We were discussing the “Derivation of relationship between young’s modulus of elasticity (E) and bulk modulus of elasticity (K)”,
“Elongation of uniformly tapering rectangular rod” and
we have also seen the “Basic principle of complementary shear stresses” and
“Volumetric strain of a rectangular body” with the help of previous posts.

Now we are going further to start our discussion to
understand the basic concept of centroid and center of gravity with the help of
this post.

###
**Let
us first see here the concept of center of gravity**

Center of gravity will be related with the
distribution of mass.

We can also say in simple words that there will be
one single point through which the complete weight of the body will be acting
irrespective of the position of the body. Center of gravity of a body will be
indicated by C.G or sometimes G.

###
**Let
us see here the basic concept of centroid **

Centroid is basically defined as a point across
which the entire area will be acting for a plane figure irrespective of the
position of the plane figure. Centroid and center of gravity will be at same
point and we must note it here that centroid of a body will also be indicated
by C.G or G.

Centroid will be related with the distribution of
area or volume.

We have mentioned here the term plane figure, before
going ahead we must need to understand the meaning of plane figure. Plane
figure could be made by straight lines or curved lines and also by both lines
i.e. curved lines and straight lines.

In simple we can easily say that plane figure
means 2D figure such as rectangle, triangle, circle or square.

Centroid term will be used only for plane figures
such as rectangle, triangle, circle or square instead of weight or mass.

###
**Determination
of centroid and center of gravity **

Centroid or center of gravity (G) will be determined
with the help of following four methods as mentioned here.

1. By method of moments

2. By integration method

3. By graphical method

4. By geometrical consideration

###
**Let
us see here the determination of centroid or center of gravity by using the
method of moments**

Let us see the following plane figure. Let us assume
that plane figure, displayed here, is made with number of small areas a

_{1}, a_{2}, a_{3}, a_{4}…etc. Let us think that the total area of the plane figure is A, now we need to determine here the centroid of this plane figure.
Where,

x

_{1}= Distance of the C.G of the area a_{1}from OY axis
x

_{2}= Distance of the C.G of the area a_{2}from OY axis
x

_{3}= Distance of the C.G of the area a_{3}from OY axis
x

_{4}= Distance of the C.G of the area a_{4}from OY axis
Similarly, we will have

y

_{1}= Distance of the C.G of the area a_{1}from OX axis
y

_{2}= Distance of the C.G of the area a_{2}from OX axis
y

_{3}= Distance of the C.G of the area a_{3}from OX axis
y

_{4}= Distance of the C.G of the area a_{4}from OX axis
Total area of the plane figure = Sum of all small
areas

A = a

_{1 }+ a_{2 }+ a_{3 }+ a_{4 }+ ----------Let us determine the moments of all small areas about the OY axis and we will have

Moments of all small areas about the OY axis = a

_{1}.x_{1 }+ a_{2}.x_{2}+ a_{3}.x_{3}+ a_{4}.x_{4 }+…….
Let us assume that G is the centroid of the entire
area A and it is located at a distance X from the axis OY as displayed in
figure.

Let us determine the moment of entire area A around
the axis OY and it will determined by multiplying the total area of the plane
figure i.e. A with the distance X.

We must note it here that moment of entire area
around the axis OY will be equal to the moments of all small areas about the OY
axis.

A. X = a

_{1}.x_{1 }+ a_{2}.x_{2}+ a_{3}.x_{3}+ a_{4}.x_{4 }+…….
X = [a

_{1}.x_{1 }+ a_{2}.x_{2}+ a_{3}.x_{3}+ a_{4}.x_{4 }+……]/A
Similarly, when we will determine the moments of all
small areas around the axis OX and equalize it with the moment of total area
around the axis OX, we will have following equation

A. Y = a

_{1}.y_{1 }+ a_{2}.y_{2}+ a_{3}.y_{3}+ a_{4}.y_{4 }+…….
Y = [a

_{1}.y_{1 }+ a_{2}.y_{2}+ a_{3}.y_{3}+ a_{4}.y_{4 }+…….]/A
Where, Y is the distance of centroid (G) of entire
area from axis OX

Therefore, we have following values of co-ordinate (X, Y) of the centroid of entire area of the plane figure and it will be as mentioned here.

*X = [a*_{1}.x_{1 }+ a_{2}.x_{2}+ a_{3}.x_{3}+ a_{4}.x_{4 }+……]/A

*Y = [a*_{1}.y_{1 }+ a_{2}.y_{2}+ a_{3}.y_{3}+ a_{4}.y_{4 }+…….]/A
Do you
have any suggestions or any amendment required in this post? Please write in
comment box.

###
**Reference:**

Strength of material, By R. K. Bansal

We will see another important topic i.e. what
is area moment of inertia? , in the category of strength of material, in
our next post.