the Center of Gravity or the Centroid: select an appropriate coordinate axes select an appropriate differential element for

integration (dL, dA, dV) express he differential element (dL, dA, dV) in

terms of the coordinates (x, y, z) determine the coordinate (x, y, z) or moment

arms for the centroid or center of gravity of the element

integrate