# Centre of Gravity in Mechanics

## Centre of Gravity:

It has been established, for a long, that every particle of a body is attracted by the earth towards its centre. The force of attraction, which is proportional to the mass of the particle acts vertically downwards and is known as the weight of the body. As the distance between the different particles of a body and the centre of the earth is the same, therefore these forces may be taken to act along parallel lines.

A point may be found out in a body, through which the resultant of all such parallel forces act. This point through which the whole weight of the body acts, irrespective of its position is known as the centre of gravity. It may be noted that everybody has one and only one centre of gravity.

### Centroid:

The plane figures (like triangle, quadrilateral, circle, etc) have only areas, but no mass. The centre area of such figures is known as the centroid. The method of finding out the centroid of a figure is the same as that of finding out the centre of gravity of a body.

## Methods for Centre of Gravity:

The centre of gravity (or centroid) may be found by any one of the following two methods:

1. By geometrical considerations

2. By moments

3. By graphical method

The graphical method is a tedious and cumbersome method for finding out the centre of gravity of simple figures. That is why it has academic value only.

### Centre of Gravity by Geometrical Considerations:

The centre of gravity of simple figures may be found from the geometry of the figure as given below:

1. The centre of gravity of a uniform rod is at its middle point.

2. The centre of gravity of a rectangle is at the point, where its diagonals meet each other. It is also a middle point of the length as well as the breadth of the rectangle as shown in the above figure.

3. The centre of gravity of a triangle is at the point, where the three medians of the triangle meet as shown in the figure.

4. The centre of gravity of a trapezium with parallel sides a and b is at a distance of h/3 x (b+2a/b+a) measured from side b as shown in the above figure.

5. The centre of gravity of a semicircle is at a distance of 4r/3π from its base measured along the vertical radius as shown below figure:

6. The centre of gravity of a circular sector makes a semi-vertical angle α at a distance or 2r/3 x sinα/α from the centre of the sector measured along the central axis as shown in the below figure:

7. The centre of gravity of a cube is at a distance of l/2 from every face.

8. The centre of gravity of a sphere is at a distance of d/2 from every point.

9. The centre of gravity of a hemisphere is at a distance of 3r/8 from its base, measured along the vertical radius as shown below figure:

10. The centre of gravity of the right circular solid cone is at a distance of h/4 from its base, measured along the vertical axis as shown below figure:

11. The centre of gravity of a segment of a sphere of a height h is at a distance of 3/4 x (2r-h)2/(3r-h) from the centre of the sphere measured along the height as shown below figure:

### Centre of Gravity by Moments:

The centre of gravity of a body may also be found by moments.

Consider a body of mass M whose centre of gravity is required to be found out. Divide the body into small masses, whose centres of gravity are known as shown below figure.