Geometric characteristics

Introduction to the geometric characteristics of plane figures

Plane figures are an essential element in general mechanics and materials strength. In this course, we will discuss geometric characteristics of these figures, namely determining the center of gravity, calculating moments of inertia and deviation moments. These calculations are necessary to calculate, among other things, stresses in rods.

The center of gravity is the point where the entire mass of the figure can be placed so that it behaves like a point mass.

Fig. 1. Division of a channel section into simple figures, determined center of gravity

Moments of inertia of plane figures

Moments of inertia describe the distribution of mass of a figure relative to the axis of rotation. They are important in the analysis of rotational motion and materials strength. We must understand the difference between such moments of inertia:

• Central moments of inertia are moments of inertia relative to axes passing through the center of gravity of the plane figure,
• Main moments of inertia are moments of inertia relative to main axes (main axes are those around which the moment of inertia of the figure is maximum or minimum),
• Main central moments of inertia are moments of inertia relative to main axes that pass through the center of gravity of the plane figure.

Let’s take a closer look at axes of symmetry and the differences in determining the center of gravity and moments of inertia for different types of figures; in this course, we have a division into:

Figures with two axes of symmetry

Fig. 2. Examples of figures with two axes of symmetry from the video course

In this case, the center of gravity is located at the intersection of the axes of symmetry, so the task is simplified; it is enough to determine the moments of inertia.

It should be noted that if the figure as a whole has an axis of symmetry, then the deviation moment of the figure is zero, and the central moments of inertia are immediately the main central moments of inertia.

Figures with one axis of symmetry

Fig. 3. Examples of figures with one axis of symmetry from the video course

If we have one axis of symmetry, the center of gravity lies on it, so we need to calculate one coordinate of the center of gravity.

The figure has an axis of symmetry, so the central moments are the main central ones.

Figures without an axis of symmetry

Fig. 4. Examples of figures without an axis of symmetry from the video course

The most difficult case, we look for both coordinates of the center of gravity, then we calculate the moments of inertia relative to the central axes. However, these are not the main central moments of inertia, because the figure does not have an axis of symmetry, and thus the deviation moment is non-zero. Therefore, we calculate the deviation moment, then the main central moments of inertia, and finally the angle of rotation by which we need to rotate the central axes to the main central axes.

You have access to video courses and a database of tasks with solutions.

We invite you to explore the topic of geometric characteristics of plane figures, which is crucial for a better understanding of structural mechanics and correct calculations, among others, of stresses in the cross-section.

Good luck! 🛠️🔍