Friday, 17 February 2017

WHAT IS THE DIFFERENCE BETWEEN CENTROID AND CENTRE OF GRAVITY?

WHAT IS THE DIFFERENCE BETWEEN CENTROID AND CENTRE OF GRAVITY?


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 of a body is basically defined as the point through which the complete weight of the body will be acting irrespective of the position of the body.

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 a1, a2, a3, a4 …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,
x1= Distance of the C.G of the area a1 from OY axis
x2= Distance of the C.G of the area a2 from OY axis
x3= Distance of the C.G of the area a3 from OY axis
x4= Distance of the C.G of the area a4 from OY axis

Similarly, we will have

y1= Distance of the C.G of the area a1 from OX axis
y2= Distance of the C.G of the area a2 from OX axis
y3= Distance of the C.G of the area a3 from OX axis
y4= Distance of the C.G of the area a4 from OX axis

Total area of the plane figure = Sum of all small areas
A = a1 + a2 + a3 + a4 + ----------
 
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 = a1.x1 + a2.x2 + a3.x3+ a4.x4 +…….

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 = a1.x1 + a2.x2 + a3.x3+ a4.x4 +…….
X = [a1.x1 + a2.x2 + a3.x3+ a4.x4 +……]/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 = a1.y1 + a2.y2 + a3.y3+ a4.y4 +…….
Y = [a1.y1 + a2.y2 + a3.y3+ a4.y4 +…….]/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 = [a1.x1 + a2.x2 + a3.x3+ a4.x4 +……]/A
Y = [a1.y1 + a2.y2 + a3.y3+ a4.y4 +…….]/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.

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