Моменты и центры масс

A flat metal plate balances at a specific point called the center of mass. This balance depends on the density of the material, which may vary from point to point within the object.

Conceptually, the center of mass is the weighted average of the plate's mass as it is distributed across the x and y coordinates. Here, the numerator is the moment about a given axis.

To find the center of mass, the region is divided into many small elements. Each element has a small area and a small mass determined by the local density.

The total mass of the plate is obtained by summing these mass elements in a double integral over the domain of the material.

To locate the center of mass, the moments about the axes are calculated.

The moment about the x-axis is the product of mass and vertical distance, while the moment about the y-axis is the product of mass and horizontal distance.

By summing these individual effects across the entire region, the total moments are given by double integrals.

Finally, dividing the moments about the coordinate axes by the total mass gives the coordinates of the center of mass.