Center of gravity of plane figures

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

Where Sx and Sy are static moments about the x and y axes , A - is the area of the figure.

The figure diagram, all calculations, and graphs of the figure with the center of gravity are generated in Section Properties Tool. You can create any figure consisting of straight figures and determine its center of gravity.

Static moment of plane figures

Static moment is an important quantity in issues related to determining the center of gravity of figures.

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

For figures consisting of simple figures for which we know the position of the centers of gravity, we will determine the static moments without using integrals (Ufff 😊).

We will use the following formulas:

Where A, x and y are the area and coordinates of the centers of gravity of the following figures.

An example of calculating the static moment for a rectangle with respect to the x-axis is shown in the figure below.

In our example, the center of gravity of the rectangle is known to us and is 0.5h. If we multiply this distance by the area of the rectangle A, we get the static moment Sx.

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

The SI unit of static moment is [m3].

According to wikipedia :

A coordinate system for which the axes of the static moments are equal to 0 is referred to as central, and its axes are the central axes.

The center of gravity does not have to be within the figure's area. An example would be a channel section.

Calculating the center of gravity of a plane figure

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

As you can see the figure can be divided into two rectangles. Let's first mark on the figure the position of the centers of gravity of each rectangle.

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

Let's move on to the calculation of individual size, we will start with the areas and static moments of the individual rectangles. Then we will calculate the sums of the static moments for the whole figure and calculate the coordinates of the center of gravity.

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