Center of gravity (centroid) of plane figures

In this entry:

1. What is center of gravity?

2. First moment of area of a plane figure

3. How to calculate the center of gravity of a plane figure?

Center of gravity (centroid) of plane figures

Center of gravity of plane figures with uniform mass density also known as geometric center or center of figure.

We calculate the center of gravity of plane figures using the following formulas:

Fig.1. Formulas for the center of gravity of plane figures

where Sx and Sy are first moment of area in the x and y directions, A - is the surface area of the figure.

All example calculations and diagrams with the center of gravity are generated in my moment of inertia calculator. You can create any figure consisting of simple figures and calculate its center of gravity (centroids).

First moment of area

The first moment of area is an important when to determining the center of gravity of figures.

For any figure, the first moment can be calculated from the following formulas:

Fig.2. Formulas for the first moment of area for any figure

The first moment of area of a shape, about a certain axis, equals the sum over all the infinitesimal parts of the shape of the area of that part times its distance from the axis.

For figures consisting of simple figures for which we know centroids, we will determine static moments without using integrals (uff.. 😊).

We will use the following equations:

Fig.3. Formulas for the static moment of a figure consisting of simple figures

where A, x and y are the surface area and centroid coordinates of subsequent figures.

The centroid of each part can be found in any list of centroids of simple shapes

An example calculation of the first moment of area for a rectangle relative to the x axis is shown in the figure below.

Fig.4. Determining the static moment of a rectangle about the x axis

In our example, the centroid of the rectangle is equal to 0.5h. If we multiply this distance by the area of the rectangle A, we obtain the static moment Sx.

The first moment of area about the axis can take positive, zero and negative values. It has a value equal to 0 when the axis about which it is determined passes through the geometric center of gravity of the figure.

The SI unit for first moment of area is a cubic metre (m3). In the American Engineering and Gravitational systems the unit is a cubic foot (ft3) or more commonly inch3.

As Wikipedia says:

The coordinate system for which the first moment of area are equal to 0 is called central and its axes are called centroids or central axes.

The position of the centroid does not have to be within the area of the figure. An example would be a U-channel profile.

Calculating the center of gravity (centroid) of a plane figure

Once we know all the formulas, let's try to calculate the center of gravity of the figure as in the figure below:

Fig.5. Determining the first moment of area of a T-shaped figure.

As you can see, the figure can be divided into two rectangles. Let us first mark the location of the centers of gravity of each rectangle in the drawing.

We can adopt the coordinate system at any point. It is worth adopting such a system that the entire figure is located in the first quadrant, thanks to which the coordinates of the centers of gravity of each figure will be positive.

Fig.5. Determining the static moment of a figure, division into simple figures.

Let's move on to the calculations of individual quantities, starting with the surface areas and centroid positions of individual rectangles. Then we will calculate the sums of first moment of area for the entire figure and calculate the coordinates of the center of gravity (centroids).

Fig.6. Calculating first moment of area and the center of gravity of a figure.

As you can see, the horizontal position of the center of gravity is in the axis of symmetry of the figure. If the figure has an axis of symmetry, the center of gravity will be located on it and there is no need to calculate it.

Fig.7. A figure with the center of gravity and central axes marked

Thank you, that's all about the center of gravity of plane figures.